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+ATM-R: An Adaptive Tradeoff Model with
+Reference Points for Constrained Multiobjective
+Evolutionary Optimization
+Bing-Chuan Wang, Yunchuan Qin, Xian-Bing Meng, Zhi-Zhong Liu
+Abstract—The goal of constrained multiobjective evolutionary
+optimization is to obtain a set of well-converged and well-
+distributed feasible solutions. To complete this goal, there should
+be a tradeoff among feasibility, diversity, and convergence.
+However, it is nontrivial to balance these three elements simulta-
+neously by using a single tradeoff model since the importance of
+each element varies in different evolutionary phases. As an alter-
+native, we adapt different tradeoff models in different phases and
+propose a novel algorithm called ATM-R. In the infeasible phase,
+ATM-R takes the tradeoff between diversity and feasibility into
+account, aiming to move the population toward feasible regions
+from diverse search directions. In the semi-feasible phase, ATM-R
+promotes the transition from “the tradeoff between feasibility and
+diversity” to “the tradeoff between diversity and convergence”,
+which can facilitate the discovering of enough feasible regions
+and speed up the search for the feasible Pareto optima in
+succession. In the feasible phase, the tradeoff between diversity
+and convergence is considered to attain a set of well-converged
+and well-distributed feasible solutions. It is worth noting that the
+merits of reference points are leveraged in ATM-R to accomplish
+these tradeoff models. Also, in ATM-R, a multiphase mating
+selection strategy is developed to generate promising solutions
+beneficial to different evolutionary phases. Systemic experiments
+on a wide range of benchmark test functions demonstrate that
+ATM-R is effective and competitive, compared against five state-
+of-the-art constrained multiobjective optimization evolutionary
+algorithms.
+Index Terms—Constrained multiobjective evolutionary opti-
+mization, adaptive tradeoff model, reference point, multiphase
+mating selection
+I. INTRODUCTION
+M
+ANY scientific or engineering problems involve the
+optimization of conflicting objectives subject to con-
+straints, which can be formulated as constrained multiobjective
+optimization problems (CMOPs) [1]:
+min
+F(x) = (f1(x), f2(x), · · · , fm(x))T ∈ Rm
+s.t.
+gj(x) < 0, j = 1, · · · , ng
+hj(x) = 0, j = ng + 1, · · · , ng + nh
+xj ≤ xj ≤ xj, j = 1, · · · , D
+,
+(1)
+B.-C. Wang is with the School of Automation, Central South University,
+Changsha 410083, China (email: bingcwang@csu.edu.cn).
+Y. Qin and Z.-Z. Liu are with the College of Information Science and
+Electronic Engineering, Hunan University, Changsha 410082, China (e-mail:
+liuzz@hnu.edu.cn; qinyunchuan@hnu.edu.cn).
+X.-B. Meng is with the School of Computer Science and Engineering,
+South China University of Technology, Guangzhou 510006, China (e-mail:
+axbmeng@scut.edu.cn).
+where F(x) denotes the objective vector consisting of m
+conflicting objectives (i.e., fi(x), i
+=
+1, · · · , m); x
+=
+(x1, · · · , xD)T is a D-dimensional decision vector/solution;
+xj and xj are the lower and upper bounds of xj, respectively;
+S = �D
+j=1[xj, xj] refers to the decision space; gj(x) and
+hj(x) represent the jth inequality and (j − ng)th equality
+constraints, respectively; ng and nh are the numbers of the
+inequality and equality constraints, respectively.
+When solving a CMOP, we always quantify constraint
+violation by the degree of constraint violation:
+G(x) =
+ng+nh
+�
+j=1
+Gj(x).
+(2)
+Gj(x) denotes the degree of constraint violation of the jth
+constraint [2]:
+Gj(x) =
+�
+max(0, gj(x)),
+1 ≤ j ≤ ng
+max(0, |hj(x)| − δ),
+ng + 1 ≤ j ≤ ng + nh
+(3)
+where δ is a small positive value used to relax an equality
+constraint to some degree. A solution x is called a feasible
+solution, if and only if G(x) = 0. All feasible solutions
+constitute the feasible region: Ω = {x ∈ RD|G(x) = 0}. For
+two solutions xu, xv ∈ Ω, xu is said to Pareto dominate xv,
+denoted as xu ≺ xv, if and only if ∀j ∈ {1, · · · , m}, fj(xu) ≤
+fj(xv) � ∃j ∈ {1, · · · , m}, fj(xu) < fj(xv). A solution
+xp ∈ Ω is considered as a Pareto optimum if and only if
+¬∃xv ∈ Ω, xv ≺ xp. The set of all Pareto optima is called
+the constrained Pareto set, and its image in the objective
+space is called the constrained Pareto front (CPF). The goal
+of constrained multiobjective evolutionary optimization is to
+pursue a set of well-converged and well-distributed feasible
+solutions to approximate the CPF.
+To complete this goal, a consensus has been reached in the
+community of constrained multiobjective optimization that a
+good tradeoff among feasibility, diversity, and convergence
+should be achieved [3]. It is worth noting that the impor-
+tance of these three elements varies in different evolutionary
+phases. Let us take the element of feasibility for example. In
+the infeasible phase, this element is very important because
+feasibility information plays an indispensable role in locating
+feasible regions, which is crucial for constrained multiobjec-
+tive optimization. However, in the feasible phase, this element
+is negligible as all solutions become feasible. We only need to
+consider the tradeoff between diversity and convergence. Due
+arXiv:2301.03317v1  [cs.NE]  9 Jan 2023
+
+2
+Convergence
+Convergence
+tradeoff
+tradeoff
+transition
+Population is 
+infeasible
+Population is 
+feasible
+Population is 
+semi-feasible
+Diversity
+Diversity
+Feasibility
+Feasibility
+Fig. 1. Task decomposition of achieving a tradeoff among feasibility, diversity,
+and convergence.
+to their varied importance, it is nontrivial to balance these three
+elements simultaneously by using a single tradeoff model.
+As an alternative, we adapt different tradeoff models in
+different evolutionary phases, proposing an adaptive tradeoff
+model with reference points (ATM-R) to handle CMOPs.
+Fig. 1 depicts the tradeoffs considered in ATM-R:
+• achieving a tradeoff between feasibility and diversity in
+the infeasible phase: when the population is entirely
+infeasible, the primary goal is to find as many feasible
+regions as possible since the Pareto optima may be scat-
+tered in different feasible regions. To this end, a tradeoff
+between feasibility and diversity should be achieved to
+move the population toward the feasible regions from
+diverse search directions.
+• promoting the transition from “the tradeoff between fea-
+sibility and diversity” to “the tradeoff between diversity
+and convergence” in the semi-feasible phase: when the
+population is semi-feasible (i.e., the population contains
+both infeasible and feasible solutions), two situations
+should be considered. In the early stage, only a few
+feasible regions are discovered. In this case, the tradeoff
+between feasibility and diversity should still be prioritized
+to find more promising feasible regions. Once enough
+feasible regions are located, in the later stage, attention
+should be paid to drive the population toward the CPF
+quickly and make them uniformly spread over the CPF
+simultaneously. Thus, the tradeoff between convergence
+and diversity should be concentrated on. In summary, in
+this phase, we should shift from “the tradeoff between
+feasibility and diversity” to “the tradeoff between diver-
+sity and convergence” [3].
+• achieving a tradeoff between diversity and convergence
+in the feasible phase: when the population is completely
+feasible, the final task is to move the feasible solutions
+toward the CPF quickly while maintaining good diversity.
+Apparently, a tradeoff between diversity and convergence
+should be realized [4].
+In summary, the core of a CMOEA is how to accomplish
+the above tradeoffs. The tradeoff in the feasible phase has
+been well studied in the community of evolutionary multiob-
+jective optimization. For convenience, in ATM-R, an off-the-
+shelf unconstrained multiobjective optimization evolutionary
+algorithm (MOEA) is utilized to achieve this tradeoff directly.
+As for the tradeoffs in the other two phases, the related
+studies remain relatively scarce. Especially for the tradeoff
+in the semi-feasible phase, little research focuses on this
+topic. Indeed, to achieve the tradeoffs in these two phases, an
+important concern is how to deal with the infeasible solutions.
+Past experience in the community of evolutionary constrained
+multiobjective optimization has shown that the infeasible so-
+lutions can not only facilitate maintaining diversity but also
+contribute to speeding up the convergence. In ATM-R, the
+merits of reference points are leveraged to select different
+kinds of infeasible solutions suitable for different evolutionary
+phases. In summary, the main contributions of this paper are
+as follows:
+• Instead of using a single tradeoff model, we adapt dif-
+ferent tradeoff models in different evolutionary phases,
+proposing a novel constrained multiobjective optimiza-
+tion algorithm (CMOEA) called ATM-R. Although it is
+inevitable for an algorithm to experience three phases
+during the evolution, few attempts have been made to
+develop alternate tradeoff models for different phases to
+facilitate a more explicit adaptation.
+• By leveraging the merits of reference points, we provide
+a new perspective that selects promising infeasible so-
+lutions suitable for different evolutionary phases. To the
+best of our knowledge, relevant work along this direction
+remains scarce.
+• A multiphase mating selection strategy is developed in
+this paper that adaptively selects suitable mating parents
+for different evolutionary phases.
+• Systemic experiments have been implemented on three
+sets of test suites including 36 benchmark CMOPs to
+validate the effectiveness of ATM-R. Comparison against
+five state-of-the-art CMOEAs suggests that ATM-R is
+significantly superior or comparable to the contender
+algorithms on most of the test problems. Additionally, the
+advantages of some important algorithmic components in
+ATM-R have been verified.
+The rest of this paper is organized as follows. Section II
+conducts a brief review of related CMOEAs. The details of
+ATM-R are described in Section III. The performance of ATM-
+R is compared with five representative CMOEAs in Section
+IV. Section V presents some further analyses of ATM-R in
+depth. The concluding remarks and future work are given in
+Section VI.
+II. RELATED WORK
+Constrained multiobjective optimization has become a hot
+topic in the community of evolutionary computation and
+numerous CMOEAs have been proposed. Based on whether
+infeasible solutions are utilized, these CMOEAs can be clas-
+sified into two categories: feasibility-driven CMOEAs and
+infeasibility-assisted CMOEAs.
+A. Feasibility-Driven CMOEAs
+A feasibility-driven CMOEA is driven by feasibility infor-
+mation, in which feasible solutions are considered to be better
+than infeasible ones. Some feasibility-driven CMOEAs use
+the constrained dominance principle (CDP) to compare two
+solutions [5]. In the CDP, a solution xu is said to be better
+
+3
+than another solution xv, if one of the following conditions is
+met:
+• both xu and xv are infeasible, and G(xu) < G(xv);
+• xu is feasible and xv is infeasible;
+• both xu and xv are feasible, and xu ≺ xv.
+Due to its preference for feasible solutions, the CDP can
+motivate the population toward feasible regions quickly. It has
+been widely integrated with different kinds of MOEAs [6],
+[7] and used in a spectrum of engineering optimization prob-
+lems [8], [9]. Liu et al. [6] combined an angle-based selection
+strategy, the shift-based density estimation strategy, and the
+CDP for constrained many-objective optimization. Jain and
+Deb [7] proposed a reference-point-based nondominated sort-
+ing approach, which is integrated with the CDP for constrained
+many-objective optimization. Jan and Khanum [10] embedded
+the CDP into the framework of MOEA/D and compared its
+performance with that of the stochastic ranking [11]. CDP-
+based CMOEAs are often used as the baseline algorithms
+when evaluating the performance of a CMOEA [12]–[14].
+The feasibility rule, which is widely used for constrained
+single-objective optimization, has been extended to solve
+CMOPs. Liu et al. [15] combined the feasibility rule with an
+indicator-based MOEA and compared its performance with
+that of some other kinds of CMOEAs. Fan et al. [16] carried
+out a comparison study on MOEA/D for constrained multiob-
+jective optimization. Different constraint-handling techniques
+including the feasibility rule are embedded into the framework
+of MOEA/D.
+Some CMOEAs put emphasis on constraints when the
+population contains no feasible solutions. Woldesenbet and
+Yen [17] presented a self-adaptive penalty method to solve
+CMOPs, in which an adaptive penalty function and a dis-
+tance measure are combined for constraint-handling. In fact,
+when the population is entirely infeasible, the self-adaptive
+penalty method compares two solutions based on constraints
+regardless of objectives. Liu and Wang [18] presented a two-
+phase CMOEA to solve CMOPs. When the population is
+entirely infeasible, all objectives are combined together and the
+feasibility rule is used to tackle constraints. Due to the superior
+capability of its search algorithm, the two-phase CMOEA can
+handle complex constraints in the decision space. Jimenez et
+al. [19] designed a CMOEA for constrained multiobjective
+optimization, in which the min-max formulation is used to
+tackle constraints. In addition, the feasibility rule is used to
+compare two solutions when an offspring is inserted into the
+new population. Miyakawa et al.
+[20] developed a two-
+stage nondominated sorting method to solve CMOPs. The
+population is divided into several fronts by the nondominated
+sorting according to constraints. The obtained fronts are further
+partitioned by the nondominated sorting based on objectives.
+In this manner, constraints are prior to objectives in environ-
+mental selection.
+B. Infeasibility-assisted CMOEAs
+An infeasibility-assisted CMOEA takes advantage of in-
+feasible solutions for constrained multiobjective optimization.
+Most state-of-the-art CMOEAs fall into this category.
+Some CMOEAs take advantage of infeasible solutions
+implicitly by using a comparison criterion that takes both
+constraints and objectives into account. Ma and Wang [3] pro-
+posed a shifted-based penalty function, in which an infeasible
+solution is penalized based on the information provided by the
+feasible solutions nearby. Jiao et al. [21] proposed a modified
+objective function method. When the population is entirely
+infeasible, the modified objective function is equivalent to
+a distance measure in which constraints and objectives are
+considered equally important. Fan et al. [?] presented an
+angle-based CDP for constrained multiobjective optimization.
+Given a feasible solution and an infeasible solution, if the
+angle between these two solutions is smaller than a predefined
+threshold, they would be nondominated each other. Thus, some
+infeasible solutions could enter into the new population instead
+of some feasible ones. Young [22] proposed a blended ranking
+measure to select solutions. By blending an individual’s rank
+in the objective space with its rank in the constraint space, an
+infeasible solution may be better than a feasible one. Similarly,
+Ma et al. [13] designed a new fitness function with two
+rankings, in which one ranking value is obtained based on the
+CDP and the other is calculated based on the Pareto dominance
+without considering constraints. The ε constrained method can
+use infeasibility information by tuning a threshold value ε [2];
+thus, it has been widely used to solve CMOPs [23]. Zapotecas-
+Mart´ınez and Ponsich [24] combined MOEA/D with the ε
+constrained method to solve CMOPs, in which the ε value
+is set according to the degree of constraint violation. Fan et
+al. [25] improved the ε constrained method by setting the ε
+value dynamically. Zhou et al. [26] extended the ε constrained
+method to solve CMOPs. When the degree of constraint
+violation of an infeasible solution is larger than the ε value, its
+diversity will be carefully maintained. The stochastic ranking
+that is popular for constrained single-objective optimization
+has also been extended to solve CMOPs [15], [27].
+Some CMOEAs leverage the advantages of infeasible so-
+lutions explicitly by archiving or coevolution. Ray et al. [28]
+proposed an infeasibility-driven EA, in which a small per-
+centage of infeasible solutions close to the constraint bound-
+aries are maintained. Li et al. [29] designed a two-archive
+EA for constrained multiobjective optimization. An archive
+is used to promote convergence, while the other is used
+to maintain diversity. The diversity archive evolves without
+considering constraints; thus, infeasible solutions with good
+objective function values can be fully used. Liu et al. [4]
+tried to solve CMOPs through bidirectional coevolution. The
+CDP is used to drive the main population toward the CPF
+from the feasible side of the search space. In addition, a
+nondominated sorting procedure and an angle-based selection
+scheme are conducted in sequence to motivate the population
+toward the CPF within the infeasible region. Tian et al. [30]
+developed a coevolutionary framework for constrained mul-
+tiobjective optimization. Similarly, one population is updated
+by the CDP, while the other is updated by an unconstrained
+MOEA. Additionally, the elites of these two populations are
+selected to generate offspring. Ishibuchi et al. [31] designed
+a dual-grid model of MOEA/D for constrained multiobjective
+optimization. Two populations are maintained and infeasible
+
+4
+solutions with good objective function values are preferred
+in the secondary population. Zhu et al. [32] employed two
+types of weight vectors in MOEA/D to solve CMOPs. The
+solutions associated with the convergence weight vectors are
+updated based on the aggregation function, while the solutions
+associated with the diversity weight vectors are renewed
+according to both the aggregation function and the degree
+of constraint violation. Peng et al. [14] used two kinds of
+weight vectors for constrained multiobjective optimization.
+Specifically, the degree of constraint violation is considered
+as another objective. Subsequently, a set of feasible weight
+vectors and a set of infeasible weight vectors are used to
+update the population. Additionally, the set of infeasible
+weight vectors is dynamically adjusted to maintain a number
+of infeasible solutions with good objective function values and
+small degrees of constraint violation.
+Some CMOEAs divide the evolutionary process into several
+phases and put emphasis on objectives in one of the phases.
+Yang et al. [33] divided the evolutionary process into a
+constrained search mode and an unconstrained search mode.
+These two search modes are executed by a dynamic constraint-
+handling mechanism. Fan et al. [12] proposed a push and pull
+search (PPS) framework to solve CMOPs, in which the evo-
+lutionary process is divided into two stages: push and pull. In
+the push stage, the population is updated by an unconstrained
+MOEA. In the pull stage, an improved ε constrained method is
+designed to tackle complex constraints. Since its proposition,
+the PPS framework has been used in various fields [34],
+[35]. Yu et al. [36] proposed a dynamic selection preference-
+assisted constrained multiobjective differential evolutionary
+(DE) algorithm. The selection preference for a solution shifts
+from infeasibility to feasibility as the optimization progresses.
+Tian et al. [37] proposed a two-stage CMOEA to balance
+objective optimization and constraint sanctification. These two
+stages are executed dynamically according to the percentage of
+feasible solutions in the population. Recently, Ming et al. [38]
+proposed a simple two-stage EA for constrained multiobjective
+optimization. The two-stage EA focuses on approaching the
+unconstrained Pareto front in the first stage and the feasible
+solutions are archived. In the second stage, the method seeks to
+approximate the CPF, where the archived feasible solutions are
+adopted as the initial population. Peng et al. [39] proposed a
+two-phase EA for constrained multiobjective optimization with
+deceptive constraints. In the first phase, two subpopulations are
+employed to explore the feasible regions and the entire space,
+respectively. The second phase aims to approach the CPF.
+Additionally, an infeasibility utilization strategy is designed
+to leverage the promising information provided by infeasible
+solutions.
+III. PROPOSED METHOD
+The general flow chart of ATM-R is shown in Fig. 2. As its
+name implies, ATM-R makes use of reference points to adap-
+tively accomplish different tradeoffs in different evolutionary
+phases, those are, the infeasible phase, the semi-feasible phase,
+and the feasible phase. The details of the update mechanisms
+in these three different phases are described in Section III-A,
+Infeasible?
+Infeasible 
+Phase
+Semi-feasible 
+Phase
+Feasible 
+Phase
+Reproduction
+Initialization
+Stop?
+Semi-feasible?
+Output the 
+Population 
+Yes
+No
+Yes
+Yes
+No
+No
+Fig. 2. Flow chart of ATM-R.
+Algorithm 1: Update Mechanism in the Infeasible
+Phase
+Input: Population P, offspring population O
+Output: New population P
+1 Q ← P ∪ O;
+2 Divide Q into k fronts based on ˆF(x): F1, · · · , Fk;
+3 P ← ∅;
+4 for l = 1 : k do
+5
+if |P| + |Fl| ≥ N then
+6
+Break;
+7
+P ← P ∪ Fl;
+8 if |P| + |Fl| > N then
+9
+Sample n uniformly distributed reference points
+and generate corresponding weight vectors:
+w1, · · · , wn;
+10
+Assign each solution in Fl to a weight vector
+according to (5)-(7);
+11
+while |P| + |Fl| > N do
+12
+Select the weight vector associated with the
+largest number of solutions: wc;
+13
+Among the solutions assigned to wc, select the
+one with the largest value of G(x): xw;
+14
+Fl ← Fl\xw;
+15 P ← P ∪ Fl;
+Section III-B, and Section III-C, respectively. Aside from the
+environmental selection procedure, another critical element
+of a CMOEA is the mating selection procedure. In ATM-
+R, a multiphase mating selection strategy is developed to
+generate promising solutions beneficial to different tradeoffs.
+The details of this strategy are illustrated in Section III-D.
+Finally, the framework of ATM-R and some discussions are
+shown in Section III-E and Section III-F, respectively.
+A. Update Mechanism in the Infeasible Phase
+In this phase, ATM-R aims to strike a balance between feasi-
+bility and diversity. In other words, it motivates the population
+toward feasibility from diverse search directions, thus locating
+as many feasible regions as possible. Algorithm 1 shows how
+ATM-R accomplishes this tradeoff. In general, it involves two
+essential elements.
+1) Nondominated Sorting in the Transformed Objective
+Space: Following the ideas in [40], we consider G(x) as
+an additional objective function, and transform (1) into an
+
+5
+unconstrained MOP:
+min ˆF(x) = (f1(x), · · · , fm(x), G(x))T ∈ Rm+1.
+(4)
+Clearly, this transformation does not introduce any extra
+parameters. In addition, both objective functions and con-
+straints are considered in (4), which can facilitate maintaining
+population diversity and enhance driving forces toward the
+feasible regions. Based on ˆF(x), the population will be divided
+into several fronts, denoted as F1, · · · , Fk, by implementing a
+nondominated sorting procedure in the transformed objective
+space. Afterward, the solutions in each front will be selected
+in turn until �l−1
+i=1 |Fi| < N ≤ �l
+i=1 |Fi| where N denotes
+the size of the final solution set.
+2) Regular Reference Point-based Selection: If �l
+i=1 |Fi|
+is larger than N, we should further select (n = N−�l−1
+i=1 |Fi|)
+solutions from the last desired front Fl. To complete this task,
+in this study, a regular reference point-based selection scheme
+is developed by taking advantage of uniformly distributed
+reference points. Its implementation is quite simple.
+• First, a set of regular (i.e., uniformly distributed) refer-
+ence points is sampled in the objective space to generate
+weight vectors denoted as {w1, · · · , wn} following the
+ideas in [41].
+• Subsequently, a solution (denoted as x) in Fl is assigned
+to the weight vector with the smallest angle to its nor-
+malized objective vector:
+I = arg min
+j∈{1,··· ,n}
+θj,
+(5)
+θj = arccos
+�����
+F′(x)Twj
+∥F′(x)∥ · ∥wj∥
+����� , j = 1, · · · , n,
+(6)
+f ′
+j(x) = fj(x) − zmin
+j
+zmax
+j
+− zmin
+j
+, j = 1, · · · , m,
+(7)
+where I indicates which weight vector the solution x is
+assigned to; θj denotes the angle between wj and the nor-
+malized objective vector F′(x) = (f ′
+1(x), · · · , f ′
+m(x))T;
+∥ · ∥ represents the function to calculate the 2-norm
+of a vector; zmax
+=
+(zmax
+1
+, · · · , zmax
+m )T and zmin
+=
+(zmin
+1 , · · · , zmin
+m )T refer to the estimated nadir point and
+ideal point, respectively.
+• Afterward, (|Fl|−n) inferior solutions are deleted one by
+one by employing a “diversity first, feasibility second”
+strategy. To be specific, it first identifies the weight
+vector associated with the largest number of solutions1.
+Intuitively, since these solutions are associated with the
+same weight vector, they will share highly similar search
+directions. To maintain diverse search directions, it is nec-
+essary to delete one of them. The feasibility information
+of these solutions is considered for the deletion. The one
+with the largest value of G(x) is discarded. These two
+steps will continue until (|Fl| − n) solutions are deleted.
+A simple example is given in Fig. 3 for better understanding
+the regular reference point-based selection scheme. We con-
+sider a CMOP with two objectives. Suppose there are seven
+1Note that the tie is broken at random
+A
+B
+D
+C
+E
+F
+Feasible 
+region
+1
+w
+2
+w
+3
+w
+4
+w
+'
+2f
+'
+1f
+G
+G
+Fig. 3. Update mechanism in the infeasible phase.
+solutions in the population, and they lie in the same front in
+the transformed objective space. According to the values of
+G(x), these individuals were ranked as F, C, E, A, G, D, and
+B in ascending order. The task is to select four solutions for
+the next generation.
+1) First, four reference points are sampled uniformly to
+generate four weight vectors denoted as {w1, · · · , w4}.
+2) Next, each solution in the population is assigned to a
+weight vector: w1 ↔ {A}, w2 ↔ {B, C}, w3 ↔ {D},
+and w4 ↔ {E, F, G}.
+3) Subsequently, three solutions are deleted one by one.
+G is first deleted since w4 is matched with the largest
+number of solutions and G is the one with the largest
+value of G(x) compared with E and F. According to
+this principle, B and E will be also removed.
+4) Finally, the solutions (i.e., A, C, D, and F will enter into
+the next generation.
+Remark 1: Both ATMES2 [40] and IDEA [28] employ non-
+dominated sorting in the transformed objective space as ATM-
+R does. The main difference lies in how to distinguish the
+solutions in the same front. Specifically, in ATMES, solutions
+are selected based on G(x) only. A solution with a smaller
+value of G(x) will be preferred. In this manner, ATMES will
+put too much emphasis on constraints. It will cause perfor-
+mance deterioration in terms of the search diversity, which
+is essential for finding as many promising feasible regions
+as possible. On the contrary, in IDEA, only the diversity in
+the transformed objective space is considered to update the
+last desired front Fl. Unfortunately, this manner will result in
+a limited driving force toward the feasible regions, which in
+turn leads to a relatively low convergence speed. Unlike these
+two methods, ATM-R takes both diversity and feasibility into
+account to update Fl, and a “diversity first, feasibility second”
+strategy is thus developed. As illustrated in Fig. 3, ATM-R can
+strike a good balance between diversity and feasibility, thereby
+motivating the population toward feasible regions from diverse
+search directions.
+B. Update Mechanism in the Semi-feasible Phase
+ATM-R intends to promote the transition from “the tradeoff
+between feasibility and diversity” to “the tradeoff between
+2Although ATMES is originally designed for constrained single-objective
+optimization, it can be directly applied to solve CMOPs.
+
+6
+Algorithm 2: Update Mechanism in the Semi-feasible
+Phase
+Input: Population P, offspring population O, FEs,
+MaxFEs
+Output: New population P
+1 Q ← P ∪ O, P ← ∅;
+2 Qf ← {x ∈ Q|G(x) = 0}, Qif ← {x ∈ Q|G(x) > 0};
+3 if |Qf| > N then
+4
+Qf ← N feasible solutions seleted from Qf by an
+unconstrained MOEA;
+5 P ← P ∪ Qf;
+6 if |Qif| > N then
+7
+if
+F Es
+MaxF Es < 0.5 or |Qf| < N then
+8
+Qif ← N infeasible solutions selected from
+Qif by using Algorithm 1;
+9
+else
+10
+Generate |Qf| weight vectors by using the
+solutions in Qf according to (8)-(9);
+11
+Assign each solution in |Qif| to a weight
+vector according to (5)-(7);
+12
+while |Qif| > N do
+13
+Select the weight vector associated with the
+largest number of solutions: wc;
+14
+Among the solutions assigned to wc, select
+the one furthest from the feasible solution
+used to generate wc: xw;
+15
+Qif ← Qif\xw;
+16 P ← P ∪ Qif;
+diversity and convergence” in the semi-feasible phase (i.e.,
+the population contains both infeasible and feasible solutions).
+The reasons for this transition are two-fold. In the early
+stage of the semi-feasible phase, ATM-R must locate as many
+feasible regions as possible. To this end, it must focus on
+the tradeoff between feasibility and diversity. After finding a
+sufficient number of feasible regions, in the later stage, ATM-
+R should steer the population rapidly toward the CPF and
+distribute it uniformly along with the CPF simultaneously.
+Thus, the tradeoff between convergence and diversity should
+be prioritized. Algorithm 2 shows how ATM-R updates the
+solutions in the semi-feasible phase.
+From Algorithm 2, it is observed that ATM-R updates
+the feasible and infeasible solutions separately. To update
+the feasible solutions, an unconstrained MOEA is used to
+truncate the feasible population Qf if its size is greater than
+N; otherwise, all feasible solutions are reserved. To update the
+infeasible solutions, ATM-R considers two situations. In the
+early stage, it aims to achieve a tradeoff between feasibility
+and diversity, which is the same as in the infeasible phase.
+Thus, the update mechanism used in the infeasible phase (i.e.,
+Algorithm 1) can be directly applied in this stage. While in
+the later stage, ATM-R shifts the emphasis to the tradeoff
+between diversity and convergence. To realize this tradeoff,
+an important task is how to preserve those infeasible solutions
+that can contribute to both diversity and convergence. ATM-R
+1
+w
+2
+w
+3
+w
+'
+1f
+'
+2f
+A
+B
+C
+D
+E
+F
+'
+1
+w
+'
+2
+w
+'
+3
+w
+'
+4
+w
+A
+B
+C
+D
+E
+F
+4
+w
+'
+1f
+'
+2f
+G
+H
+I
+J
+G
+H
+I
+J
+Feasible solution
+Infeasible solution
+CPF
+Feasible solution
+Infeasible solution
+CPF
+Feasible 
+region
+Feasible 
+region
+(a)
+1
+w
+2
+w
+3
+w
+'
+1f
+'
+2f
+A
+B
+C
+D
+E
+F
+'
+1
+w
+'
+2
+w
+'
+3
+w
+'
+4
+w
+A
+B
+C
+D
+E
+F
+4
+w
+'
+1f
+'
+2f
+G
+H
+I
+J
+G
+H
+I
+J
+Feasible solution
+Infeasible solution
+CPF
+Feasible solution
+Infeasible solution
+CPF
+Feasible 
+region
+Feasible 
+region
+(b)
+Fig. 4.
+Illustration of difference between the weight vectors in the regular
+reference point-based selection and those in the adaptive reference point-based
+selection: (a) weight vectors in regular reference point-based selection and (b)
+weight vectors in adaptive reference point-based selection.
+designs the following two steps to accomplish this task.
+1) Discovery of the Nondominated Infeasible Solutions:
+Compared with the feasible solutions in the current population,
+the nondominated infeasible solutions usually have smaller ob-
+jective function values. It is natural to leverage their benefits to
+promote convergence. To distinguish these infeasible solutions,
+we first employ a nondominated sorting procedure to divide
+the union population (i.e., Q in Algorithm 2) into several
+fronts based on ˆF(x) in
+(4). Subsequently, the infeasible
+solutions in the first front are picked out. If the number of these
+nondominated infeasible solutions (denoted as M) is smaller
+than N, all of them will be kept; otherwise, they will be further
+distinguished by the following adaptive reference point-based
+selection.
+2) Adaptive Reference Point-based Selection: Herein, the
+regular reference points are no longer used to assist the
+selection. The reason is that the CPF might be disconnected
+(see Fig. 4), and some weight vectors (i.e., w1 and w3 in
+Fig. 4(a)) generated using the uniformly distributed reference
+points cannot point to any parts of the CPF. As a result, the
+solutions preserved by making use of such weight vectors
+(i.e., C and F) are far away from the CPF and hardly con-
+tribute to convergence speed, which is not desirable. Instead,
+we use adaptive reference points for solution selection. For
+convenience, in our study, the feasible solutions are considered
+as adaptive reference points since they can deliver important
+clues for the localization of the CPF (see Fig. 4(b)). Based
+on these reference points, a set of adaptive weight vectors
+can be obtained conveniently. To be specific, for the ith
+
+7
+feasible solution xi, the corresponding weight vector (denoted
+as w
+′
+i = (w
+′
+i,1, · · · , w
+′
+i,m)T) is generated as follows:
+w
+′
+i,j =
+f
+′
+j(x)
+�m
+j=1 f
+′
+j(x), j = 1, · · · , m,
+(8)
+f ′
+j(x) = fj(x) − zmin
+j
+zmax
+j
+− zmin
+j
+, j = 1, · · · , m,
+(9)
+where (f ′
+1(x), · · · , f ′
+m(x))T is the normalized objective vector,
+(zmax
+1
+, · · · , zmax
+m )T and (zmin
+1 , · · · , zmin
+m )T denote the estimated
+nadir point and ideal point, respectively. Fig. 4 shows the
+difference between the weight vectors generated using regular
+reference points and those using adaptive reference points.
+It is evident that the weight vectors obtained using adaptive
+reference points fit better to the characteristics of the CPF.
+Once the adaptive weight vectors are prepared, the next
+procedures in the adaptive reference point-based selection
+scheme are quite simple. First, each nondominated infeasi-
+ble solution is assigned to a weight vector following the
+ideas in the regular reference point-based selection scheme
+(see Eqs. (5)-(7)). Afterward, (M-N) infeasible solutions are
+deleted one by one in a two-step manner. The first step
+is to identify the weight vector associated with the largest
+number of solutions. In the second step, among the solutions
+assigned to this weight vector, the one furthest from the
+feasible solution corresponding to the weight vector in the
+objective space will be deleted. In general, the first step is
+similar to many decomposition-based approaches and can help
+to maintain population diversity. As for the second step, it can
+help to retain those infeasible solutions close to the feasible
+solutions and thus offer a driving force toward the CPF from
+the infeasible side of the search space. Intuitively, this way
+can speed up the convergence.
+Remark 2: In the semi-feasible phase, the population size
+in ATM-R is larger than or equal to N. The reason is that a
+larger population can enhance population diversity, which is
+critical to both “the tradeoff between feasibility and diversity”
+and “the tradeoff between diversity and convergence”. As for
+how to determine whether the algorithm has entered the later
+stage of the semi-feasible phase, we considered two simple
+conditions which should be satisfied simultaneously. The first
+condition is that
+F Es
+MaxF Es should be larger than 0.5. Note
+that FEs and MaxFEs denote the function evaluations and
+the maximum function evaluations, respectively. The second
+condition relies on the number of feasible solutions which
+should be equal to N. The first condition implies that enough
+search efforts have been devoted to finding feasible regions,
+while the second condition is set to ensure a sufficient number
+of reference points. In the later stage of the semi-infeasible
+phase, if no nondominated infeasible solutions are discovered,
+the algorithm will enter the feasible phase.
+C. Update Mechanism in the Feasible Phase
+In this phase, all solutions are feasible. Under this condition,
+only the tradeoff between diversity and convergence should be
+considered, thus motivating the feasible solutions toward the
+CPF quickly while maintaining good diversity. Apparently,
+Algorithm 3: Multiphase Mating Selection Strategy
+Input: Population P, N
+Output: Mating population C
+1 C ← ∅;
+2 for i = 1 : N do
+3
+Randomly select two different solutions denoted as
+xa and xb from P;
+4
+if P is entirely infeasible then
+5
+if rand < 0.5 then
+6
+xm ← the better one between xa and xb
+based on the degree of constraint
+violation;
+7
+else
+8
+xm ← the better one between xa and xb
+based on the diversity;
+9
+else if P is feasible then
+10
+if xa ≺ xb then
+11
+xm ← xa;
+12
+else if xb ≺ xa then
+13
+xm ← xb;
+14
+else
+15
+xm ← the better one between xa and xb
+based on the diversity;
+16
+else if P is semi-feasible then
+17
+if i < N/2 then
+18
+xm ← the better one between xa and xb by
+using the method in the infeasible phase;
+19
+else
+20
+xm ← the better one between xa and xb
+by using the method in the feasible phase;
+21
+C ← C ∪ xm;
+a current effective unconstrained MOEA can be applied to
+achieve this balance. Thus, in ATM-R, an off-the-shelf uncon-
+strained MOEA is employed in this phase straightforwardly.
+D. Multiphase Mating Selection Strategy
+In addition to the multi-phase strategy in environmental
+selection, ATM-R uses a multi-phase strategy for mating selec-
+tion. It selects appropriate mating parents suitable for different
+evolutionary phases. The details of this multiphase mating
+selection strategy are described in Algorithm 3. Similarly,
+three different phases are considered in this strategy.
+• In the infeasible phase, population diversity and feasi-
+bility should be focused on simultaneously. Thus, in the
+tournament selection, solutions are compared based on
+the diversity and the degree of constraint violation with
+the same probability (i.e., 0.5). Note that the diversity is
+quantified by the same way as in [30].
+• In the feasible phase, population diversity and conver-
+gence should be taken into account. Following the ideas
+in NSGA-II [5], in the tournament selection, solutions are
+compared based on the Pareto dominance relationship.
+
+8
+Algorithm 4: ATM-R
+Input: A CMOP, N, MaxFEs
+Output: Final population P
+1 P ← a population initialized from the decision space;
+2 FEs ← N;
+3 while FEs < MaxFEs do
+4
+C ← a mating population selected from P by using
+Algorithm 3;
+5
+O ← an offspring population generated by
+executing genetic operators on C;
+6
+FEs ← FEs + N;
+7
+Q ← P ∪ O;
+8
+if Q is entirely infeasible then
+9
+P ← the solutions seleted from Q by using
+Algorithm 1;
+10
+else if Q is semi-feasible then
+11
+P ← the solutions selected from Q by using
+Algorithm 2;
+12
+else if Q is feasible then
+13
+P ← the solutions selected from Q by using an
+unconstrained MOEA;
+Also, if two solutions do not dominate each other, they
+are compared based on the diversity.
+• The semi-feasible phase needs to bridge the gap between
+the feasible phase and the infeasible phase. Thus, in this
+phase, the first half of the mating population is selected
+by using the method in the infeasible phase, while the
+other half is selected by using the method in the feasible
+phase.
+E. ATM-R
+In summary, the details of ATM-R are given in Algorithm 4.
+At the beginning, a population of N solutions is sampled
+uniformly in the decision space (Lines 1-2). Afterward, the
+population is employed to search for the CPF until the
+maximum number of function evaluations is exhausted (Lines
+3-15). In the search process, first, N mating parents are
+selected for offspring generation by using the multiphase
+mating selection strategy in Algorithm 3 (Line 4). Next,
+N offspring are produced by the simulated binary crossover
+(SBX) [42] and the polynomial mutation (PM) [43] (Lines
+5-6). Afterward, promising solutions are selected based on
+population feasibility (Lines 7-14) where Algorithm 1 and
+Algorithm 2 are used in the infeasible phase and the semi-
+feasible phase, respectively. Note that if FEs ≥ MaxFEs,
+the final population P would be output.
+F. Discussion
+In essence, ATM-R is a multiphase CMOEA. ATM-R in-
+tends to achieve a tradeoff between diversity and feasibility in
+the infeasible phase, promote the transition from “the tradeoff
+between feasibility and diversity” to “the tradeoff between
+diversity and convergence” in the semi-feasible phase, and
+accomplish the tradeoff between diversity and convergence in
+the feasible phase. To the best of our knowledge, ATM-R is
+the first algorithm considering these tradeoffs simultaneously
+during different evolution phases. Also, ATM-R is interesting
+in that it selects promising infeasible solutions suitable for
+different evolutionary phases by using two kinds of reference
+points. As far as we know, relevant studies in this direction
+are almost absent. From our analysis, it is apparent that ATM-
+R is a brand-new CMOEA for constrained multiobjective
+optimization.
+The computational time complexity of ATM-R is mainly
+determined by the nondominated sorting and the unconstrained
+MOEA. Suppose the fast nondominated sorting and NS-
+GAII [5] are adopted in ATM-R. In the worst case of the
+infeasible phase, no solutions nondominated another in the
+transformed objective space. The time complexity of this
+nondominated sorting is O((m+1)·N 2). The time complexity
+of assigning each solution to a weight vector is O(m·N 2). The
+time complexity of selecting N solutions is O(N 2). Thus, the
+time complexity of the infeasible phase is O((m + 1) · N 2) +
+O(m · N 2) + O(N 2) = O((m + 1) · N 2). In the semi-feasible
+phase, the worst-case time complexity of selecting feasible
+solutions is O(m·N 2). In the early stage of the semi-feasible
+phase, the worst-case time complexity is the same as that of
+the infeasible phase: O((m + 1) · N 2). In the worst case of
+the later stage, no infeasible solutions nondominated another.
+It is the same as that of the infeasible phase. Thus, its time
+complexity is O((m + 1) · N 2). The time complexity of the
+semi-feasible phase is O(m·N 2)+O((m+1)·N 2)+O((m+
+1) · N 2) = O((m + 1) · N 2). In the feasible phase, the time
+complexity is the same as that of NSGAII: O(m · N 2). In
+summary, the computational time complexity of ATM-R is
+O((m+1)·N 2)+O((m+1)·N 2)+O(m·N 2) = O(m·N 2),
+which is indeed acceptable.
+IV. PERFORMANCE COMPARISON
+In this section, we assess the performance of ATM-R based
+on a wide range of benchmark test functions. Specifically,
+ATM-R was used to solve three test suites and its performance
+was compared with that of five representative CMOEAs.
+Note that all experiments were implemented by the PlatEMO
+toolbox [44].
+A. Experimental Settings
+1) Test Functions: Three test suites consisting of 36 bench-
+mark test functions (e.g., MW [45], CTP [46], and LIRC-
+MOP [25]) were adopted in our study. These test functions
+own various challenging characteristics; thus, they can assess
+the performance of a CMOEA adequately. Most state-of-the-
+art CMOEAs adopt these test functions for empirical study.
+Note that the number of decision variables in MW and
+LIRCMOP was set to 15 and 10, respectively. Please see [25],
+[45], [46] for the details of these test functions.
+2) Peer Algorithms:
+For performance comparison, five
+representative
+CMOEAs
+were
+taken
+into
+consideration:
+NSGAII-CDP [5], PPS [12], the constrained two-archive EA
+(CTAEA) [29], the coevolutionary constrained multiobjective
+
+9
+TABLE I
+THE IGD VALUES OF NSGAII-CDP, PPS, CTAEA, CCMO, TOP, AND ATM-R ON THREE SETS OF BENCHMARK TEST FUNCTIONS.
+Test Functions
+NSGAII-CDP
+mean IGD (std)
+PPS
+mean IGD (std)
+CTAEA
+mean IGD (std)
+CCMO
+mean IGD (std)
+ToP
+mean IGD (std)
+ATM-R
+mean IGD (std)
+MW1
+4.0545e-2 (1.02e-1) -
+2.3190e-2 (4.01e-2) -
+2.1884e-3 (9.96e-4) -
+1.8990e-3 (1.42e-3) +
+NaN (NaN) -
+2.1748e-3 (1.70e-3)
+MW2
+2.3926e-2 (7.65e-3) -
+4.2401e-2 (3.25e-2) -
+1.7953e-2 (6.74e-3) ≈
+2.1515e-2 (8.20e-3) -
+2.3108e-1 (1.89e-1) -
+1.9130e-2 (9.76e-3)
+MW3
+7.4318e-2 (2.31e-1) -
+7.5935e-3 (9.94e-4) -
+5.4804e-3 (4.86e-4) ≈
+5.2178e-3 (4.41e-4) ≈
+5.9698e-1 (2.78e-1) -
+5.3646e-3 (4.08e-4)
+MW4
+5.5780e-2 (2.97e-3) -
+5.3955e-2 (1.73e-3) -
+4.6413e-2 (4.99e-4) -
+4.1285e-2 (3.48e-4) ≈
+NaN (NaN) -
+4.1255e-2 (3.45e-4)
+MW5
+4.2761e-1 (3.35e-1) -
+1.4507e-1 (1.97e-1) -
+1.5758e-2 (3.38e-3) -
+4.6474e-3 (7.30e-3) -
+NaN (NaN) -
+4.0638e-3 (1.06e-2)
+MW6
+8.0099e-2 (1.51e-1) -
+1.0037e-1 (1.61e-1) -
+1.1188e-2 (6.68e-3) ≈
+5.2473e-2 (1.26e-1) -
+1.0872e+0 (1.81e-1) -
+1.5369e-2 (8.69e-3)
+MW7
+1.0205e-1 (1.93e-1) -
+2.5520e-2 (1.88e-2) -
+7.2156e-3 (5.22e-4) -
+4.8994e-3 (4.80e-4) +
+4.7226e-1 (2.39e-1) -
+5.2004e-3 (4.73e-4)
+MW8
+6.1793e-2 (8.78e-3) -
+7.4112e-2 (2.69e-2) -
+5.5531e-2 (2.47e-3) -
+4.9189e-2 (1.58e-2) ≈
+9.5949e-1 (2.01e-1) -
+4.6368e-2 (5.74e-3)
+MW9
+2.1737e-1 (3.10e-1) -
+7.4032e-2 (1.81e-1) -
+8.8691e-3 (9.23e-4) ≈
+5.1927e-2 (1.79e-1) -
+NaN (NaN) -
+9.9563e-3 (2.88e-3)
+MW10
+2.3341e-1 (2.36e-1) -
+1.3321e-1 (1.48e-1) -
+1.7599e-2 (1.22e-2) ≈
+4.2867e-2 (2.54e-2) -
+NaN (NaN) -
+2.7242e-2 (2.33e-2)
+MW11
+4.7335e-1 (3.24e-1) -
+1.3565e-2 (2.15e-2) -
+1.6564e-2 (2.80e-3) -
+6.3416e-3 (5.30e-4) ≈
+9.2141e-1 (1.37e-1) -
+6.1791e-3 (2.34e-4)
+MW12
+8.2766e-2 (2.23e-1) -
+2.9552e-2 (1.20e-1) +
+8.0645e-3 (6.84e-4) +
+3.0553e-2 (1.40e-1) ≈
+NaN (NaN) -
+7.8769e-2 (2.24e-1)
+MW13
+2.0642e-1 (2.77e-1) -
+1.3735e-1 (6.32e-2) -
+3.8211e-2 (2.66e-2) ≈
+8.2172e-2 (4.41e-2) -
+8.3328e-1 (5.33e-1) -
+5.2527e-2 (3.23e-2)
+MW14
+1.2974e-1 (1.27e-2) -
+2.5313e-1 (9.28e-2) -
+1.1279e-1 (6.91e-3) +
+9.8349e-2 (2.41e-3) +
+4.9059e-1 (5.81e-1) -
+1.1492e-1 (4.72e-2)
+CTP1
+8.1699e-2 (6.62e-2) -
+1.9234e-2 (1.80e-2) -
+1.8672e-2 (3.64e-2) -
+4.4317e-3 (1.05e-3) -
+3.9400e-3 (1.48e-4) -
+3.2367e-3 (7.38e-5)
+CTP2
+2.4408e-3 (1.89e-3) -
+3.7003e-3 (6.96e-4) -
+4.6860e-2 (1.25e-2) -
+1.6836e-3 (1.65e-4) -
+4.4453e-3 (1.08e-3) -
+1.4735e-3 (5.94e-5)
+CTP3
+6.2833e-2 (9.95e-2) -
+3.1094e-2 (4.07e-3) -
+5.8093e-2 (5.79e-3) -
+2.2180e-2 (2.24e-3) -
+3.2847e-2 (6.99e-3) -
+1.0066e-2 (1.60e-3)
+CTP4
+2.4494e-1 (1.29e-1) -
+1.4930e-1 (1.84e-2) -
+1.5350e-1 (1.92e-2) -
+1.3538e-1 (2.18e-2) -
+1.8414e-1 (3.29e-2) -
+7.9556e-2 (1.13e-2)
+CTP5
+7.2574e-3 (2.92e-3) -
+1.8168e-2 (6.11e-3) -
+1.8209e-2 (4.61e-3) -
+7.6639e-3 (1.76e-3) -
+1.2167e-2 (3.45e-3) -
+3.3142e-3 (4.13e-4)
+CTP6
+1.1404e-2 (4.04e-4) -
+1.3061e-2 (7.81e-4) -
+3.8535e-2 (5.23e-3) -
+1.0141e-2 (3.56e-4) -
+1.5214e-2 (2.78e-3) -
+9.7103e-3 (3.13e-4)
+CTP7
+1.6882e-3 (1.39e-3) -
+1.6825e-3 (7.14e-5) -
+1.6364e-3 (1.31e-4) -
+1.1669e-3 (4.52e-5) ≈
+1.5176e-3 (5.54e-5) -
+1.1599e-3 (4.62e-5)
+CTP8
+1.2019e-1 (1.45e-1) -
+1.1932e-2 (5.26e-3) -
+3.4505e-2 (4.79e-3) -
+5.5516e-3 (6.49e-4) -
+8.0925e-2 (1.38e-1) -
+4.7357e-3 (2.32e-4)
+LIRCMOP1
+2.6010e-1 (8.10e-2) -
+1.1024e-1 (3.40e-2) -
+3.7900e-1 (1.66e-1) -
+2.0503e-1 (6.82e-2) -
+1.2547e-1 (1.36e-1) -
+3.5295e-2 (1.26e-2)
+LIRCMOP2
+1.9890e-1 (7.22e-2) -
+7.3024e-2 (2.79e-2) -
+1.2324e-1 (6.12e-2) -
+1.1419e-1 (3.19e-2) -
+6.8227e-2 (5.38e-2) -
+3.1146e-2 (9.36e-3)
+LIRCMOP3
+2.4894e-1 (8.38e-2) -
+1.7697e-1 (5.80e-2) -
+3.4751e-1 (1.14e-1) -
+2.0960e-1 (7.98e-2) -
+3.6351e-1 (5.72e-2) -
+2.3380e-2 (1.06e-2)
+LIRCMOP4
+2.3080e-1 (6.23e-2) -
+1.4996e-1 (5.59e-2) -
+2.7661e-1 (1.38e-1) -
+1.9069e-1 (7.18e-2) -
+3.2442e-1 (5.76e-2) -
+2.3928e-2 (1.11e-2)
+LIRCMOP5
+7.3176e-1 (4.81e-1) -
+8.4362e-2 (2.44e-2) -
+1.3918e-1 (4.41e-2) -
+1.4046e-2 (8.21e-3) ≈
+1.2091e-1 (3.48e-1) -
+1.3635e-2 (6.48e-3)
+LIRCMOP6
+5.7447e-1 (4.57e-1) -
+9.5258e-2 (6.31e-2) -
+1.3633e-1 (1.13e-1) -
+1.1357e-2 (7.75e-3) ≈
+6.4593e-3 (3.45e-4) +
+5.7790e-2 (1.51e-1)
+LIRCMOP7
+1.7441e-2 (1.32e-2) ≈
+5.6488e-2 (5.84e-2) -
+2.5246e-2 (9.12e-3) -
+1.1404e-2 (6.38e-3) ≈
+8.6357e-3 (2.52e-4) +
+1.2400e-2 (5.03e-3)
+LIRCMOP8
+3.6946e-2 (4.51e-2) -
+6.6479e-2 (7.05e-2) -
+3.6096e-2 (6.64e-2) -
+9.1531e-3 (5.01e-3) ≈
+8.6820e-3 (4.53e-4) +
+9.4267e-3 (3.85e-3)
+LIRCMOP9
+5.3564e-1 (1.24e-1) -
+1.4063e-1 (8.94e-2) ≈
+1.1622e-1 (5.42e-2) ≈
+3.4398e-2 (3.91e-2) +
+2.4115e-1 (1.73e-1) -
+1.1216e-1 (7.48e-2)
+LIRCMOP10
+3.6496e-1 (9.66e-2) -
+8.2848e-3 (1.47e-2) -
+6.0919e-2 (6.50e-2) -
+5.4399e-3 (3.36e-4) +
+5.4878e-3 (2.21e-4) +
+6.9018e-3 (6.27e-4)
+LIRCMOP11
+2.4114e-1 (1.80e-1) -
+8.1119e-3 (7.48e-3) -
+1.3778e-1 (3.83e-2) -
+2.4538e-3 (8.89e-5) +
+1.2447e-1 (6.37e-2) -
+5.3691e-3 (1.45e-2)
+LIRCMOP12
+1.5180e-1 (8.66e-2) -
+1.5216e-2 (2.43e-2) ≈
+3.1152e-2 (1.68e-2) -
+4.6113e-3 (2.58e-3) ≈
+2.9104e-2 (5.29e-2) ≈
+7.8014e-3 (7.28e-3)
+LIRCMOP13
+2.3757e-1 (3.69e-1) -
+1.1968e-1 (3.45e-3) -
+1.0834e-1 (3.97e-4) -
+9.3972e-2 (1.13e-3) -
+1.2450e-1 (3.78e-3) -
+9.3120e-2 (9.31e-4)
+LIRCMOP14
+2.0248e-1 (2.93e-1) -
+1.1859e-1 (3.97e-3) -
+1.1126e-1 (7.98e-4) -
+9.5773e-2 (7.40e-4) -
+1.1883e-1 (4.04e-3) -
+9.4848e-2 (7.79e-4)
++/-/≈
+0/35/1
+1/33/2
+2/27/7
+6/19/11
+4/25/1
+optimization (CCMO) [30], and the two-phase EA (ToP) [18].
+NSGAII-CDP is a classic CMOEA that is usually adopted
+as a baseline algorithm, while the other four CMOEAs are
+state-of-the-art algorithms proposed recently. NSGAII-CDP is
+a feasibility-driven CMOEA and the others are infeasibility-
+assisted CMOEAs. Among these four infeasibility-assisted
+CMOEAs, CTAEA and CCMO are multi-population methods
+which take advantage of infeasible solutions explicitly by
+an archive or an additional population. PPS and ToP are
+multiphase methods which divide the evolutionary process into
+several phases and put emphasis on objectives in some phases.
+Note that ATM-R is also a multiphase method.
+3) Performance Metrics: Two frequently used performance
+metrics were adopted to assess the performance of a CMOEA:
+inverted generational distance (IGD) and hyper-volume (HV).
+Both IGD and HV can measure the convergence and coverage
+of a solution set. More details of these two metrics can be
+found in [47].
+4) Parameter Settings: The parameters involved in the
+experiments are given as follows:
+• Size of the final solution set: N = 100 for all comparison
+CMOEAs;
+• MaxFEs: MaxFEs = 60, 000 for the MW and CTP
+test suites, and MaxFEs = 300, 000 for the LIRCMOP
+test suite;
+4.6
+3.89
+3.68
+2.15
+5
+1.68
+4.65
+3.69
+3.78
+2.15
+4.85
+1.87
+0
+1
+2
+3
+4
+5
+6
+NSGAII-CDP
+PPS
+CTAEA
+CCMO
+ToP
+ATM-R
+IGD
+HV
+Fig. 5. Average rankings of six CMOEAs on 36 test functions in terms of
+the IGD/HV value. A lower ranking value denotes a better performance.
+• Number of independent runs: 30.
+The SBX and PM were used as genetic operators in all
+CMOEAs except ToP. The parameters of SBX and PM are as
+follows:
+• Crossover probability of SBX: 1;
+• Mutation probability of PM: 1/D;
+• Distribution index of SBX and PM: 20.
+In addition, the algorithm-specific parameters of the five
+peer CMOEAs were obtained from their original papers.
+
+10
+TABLE II
+THE HV VALUES OF NSGAII-CDP, PPS, CTAEA, CCMO, TOP, AND ATM-R ON THREE SETS OF BENCHMARK TEST FUNCTIONS.
+Test Functions
+NSGAII-CDP
+mean HV (std)
+PPS
+mean HV (std)
+CTAEA
+mean HV (std)
+CCMO
+mean HV (std)
+ToP
+mean HV (std)
+ATM-R
+mean HV (std)
+MW1
+4.5445e-1 (8.09e-2) -
+4.6529e-1 (3.63e-2) -
+4.8849e-1 (2.03e-3) -
+4.8927e-1 (3.04e-3) +
+NaN (NaN) -
+4.8853e-1 (3.60e-3)
+MW2
+5.4798e-1 (1.15e-2) -
+5.2241e-1 (4.44e-2) -
+5.5765e-1 (1.14e-2) ≈
+5.5199e-1 (1.30e-2) -
+3.2482e-1 (1.46e-1) -
+5.5635e-1 (1.54e-2)
+MW3
+5.0168e-1 (1.37e-1) -
+5.4398e-1 (4.88e-4) +
+5.4413e-1 (6.14e-4) +
+5.4368e-1 (7.81e-4) +
+1.2745e-1 (1.27e-1) -
+5.4292e-1 (7.86e-4)
+MW4
+8.2309e-1 (5.63e-3) -
+8.2478e-1 (2.49e-3) -
+8.3814e-1 (4.04e-4) -
+8.4116e-1 (4.35e-4) +
+NaN (NaN) -
+8.4001e-1 (7.93e-4)
+MW5
+1.7725e-1 (9.80e-2) -
+2.5212e-1 (6.86e-2) -
+3.1449e-1 (2.61e-3) -
+3.2205e-1 (5.38e-3) -
+NaN (NaN) -
+3.2214e-1 (6.45e-3)
+MW6
+2.8267e-1 (4.89e-2) -
+2.5928e-1 (6.08e-2) -
+3.1251e-1 (9.93e-3) ≈
+2.9009e-1 (5.16e-2) -
+1.2194e-2 (2.75e-2) -
+3.0911e-1 (1.20e-2)
+MW7
+3.7706e-1 (6.78e-2) ≈
+4.0647e-1 (2.09e-3) -
+4.0868e-1 (1.03e-3) -
+4.1205e-1 (5.95e-4) +
+1.9015e-1 (7.70e-2) -
+4.1019e-1 (9.75e-4)
+MW8
+4.9733e-1 (2.20e-2) -
+4.7275e-1 (5.64e-2) -
+5.2198e-1 (1.16e-2) -
+5.2798e-1 (3.48e-2) -
+4.6501e-2 (7.81e-2) -
+5.3338e-1 (1.72e-2)
+MW9
+2.6792e-1 (1.71e-1) ≈
+3.4455e-1 (1.00e-1) -
+3.9100e-1 (2.43e-3) +
+3.7160e-1 (1.01e-1) -
+NaN (NaN) -
+3.8287e-1 (4.60e-3)
+MW10
+3.1175e-1 (1.18e-1) -
+3.5982e-1 (7.57e-2) -
+4.3564e-1 (1.30e-2) ≈
+4.1378e-1 (1.88e-2) -
+NaN (NaN) -
+4.2764e-1 (1.94e-2)
+MW11
+3.2816e-1 (8.07e-2) -
+4.4157e-1 (9.48e-3) -
+4.4127e-1 (1.39e-3) -
+4.4609e-1 (2.03e-3) -
+2.2321e-1 (4.19e-2) -
+4.4746e-1 (2.05e-4)
+MW12
+5.4172e-1 (1.81e-1) -
+5.8181e-1 (1.06e-1) +
+6.0052e-1 (7.80e-4) +
+5.8415e-1 (1.10e-1) ≈
+NaN (NaN) -
+5.4377e-1 (1.82e-1)
+MW13
+4.0153e-1 (5.63e-2) -
+4.1137e-1 (4.30e-2) -
+4.6130e-1 (1.23e-2) ≈
+4.3974e-1 (2.53e-2) -
+2.3054e-1 (1.15e-1) -
+4.5371e-1 (1.66e-2)
+MW14
+4.5123e-1 (5.66e-3) -
+4.2008e-1 (2.54e-2) -
+4.6575e-1 (3.90e-3) ≈
+4.7246e-1 (1.53e-3) +
+3.4138e-1 (1.53e-1) -
+4.6217e-1 (1.49e-2)
+CTP1
+3.5920e-1 (1.97e-2) -
+3.7510e-1 (5.31e-3) -
+3.7588e-1 (1.03e-2) -
+3.8065e-1 (3.93e-4) -
+3.8036e-1 (1.15e-4) -
+3.8106e-1 (1.09e-4)
+CTP2
+4.3083e-1 (1.66e-3) -
+4.2928e-1 (7.86e-4) -
+3.9367e-1 (8.22e-3) -
+4.3073e-1 (2.94e-4) -
+4.2689e-1 (1.40e-3) -
+4.3128e-1 (2.45e-4)
+CTP3
+3.7267e-1 (5.45e-2) -
+3.8376e-1 (4.03e-3) -
+3.5588e-1 (7.25e-3) -
+3.9219e-1 (2.18e-3) -
+3.8119e-1 (6.97e-3) -
+4.0507e-1 (1.75e-3)
+CTP4
+2.2838e-1 (6.88e-2) -
+2.5406e-1 (1.86e-2) -
+2.4611e-1 (2.00e-2) -
+2.7265e-1 (2.41e-2) -
+2.2246e-1 (2.88e-2) -
+3.3430e-1 (1.20e-2)
+CTP5
+3.9329e-1 (2.59e-2) -
+3.8822e-1 (4.53e-3) -
+3.5629e-1 (9.01e-3) -
+3.9643e-1 (2.52e-3) -
+3.8643e-1 (5.29e-3) -
+4.0665e-1 (1.86e-3)
+CTP6
+4.6359e-1 (3.75e-4) -
+4.6198e-1 (6.48e-4) -
+4.4896e-1 (2.68e-3) -
+4.6381e-1 (3.03e-4) -
+4.6034e-1 (1.83e-3) -
+4.6468e-1 (2.22e-4)
+CTP7
+5.6701e-1 (2.23e-3) -
+5.6676e-1 (9.22e-4) -
+5.6637e-1 (4.11e-4) -
+5.6745e-1 (1.69e-4) ≈
+5.6692e-1 (1.86e-4) -
+5.6721e-1 (1.62e-3)
+CTP8
+3.4937e-1 (2.45e-2) -
+3.6598e-1 (2.71e-3) -
+3.5213e-1 (3.79e-3) -
+3.6932e-1 (8.68e-4) -
+3.5503e-1 (2.42e-2) -
+3.7069e-1 (4.25e-4)
+LIRCMOP1
+1.3114e-1 (2.17e-2) -
+1.9042e-1 (1.06e-2) -
+1.0593e-1 (3.85e-2) -
+1.4954e-1 (1.82e-2) -
+1.8833e-1 (4.61e-2) -
+2.2304e-1 (6.36e-3)
+LIRCMOP2
+2.5580e-1 (2.95e-2) -
+3.2332e-1 (1.35e-2) -
+2.9229e-1 (3.67e-2) -
+2.9325e-1 (2.05e-2) -
+3.2282e-1 (2.82e-2) -
+3.4702e-1 (3.44e-3)
+LIRCMOP3
+1.1697e-1 (2.61e-2) -
+1.4007e-1 (1.86e-2) -
+9.9083e-2 (2.08e-2) -
+1.2942e-1 (2.47e-2) -
+9.1646e-2 (1.42e-2) -
+1.9947e-1 (4.24e-3)
+LIRCMOP4
+2.1773e-1 (2.72e-2) -
+2.4241e-1 (3.29e-2) -
+1.8974e-1 (4.86e-2) -
+2.3242e-1 (3.14e-2) -
+1.8379e-1 (2.31e-2) -
+3.0693e-1 (3.76e-3)
+LIRCMOP5
+8.9983e-2 (1.05e-1) -
+2.4475e-1 (1.22e-2) -
+2.4215e-1 (1.32e-2) -
+2.8700e-1 (5.09e-3) ≈
+2.6214e-1 (8.89e-2) -
+2.8657e-1 (5.90e-3)
+LIRCMOP6
+8.6641e-2 (5.68e-2) -
+1.7269e-1 (1.24e-2) -
+1.4582e-1 (3.86e-2) -
+1.9402e-1 (3.37e-3) ≈
+1.9677e-1 (1.59e-4) +
+1.8377e-1 (3.56e-2)
+LIRCMOP7
+2.8752e-1 (6.87e-3) ≈
+2.6957e-1 (2.12e-2) -
+2.8582e-1 (3.18e-3) -
+2.9114e-1 (4.01e-3) ≈
+2.9389e-1 (1.55e-4) +
+2.8957e-1 (4.05e-3)
+LIRCMOP8
+2.8236e-1 (1.47e-2) -
+2.6950e-1 (1.80e-2) -
+2.8424e-1 (1.40e-2) -
+2.9321e-1 (3.40e-3) ≈
+2.9387e-1 (2.04e-4) ≈
+2.9235e-1 (3.25e-3)
+LIRCMOP9
+3.5772e-1 (8.15e-2) -
+5.2821e-1 (2.63e-2) ≈
+4.9955e-1 (2.81e-2) -
+5.5712e-1 (8.54e-3) +
+4.9582e-1 (5.19e-2) -
+5.3708e-1 (2.00e-2)
+LIRCMOP10
+5.1193e-1 (6.34e-2) -
+7.0668e-1 (5.59e-3) +
+6.7264e-1 (2.74e-2) -
+7.0659e-1 (3.77e-4) +
+7.0755e-1 (1.24e-4) +
+7.0630e-1 (4.01e-4)
+LIRCMOP11
+5.3737e-1 (1.32e-1) -
+6.9062e-1 (4.88e-3) -
+6.4145e-1 (1.47e-2) -
+6.9392e-1 (7.61e-5) ≈
+6.1686e-1 (4.29e-2) -
+6.9393e-1 (5.71e-5)
+LIRCMOP12
+5.5161e-1 (4.68e-2) -
+6.1582e-1 (9.12e-3) ≈
+6.0522e-1 (7.22e-3) -
+6.1952e-1 (1.29e-3) ≈
+6.0839e-1 (2.46e-2) -
+6.1811e-1 (3.04e-3)
+LIRCMOP13
+4.7950e-1 (1.63e-1) -
+5.3426e-1 (4.14e-3) -
+5.4704e-1 (3.37e-4) -
+5.5421e-1 (1.49e-3) -
+5.1626e-1 (3.56e-3) -
+5.5578e-1 (1.23e-3)
+LIRCMOP14
+4.9121e-1 (1.36e-1) -
+5.3744e-1 (4.86e-3) -
+5.4656e-1 (7.33e-4) -
+5.5357e-1 (1.22e-3) -
+5.2944e-1 (4.28e-3) -
+5.5604e-1 (1.16e-3)
++/-/≈
+0/33/3
+3/31/2
+3/28/5
+7/20/9
+3/32/1
+B. Comparison Results
+First, we compared the performance of ATM-R with that of
+the other five CMOEAs. The mean IGD values and standard
+deviations of 36 test functions over 30 independent runs are
+summarized in Table I. The results in terms of the HV value
+are collected in Table II. In each table, “std” represents the
+standard deviation of the IGD/HV values over 30 independent
+runs. “NaN” denotes that a CMOEA cannot find a feasible
+solution of a test function over all 30 independent runs.
+For a given test function, ATM-R was compared with each
+competitor by the Friedman test with Bonferroni correction at
+a significance level of 0.05. For convenience, “+”, “-”, and
+“≈” are used to represent that a competitor is better than,
+worse than, and similar to ATM-R, respectively. In addition,
+for each test function, the best result among the six CMOEAs
+is highlighted in gray. To visualize the results, we plotted the
+CPFs obtained by the six CMOEAs in a typical run on three
+representative CMOPs in Figs. 6-8. A typical run denotes the
+one producing the median IGD value among all runs.
+1) General Performance: In general, as shown in Table I
+and Table II, ATM-R obtained the best results of most of the
+test functions in terms of both the IGD and the HV values.
+Additionally, it performed significantly better than the other
+five competitors on most of the test functions. The multi-
+problem Friedman’s test [?] was implemented to compare
+these six CMOEAs simultaneously. As shown in Fig. 5, ATM-
+R achieved the lowest ranking value among six CMOEAs.
+Furthermore, the results in Figs. 6-8 show that ATM-R can
+obtain a set of well-converged and well-distributed solutions.
+A more detailed discussion on different test suites is given
+next.
+2) Performance on MW Test Suite: In terms of the IGD
+value, ATM-R performed better than NSGAII-CDP, PPS,
+CTAEA, CCMO, and ToP on 14, 13, six, six, and 14 test
+functions, respectively. Inversely, these peer CMOEAs were
+better than ATM-R on zero, one, two, three, and zero test
+functions, respectively. ATM-R obtained the best results of
+four test functions on which it performed better than the other
+five competitors. CCMO obtained the best results of four test
+functions, on one of which it performed similarly to ATM-
+R. Although CTAEA obtained the best results of six test
+functions, it performed similarly to ATM-R on five of these
+test functions.
+In terms of the HV value, ATM-R performed better than
+NSGAII-CDP, PPS, CTAEA, CCMO, and ToP on 12, 12, six,
+eight, and 14 test functions, respectively. On the contrary, these
+peer CMOEAs revealed better results than ATM-R on zero,
+two, three, five, and zero test functions, respectively. ATM-
+R obtained the best results of three test functions on which
+it performed better than the other five competitors. CCMO
+
+11
+0.5
+1
+1.5
+1
+2
+3
+NSGAII-CDP on MW13
+0.5
+1
+1.5
+1
+2
+3
+PPS on MW13
+0.5
+1
+1.5
+1
+2
+3
+CTAEA on MW13
+0.5
+1
+1.5
+1
+2
+3
+CCMO on MW13
+2
+3
+4
+5
+1
+2
+3
+ToP on MW13
+0.5
+1
+1.5
+1
+2
+3
+ATM-R on MW13
+Fig. 6. The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on MW13.
+0
+0.2
+0.4
+0.6
+0.8
+0.6
+0.8
+1
+1.2
+NSGAII-CDP on CTP4
+0
+0.2
+0.4
+0.6
+0.8
+0.6
+0.8
+1
+1.2
+PPS on CTP4
+0
+0.5
+1
+5
+10
+15
+CTAEA on CTP4
+0
+0.2
+0.4
+0.6
+0.8
+0.6
+0.8
+1
+1.2
+CCMO on CTP4
+0
+0.2
+0.4
+0.6
+0.8
+0.6
+0.8
+1
+1.2
+ToP on CTP4
+0
+0.2
+0.4
+0.6
+0.8
+0.4
+0.6
+0.8
+1
+ATM-R on CTP4
+Fig. 7. The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on CTP4.
+0
+0
+0.5
+0.5
+0.5
+NSGAII-CDP on LIRCMOP14
+1
+1
+1
+1.5
+1.5
+1.5
+0
+0
+0.5
+0.5
+0.5
+PPS on LIRCMOP14
+1
+1
+1
+1.5
+1.5
+1.5
+0
+0
+0.5
+0.5
+0.5
+CTAEA on LIRCMOP14
+1
+1
+1
+1.5
+1.5
+1.5
+0
+0
+0.5
+0.5
+0.5
+1
+CCMO on LIRCMOP14
+1.5
+1
+1
+1.5
+1.5
+0
+0
+0.5
+0.5
+0.5
+ToP on LIRCMOP14
+1
+1
+1
+1.5
+1.5
+1.5
+0
+0
+0.5
+0.5
+0.5
+ATM-R on LIRCMOP14
+1
+1
+1
+1.5
+1.5
+1.5
+Fig. 8. The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on LIRCMOP14.
+obtained the best results of four test functions. Although
+CTAEA obtained the best results of seven test functions, it
+performed similarly to ATM-R on four of these test functions.
+Furthermore, as shown in Fig 6, ATM-R obtained a set of
+well-converged and well-distributed feasible solutions that is
+close to the CPF. However, ToP failed to converge to the CPF.
+NSGAII-CDP, PPS, CTAEA, and CCMO lost some parts of
+the CPF. The results reflect that ATM-R performs better than
+the other five competitors on the MW test suite.
+3) Performance on CTP Test Suite: For the CTP test suite,
+ATM-R performed better than the other five competitors on
+most of the test functions in terms of both the IGD and HV
+values. Additionally, it obtained the best IGD/HV values on
+most of the test functions.
+For CTP1, some parts of the CPF come from the un-
+constrained Pareto front. For CTP6, the objective space has
+infeasible holes of differing widths toward the Pareto-optimal
+regions. For CTP2-CTP5, CTP7, and CTP8, the CPFs are
+divided into several disconnected segments. To solve these
+problems effectively, infeasibility information should be used
+carefully. Thus, NSGAII-CDP and ToP, which only consider
+constraints in the infeasible phase, performed worse than
+ATM-R. As stated in [4], due to the complex (i.e., disconnected
+and discrete) characteristics of the CPFs, the CMOEAs using
+reference points or vectors would have inferior performance.
+Therefore, CTAEA performed worse than ATM-R. PPS puts
+emphasis on objectives in the early stage, while CCMO adopts
+a specific population to make use of infeasibility information.
+Compared with the MW test suite, the test functions in the
+CTP test suite have larger feasibility ratios. Thus, too much
+infeasibility information would impair the performance of a
+CMOEA. This may be why PPS and CCMO performed worse
+than ATM-R.
+Furthermore, as shown in Fig. 7, ATM-R can converge to
+the CPF of CTP4 more quickly than the other five competitors.
+Additionally, it can cover more parts of the CPF than the other
+five competitors. The results reflect that ATM-R performs
+better than the other five competitors on the CTP test suite.
+4) Performance on LIRCMOP Test Suite: For the LIRC-
+MOP test suite, ATM-R obtained the best results on half of the
+test functions in terms of both the IGD and HV values. Similar
+to the CTP test suite, the test functions in the LIRCMOP
+test suite have infeasible holes in the objective space and the
+CPFs of some test functions are disconnected. To solve these
+test functions effectively, infeasibility information should be
+used carefully. NSGAII-CDP and ToP performed worse than
+ATM-R on most of the test functions because they ignore
+the infeasibility information to a great extent. ToP performed
+better than ATM-R on four and three test functions in terms
+of the IGD and HV values, respectively. This is attributed to
+the powerful genetic operator (i.e., differential evolution) used
+in ToP.
+Among the infeasibility-assisted CMOEAs, PPS motivates
+the population toward the unconstrained Pareto front in the
+early stage. In CTAEA and CCMO, an additional popula-
+tion is employed to approach the unconstrained Pareto front.
+Thus, these three CMOEAs will fail to solve a CMOP (i.e.,
+LIRCMOP1-LIRCMOP4) in which the CPF is far away from
+
+12
+the unconstrained Pareto front. Regarding LIRCMOP5 and
+LIRCMOP6, the CPFs are the same as the unconstrained
+Pareto fronts. Regarding LIRCMOP7 and LIRCMOP8, the
+CPFs are near the unconstrained Pareto fronts. For these four
+test functions, an infeasibility-assisted CMOEA can approach
+the CPF easily; thus, uniformity is the key factor affecting its
+performance. ATM-R and CCMO performed better than PPS
+and CTAEA on these test functions because they can preserve
+diversity more effectively. For LIRCMOP9-LIRCMOP12, the
+CPFs are divided into several disconnected segments. To solve
+these test functions effectively, diversity should be maintained
+carefully. Due to the weak cooperation of two populations,
+CCMO can maintain diversity effectively during the evolution-
+ary process. Thus, it performed better than ATM-R on these
+four test functions. For the two three-objective test functions
+(i.e., LIRCMOP13-LIRCMOP14), ATM-R performed better
+than the other five competitors in terms of both the IGD
+and HV values. It implies that ATM-R can achieve a bet-
+ter tradeoff among feasibility, diversity, and convergence for
+three-objective CMOPs.
+Furthermore, as shown in Fig. 8, all six CMOEAs can
+converge to the constrained Pareto front successfully. ATM-
+R performed better than the other five competitors in terms of
+the uniformity since it can achieve a better tradeoff among fea-
+sibility, diversity, and convergence for three-objective CMOPs.
+The results reflect that ATM-R performs better than the other
+five competitors on the LIRCMOP test suite.
+In summary, the extensive experiments on 36 test func-
+tions with various challenging characteristics demonstrate that
+ATM-R is able to solve complex CMOPs successfully.
+V. FURTHER ANALYSES
+A. Advantages of ATM-R over ATM
+As discussed in Remark 1, it is not effective to extend
+ATM [40] to solve CMOPs straightforwardly. In this subsec-
+tion, the advantages of ATM-R over ATM were demonstrated
+through experiments. The comparison results on 36 test func-
+tions in terms of the IGD and the HV values are summarized in
+Table III, where “+”, “-”, and “≈” denote that ATM performs
+better than, worse than, and similarly to ATM-R in terms of the
+IGD/HV value, respectively. As shown in Table III, ATM-R
+performed better than ATM on these three test suites in terms
+of both the IGD and the HV values. Specifically, in terms of
+the IGD value, ATM-R was better than ATM on 9, 5, and 12
+test functions of the MW, the CTP, and the LIRCMOP test
+suites, respectively. In contrast, ATM was better than ATM-
+R on no more than three test functions of these test suites.
+With regard to the HV value, ATM-R performed better than
+ATM on 8, 6, and 12 test functions, respectively. Inversely,
+ATM outperformed ATM-R on no more than 4 test functions
+of these test suites. In summary, the experimental results on
+these test functions with various characteristics demonstrate
+that ATM-R has an edge over ATM.
+B. Effectiveness of the Infeasible Phase
+To validate the effectiveness of the update mechanism in the
+infeasible phase, we implemented three variants (denoted as
+TABLE III
+RESULTS OF ATM VS ATM-R ON 36 TEST FUNCTIONS.
+Test Functions
+IGD
++/-/≈
+HV
++/-/≈
+MW1-MW14
+2/9/3
+4/8/2
+CTP1-CTP8
+3/5/0
+2/6/0
+LIRCMOP1-LIRCMOP14
+1/12/1
+1/12/1
+TABLE IV
+RESULTS OF ATM-RICDP VS ATM-R, ATM-RIOBJ VS ATM-R, AND
+ATM-RIDIV VS ATM-R ON 36 TEST FUNCTIONS.
+Algorithms
+IGD
++/-/≈
+HV
++/-/≈
+ATM-RICDP vs ATM-R
+2/19/15
+2/23/11
+ATM-RIobj vs ATM-R
+5/11/20
+7/11/18
+ATM-RIdiv vs ATM-R
+8/14/14
+10/14/12
+ATM-RICDP, ATM-RIobj, and ATM-RIdiv) by using different
+update mechanisms in this phase. Specifically, in ATM-RICDP,
+the CDP is used for solution selection. In ATM-RIobj, the
+solutions are selected based on Pareto dominance regardless
+of constraints. In ATM-RIdiv, the diversity is quantified and
+used to select promising solutions. By comparing ATM-R
+with each of ATM-RICDP, ATM-RIobj, and ATM-RIdiv, the
+effectiveness of the update mechanism in the infeasible phase
+can be validated. The comparison results on 36 test functions
+mentioned above are summarized in Table IV, where “+”, “-”,
+and “≈” denote that a competitor performs better than, worse
+than, and similarly to ATM-R in terms of the IGD/HV value,
+respectively.
+As shown in Table IV, ATM-R performed better than ATM-
+RICDP, ATM-RIobj, and ATM-RIdiv in terms of both the
+IGD and the HV values. Specifically, with regard to the IGD
+value, ATM-R was better than ATM-RICDP, ATM-RIobj, and
+ATM-RIdiv on 19, 11, and 14 test functions, respectively.
+Inversely, ATM-RICDP, ATM-RIobj, and ATM-RIdiv outper-
+formed ATM-R on 2, 5, and 8 test functions, respectively. In
+terms of the HV value, ATM-R performed better than ATM-
+RICDP, ATM-RIobj, and ATM-RIdiv on 23, 11, and 14 test
+functions, respectively. In contrast, ATM-RICDP, ATM-RIobj,
+and ATM-RIdiv outperformed ATM-R on 2, 7, and 4 test
+functions, respectively. The experimental results show that the
+update mechanism in the infeasible phase is critical to ATM-R.
+C. Effectiveness of the Semi-feasible Phase
+To validate the effectiveness of the update mechanism
+in the semi-feasible phase, we implemented three variants
+(i.e., ATM-RSCDP, ATM-RSobj, and ATM-RSdiv) by using
+different update mechanisms in this phase. Specifically, in
+ATM-RSCDP, ATM-RSobj, and ATM-RSdiv, the CDP, the
+Pareto dominance, and the diversity are used for solution
+selection, respectively. By comparing ATM-R with each of
+ATM-RICDP, ATM-RIobj, and ATM-RIdiv, the effectiveness
+of the update mechanism in the semi-feasible phase can
+be validated. Specifically, the comparison results on 36 test
+functions mentioned above are summarized in Table V, where
+
+13
+TABLE V
+RESULTS OF ATM-RSCDP VS ATM-R, ATM-RSOBJ VS ATM-R, AND
+ATM-RSDIV VS ATM-R ON 36 TEST FUNCTIONS.
+Algorithms
+IGD
++/-/≈
+HV
++/-/≈
+ATM-RSCDP vs ATM-R
+3/24/9
+5/23/8
+ATM-RSobj vs ATM-R
+3/32/1
+2/31/3
+ATM-RSdiv vs ATM-R
+0/36/0
+0/36/0
+“+”, “-”, and “≈” denote that a competitor performs better
+than, worse than, and similarly to ATM-R in terms of the
+IGD/HV value, respectively.
+As shown in Table V, ATM-R performed better than ATM-
+RSCDP, ATM-RSobj, and ATM-RSdiv in terms of both the
+IGD and the HV values. With regard to the IGD value,
+ATM-R was better than ATM-RSCDP, ATM-RSobj, and ATM-
+RSdiv on 24, 32, and 36 test functions, respectively. In
+contrast, ATM-RSCDP, ATM-RSobj, and ATM-RSdiv out-
+performed ATM-R on no more than three test functions. In
+terms of the HV value, ATM-R performed better than ATM-
+RSCDP, ATM-RSobj, and ATM-RSdiv on 23, 31, and 36 test
+functions, respectively. Inversely, ATM-RSCDP, ATM-RSobj,
+and ATM-RSdiv outperformed ATM-R on no more than five
+test functions. The experimental results show that the update
+mechanism in the semi-feasible phase is critical to ATM-R.
+D. Effectiveness of the Multiphase Mating Selection Strategy
+In order to verify the effectiveness of the multiphase mating
+selection strategy, we implemented three variants (i.e., ATM-
+RMCDP, ATM-RMobj, and ATM-RMdiv) by using different
+selection methods to select mating solutions. Specifically, in
+ATM-RMCDP, ATM-RMobj, and ATM-RMdiv, the CDP, the
+Pareto dominance, and the diversity are used for solution selec-
+tion, respectively. By comparing ATM-R with each of ATM-
+RMCDP, ATM-RMobj, and ATM-RMdiv, the effectiveness of
+the multiphase mating selection strategy can be demonstrated.
+Specifically, the comparison results on 36 test functions men-
+tioned above are summarized in Table VI, where “+”, “-”,
+and “≈” denote that a competitor performs better than, worse
+than, and similarly to ATM-R in terms of the IGD/HV value,
+respectively.
+As shown in Table VI, ATM-R performed better than ATM-
+RMCDP, ATM-RMobj, and ATM-RMdiv in terms of both
+the IGD and the HV values. With regard to the IGD value,
+ATM-R was better than ATM-RMCDP, ATM-RMobj, and
+ATM-RMdiv on 19, 10, and 9 test functions, respectively. In
+contrast, ATM-RMCDP, ATM-RMobj, and ATM-RMdiv out-
+performed ATM-R on 5, 7, and 5 test functions, respectively. In
+terms of the HV value, ATM-R performed better than ATM-
+RMCDP, ATM-RMobj, and ATM-RMdiv on 22, 13, and 8
+test functions, respectively. Inversely, ATM-RMCDP, ATM-
+RMobj, and ATM-RMdiv outperformed ATM-R on 3, 7, and
+4 test functions, respectively. The experimental results show
+that the multiphase selection strategy is critical to ATM-R.
+TABLE VI
+RESULTS OF ATM-RMCDP VS ATM-R, ATM-RMOBJ VS ATM-R, AND
+ATM-RMDIV VS ATM-R ON 36 TEST FUNCTIONS.
+Algorithms
+IGD
++/-/≈
+HV
++/-/≈
+ATM-RMCDP vs ATM-R
+5/19/12
+3/22/11
+ATM-RMobj vs ATM-R
+7/10/19
+7/13/16
+ATM-RMdiv vs ATM-R
+5/9/22
+4/8/24
+VI. CONCLUSIONS
+This paper has analyzed the key task of constrained multi-
+objective optimization in depth and decomposed it into three
+subtasks explicitly for the first time. To accomplish these
+three subtasks in different evolutionary phases, an adaptive
+tradeoff model with reference points (ATM-R) was designed.
+Specifically, ATM-R takes advantage of infeasible solutions
+to achieve different tradeoffs in these three subtasks. In the
+infeasible phase, ATM-R distinguishes and uses infeasible
+solutions with good diversity to enhance the diversity loss
+due to its pursuit of feasibility. Thus, the population can
+move toward feasible regions from diverse search directions. In
+the semi-feasible phase, ATM-R leverages infeasible solutions
+with good diversity/objective function values to promote the
+transition from “the tradeoff between feasibility and diversity”
+to “the tradeoff between convergence and diversity”. Thus, the
+population can locate enough feasible solutions and approach
+the CPF quickly. In the feasible phase, ATM-R employs
+NSGAII to seek a set of well-converged and well-distributed
+solutions close to the constrained Pareto front. Moreover, a
+multiphase mating selection strategy is proposed to select
+appropriate mating parents adaptively. Experimental studies on
+a wide range of CMOPs demonstrate that:
+• ATM-R achieves better or at least highly competitive
+performance against other representative CMOEAs.
+• ATM-R has a significant advantage over ATM for con-
+strained multiobjective optimization.
+• The update mechanisms in the infeasible phase and the
+semi-feasible phase are both critical to the performance
+of ATM-R.
+• The multiphase mating selection strategy is significant to
+the performance of ATM-R.
+In the future, we will extend ATM-R to solve constrained
+expensive multiobjective optimization problems.
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+

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+X-ray properties of high-redshift Radio Loud and Radio Quiet
+Quasars observed by Chandra
+F. Shabana,∗, A. Siemiginowskab, R.M. Suleimanb, M.S. El-Nawawya, A. Alia
+aAstronomy, Space Science and Meteorology Department, Faculty of Science, Cairo University, Giza, EGYPT
+bCenter for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA
+Abstract
+We performed a study of high redshift (z > 2) quasars, looking for the main differences be-
+tween Radio Loud Quasars (RLQ) and Radio Quiet Quasars (RQQ) in the X-ray band. Our
+sample of 472 RQQ and 81 RLQ was selected by cross-matching the SDSS DR7 quasars
+catalog with the Chandra Source Catalog. We computed the X-ray luminosity for the two
+samples and confirmed the X-ray luminosity excess of RLQ over RQQ. We fit the X-ray
+spectra assuming the absorbed power law model and obtained the photon index (Γ) values
+for all the sources in the sample. We excluded quasars with a low number of counts (< 10)
+and large uncertainty on the best-fit photon index (Γerr > 1), and obtained the mean values
+of ΓRLQ = 1.70 +0.36
+−0.33 and ΓRQQ = 2.19 +0.46
+−0.44 for the RLQ and RQQ samples, respectively,
+showing that the RLQ have flatter (harder) X-ray spectra than RQQ. The Kuiper-two test
+confirms this result with the significant difference between the RLQ and RQQ photon in-
+dex distributions (Dk = 0.37 and P-value = 10−6). We also evaluated the hardness ratio
+distributions and confirmed that the spectra of RLQ are flatter than the spectra of the RQQ.
+The RLQ’s hard-to-soft ratio distribution is skewed towards the hard X-ray band, while the
+RQQ is towards the soft X-ray band. The hard-to-medium and medium-to-soft ratios show
+no difference.
+Keywords: Radio Loud Quasars, Radio Quiet Quasars, X-ray Astrophysics, X-ray photon
+index, Hardness Ratio
+1. Introduction
+There are two main classes of quasars: the Radio Quiet Quasars (RQQ) and the Radio
+Loud Quasars (RLQ). They have been identified based on the orientation and presence of
+∗corresponding author
+Email address: fshaban@sci.cu.edu.eg (F. Shaban )
+Preprint submitted to JHEAP
+January 10, 2023
+arXiv:2301.02866v1  [astro-ph.HE]  7 Jan 2023
+
+a radio jet (Antonucci, 1993; Wilson and Colbert, 1995; Urry and Padovani, 1995). RLQ
+have optical and X-rays luminosity about three times greater than their RQQ counterparts
+(Zamorani et al., 1981; Worrall et al., 1987; Miller et al., 2010; Zhu et al., 2020). The X-
+ray radiation could be due to the Compton scattering of UV photons by energetic electrons
+or due to synchrotron radiation from highly relativistic electrons. (Mushotzky et al., 1993;
+Nowak, 1995; Turner and Miller, 2009; Worrall, 2009; Fabian, 2012).
+Majority of quasars are RQQ with the X-ray radiation attributed to a hot corona formed in
+the accretion flow (Haardt and Maraschi, 1993; Fabian et al., 2015; Zhu et al., 2020). RLQ
+are a small minority, about ≈ 10% of all quasars, and are characterized by their relativis-
+tic jets generated by an accreting supermassive black hole (SMBH) (Padovani et al., 2017;
+Blandford et al., 2019). The amount of jet radiation contributing to the X-ray spectrum in
+RLQ is still not fully understood. However, identifying the main radiation components in the
+X-ray spectrum is important to the estimates of the quasar power.
+The RLQ have flatter X-rays spectra (lower photon index value) than those of the RQQ
+(Reeves et al., 1997; Page et al., 2005; Miller et al., 2010). The quasar’s hardness ratio
+is consistent with the spectral slope (Freeman et al., 2001; Evans et al., 2010; Peca et al.,
+2021). The RLQ are divided into Core Dominant (CD) and Lobe Dominant (LD) (Haardt and
+Maraschi, 1993; Wilson and Colbert, 1995). The radio emission of CD quasars is dominated
+by the relativistic jet, while the LD quasars show significant radio emission from the large-
+scale components in comparison to the core (Falcke et al., 1995; Boroson, 2002). These two
+populations might have different X-ray radiation processes, which was noted recently by Zhu
+et al. (2020).
+During the past decades, quasar data from X-ray surveys have become available, allowing
+for statistical studies of relatively large samples. Many recent studies considered the high
+redshift quasars for survey (Kelly et al., 2007; Vito et al., 2019; Pons et al., 2020; Li et al.,
+2021). Some studies were focusing on deducing RLQ properties using correlations between
+X-ray, radio, and optical (or UV) luminosities to investigate the quasars physical model,
+Miller et al. (2010) investigate the disk-jet model, Zhu et al. (2020) deduced the disk-corona-
+jet model. Interestingly, Lusso and Risaliti (2017) were studying RQQ and showed that RQQ
+could be used as standard candles at high redshifts (z > 2), which is important for distance
+measurement and cosmological tests.
+In this research, we investigate the differences in the X-ray spectral properties (photon
+index, intrinsic absorption, hardness ratios, and X-ray luminosity) between RQQ and RLQ
+samples using the data available in the Chandra Source Catalog (CSC2) (Evans et al., 2019).
+We study quasars at a high redshift near the peak of cosmic quasar activity, at z > 2. Our
+sample contains the largest number of RLQ at high redshift observed with Chandra and
+2
+
+include faint sources with [10−15 - 10−13] ergcm−2 s−1 1 for the energy range [0.5 - 7.0]
+keV. We calculate the photon index by fitting the faint X-ray spectra, thus expanding the
+number of quasars with this parameter. The observed Chandra effective energy range is [0.5
+- 7.0] keV and corresponds to the rest frame energy greater than [1.5 - 21.0] keV at redshift
+z > 2, so at the higher redshifts we are able to study the X-ray spectra, which are the most
+sensitive to the properties of the corona and relativistic jet.
+In section 2, we briefly describe the data catalogs, the sample selection criteria, and our
+constraints. In section 3, we show the distributions of RLQ and RQQ as functions of X-
+ray parameters and illustrate the photon index calculations and constraints, furthermore, we
+analyze extreme cases for the photon index. In section 4, we discuss our results and compare
+them with previous studies, and conclude with a discussion and outlook for future work.
+2. Sample selection
+DR7+ CSC2
+(2561)
+Z > 2  
+(595)
+DR7
+105,783 
+quasars
+First_flag= 2;
+Lobe-
+Dominant
+First_flag= -1;
+Out of First 
+field
+First_flag= 0; 
+Radio Quiet
+First_flag= 1;
+Core-
+Dominant
+Radio Loud
+R ≥ 10
+Radio intermediate
+R < 10
+19
+71
+472
+10
+23
+Radio Loud
+R ≥ 10
+Figure 1: The sample selection is based on the DR7, CSC2, redshift and radio-loudness. The filters in the fourth column are showing the
+FIRST flag values (-1, 0, 1, and 2) representing quasars (not in the FIRST field, Radio Quiet, Core dominant, and Lobe dominant). The
+next column filters the radio-loudness (R) into RQQ, RLQ, and RIQ. The circles represent the quasar’s number in each category.
+1https://cxc.cfa.harvard.edu/csc/char.html
+3
+
+We study the X-ray properties of quasars using archival data from two quasar catalogs.
+We use DR7 quasars catalog (Shen et al., 2011), which contains 105,783 quasars with optical
+spectra and redshift measurements. Shen et al. (2011) quasars were selected from the SDSS
+DR7 sample compiled by Schneider et al. (2010) and all have spectroscopic redshift mea-
+surements. Schneider et al. (2010) rejected the pipeline redshift measurements for the quasar
+candidates with images exceeding the PSF size in the r-band. They provide the uncertainty
+on the redshift measurement to be +
+− 0.004.
+We use the X-ray data obtained by the Chandra X-ray Observatory (Chandra) during the
+first 15 years of the mission available in the Chandra Source Catalog release 2.0 (CSC22).
+There are more than 315,000 unique X-ray sources in the CSC2 (Evans et al., 2019). Chandra
+has a high-quality angular resolution (better than 5′′), which is important for detecting faint
+sources, at high redshift, with good source positions. We cross-matched the 105,783 DR7
+quasars with sources in CSC2, using TOPCAT (Taylor, 2017), and set a search cone radius of
+30′′, consistent with the range of the sources offset uncertainty given by Evans et al. (2019).
+We found 2,561 sources corresponding to X-ray sources in CSC2. We study the sources at
+high redshift (z > 2). After applying the (z > 2) filter, we obtained 595 out of 2,561 quasars.
+The details of our full sample selection are presented in Fig.1.
+Shen et al. (2011) matched DR7 optical quasars catalog with Faint Images of the Radio
+Sky at Twenty Centimeters (FIRST) catalog (White et al., 1997), and estimated the quasar
+radio loudness parameter (R) defining RLQ and RQQ based on the following equation
+R =
+�f6 cm
+f2500
+�
+(1)
+where f6 cm and f2500 are the fluxes density (fν) at rest-frame 6 cm and 2500 ˚A, respec-
+tively. The flux density in DR7 is determined from the FIRST integrated flux density at 20 cm
+assuming a power-law slope of αν = − 0.5. The flux density at the rest frame of 2500 ˚A is
+determined by fitting the optical spectrum with a power-law continuum (Shen et al., 2011).
+Similar to Jiang et al. (2007), Shen et al. (2011) have divided RLQ in DR7 into lobe
+dominant (LD) and core dominant (CD) with FIRST cone radius of 30′′ and 5′′, respectively.
+Shen et al. (2011) have removed the effects of galactic extinction in the SDSS spectra
+using the Schlegel et al. (1998) map, and the Milky Way extinction curve by Cardelli et al.
+(1989). Furthermore, Shen et al. (2011) shifts the spectra to the rest frame using the cataloged
+redshift as the systematic redshift (Hewett and Wild, 2010).
+We select the Radio Intermediate quasars (RIQ) to have R < 10 and RLQ with (R ≥ 10)
+(Miller et al., 2010). We applied the above selection categories to our initial sample of 595
+2https://cxc.cfa.harvard.edu/csc/
+4
+
+quasars in CSC2 and divided them into different radio-loudness categories as given in Figure
+1. Because we focus on strong differences between the RLQ and RQQ, therefore we exclude
+the intermediate sample and only include RLQ and RQQ in our analysis. Our final sample
+contains 81 RLQ and 472 RQQ.
+3. Data Analysis and Results
+We study several parameters representing the X-ray properties of the quasars in our sam-
+ples. The redshift (z), and the radio loudness (R) are provided from DR7 (Shen et al., 2011),
+while the X-ray flux (fX), the hardness ratios (HRh/m), the hydrogen column density (NH),
+and the X-ray spectral files are given in CSC2 (Evans et al., 2019). We calculate the X-ray
+luminosity (LX) and the X-ray photon index (Γ).
+In order to evaluate the difference between RLQ and RQQ samples in all X-ray param-
+eters we use the Kuiper-two sample test (Watson, 1961). The Kuiper test is a test for the
+difference between two samples based on their observed Cumulative Distribution Functions
+(CDF). It is an extension of the Kolmogorov–Smirnov test, but it is more sensitive to the
+shift between the two distributions and the difference in the tails of the distributions. The
+Kuiper test is non-parametric and does not assume any functional form of the sample’s true
+distribution and it is appropriate when true distributions are unknown. The test returns DK
+and Fk, which are the maximum difference between the two samples and the probability
+p-value of the test, respectively. The Fk < 0.05 rejects the hypothesis that the two samples
+are drawn from the same distribution, so the smaller the value the stronger the significance
+of the difference between the two samples. All the Kuiper-two test values of this study are
+given in (Table 2).
+In our figures, we use normalized density histograms because we have different samples
+size. The histograms represent the probability density function of the parameter distributions
+(Hunter, 2007), (i.e., m/M × b), where m is the number of quasars in each specific bin, M
+is the total number of quasars in the sample, and b is the bin bandwidth. So the area under
+the bins integrates into one. We apply the same number of bins to RLQ and RQQ. The RQQ
+sample appears to have a smaller bin bandwidth than the RLQ sample because the bin band-
+width is affected by the sample number in the probability density function. We also apply
+the Kernel Density Estimation (KDE) smoothing function to account for the small sample
+size and different bin sizes (Rosenblatt, 1956). The small sample size may contribute to the
+gaps within the histograms, and different binning could lead to statistical biases (Waskom,
+2021). We use the following KDE equation:
+P(x) =
+1
+M × h
+M
+�
+i=1
+k
+�x − xi
+h
+�
+(2)
+5
+
+Where M is the total number of quasars in the sample, h the Kernel bandwidth, k the
+chosen kernel weight function in our estimate (Gaussian), x is the point where to calculate
+the function, and xi is the parameter value in bin i. The seaborn package 3 for fitting KDE
+has a built-in kernel bandwidth optimal estimation using Silverman methods, which are used
+for random normally distributed samples (Silverman, 1981).
+Figure 2 shows the redshift distributions of RLQ and RQQ samples. We apply the Kuiper-
+two test which returns a small difference between the RLQ and RQQ samples with Dk =
+0.19 and Fk = 0.08. This confirms that the RLQ and RQQ samples in our studies have
+consistent redshift distributions.
+3.1. X-ray Luminosity
+We calculated the X-ray luminosity using the equation given by:
+LX = 4πdL
+2fX
+(3)
+Where LX is X-ray luminosity, dL is the distance luminosity, and fX is the X-ray flux
+in [0.5 - 7.0] keV broadband energy band. The cosmological model used in this study is the
+WMAP9 with (Ho = 69.33, Ωo = 0.287, ΩΛ = 0.712) parameters (Hinshaw et al., 2013).
+We use the WMAP9 under the astropy.cosmology package to obtain the distance lu-
+minosity (dL) (Astropy Collaboration et al., 2018). Using fX and dL, and Eq.3 we calculate
+the X-ray luminosity (Harris et al., 2020).
+Figure 2 shows the X-ray luminosity distributions of RLQ and RQQ samples. The RQQ
+KDE (blue) shows a shape consistent with a Gaussian distribution and the RLQ KDE (green)
+is skewed to the higher X-ray luminosities. The X-ray luminosity range, given in log scale,
+for RLQ is LXmin. = 44.5 and LXmax. = 47.07, while for RQQ are LXmin. = 43.68 and
+LXmax. = 46.30. The differences between minimum and maximum luminosities are similar,
+2.57 and 2.62 for RLQ and RQQ, respectively. However, the median of LX is higher in the
+RLQ sample by 0.53 compared to the RQQ’s median. This difference in the median between
+RLQ and RQQ is significant and indicates a reliable difference between the intrinsic proper-
+ties of the two samples, RLQ and RQQ. The Kuiper-two test returns a significant difference
+in X-ray luminosity distributions between RLQ and RQQ, Dk = 0.42, and Fk = 2.18×10−9.
+The Fk value validates the remarkable difference in the X-ray luminosity between the radio-
+quiet and radio-loud quasars (see Table 1).
+3https://seaborn.pydata.org/generated/seaborn.kdeplot.html
+6
+
+Figure 2: The two panels show the redshift (left), and the X-ray luminosity (right) distributions comparison between RLQ and RQQ. The
+green histogram represents all of the RLQ as a function of X-ray Luminosity, and the solid green curve represents the KDE for the RLQ.
+The blue histogram represents the RQQ as a function of the X-ray Luminosity, and the dashed-blue line represents its KDE. The
+histograms are normalized.
+3.2. The Hardness ratios
+The hardness ratio is defined as the flux ratio between two different Chandra energy
+bands. The X-ray energy bands in the CSC2 are divided into several categories 4:
+• Broad (0.5-7.0) keV
+• Hard (2.0-7.0) keV
+• Medium (1.2-2.0) keV
+• Soft (0.5-1.2) keV
+The hardness ratios for hard to medium (HRh/m), medium to soft (HRm/s), and hard to
+soft (HRh/s) 5 are given in CSC2 for each detected source. CSC2 provides f(h), f(m), and
+f(s) the X-ray fluxes in the hard, medium, and soft energy bands, respectively.
+HRh/m
+=
+f(h) − f(m)
+f(h) + f(m)
+(4)
+The HRm/s and HRh/s are defined similar to equation 4. When the hardness ratio exceeds
+zero, the flux of the higher energy band dominates over the flux of the lower energy band.
+For a general comparison between RLQ and RQQ samples, we investigate the distributions
+for HRh/m, HRm/s and HRh/s shown in Figure 3 and Figure 3. The distribution plots were
+normalized and smoothed with KDE.
+4https://cxc.cfa.harvard.edu/csc/columns/ebands.html
+5https://cxc.cfa.harvard.edu/csc/columns/spectral properties.html
+7
+
+1.75
+RQQ
+RLQ
+1.50
+1.25
+#1.00
+Densit
+0.75
+0.50
+0.25
+0.00
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+6.0
+Redshift-
+RQQ
+1.0
+RLQ
+0.8
+L
+Density
+0.6
+0.4 
+0.2
+0.0
+44
+45
+46
+47
+48
+log (x-rays_luminosity)Figure 3: The three panels show the HRh/m, HRm/s, and HRh/s distributions. The green solid line represents the RLQ, while the blue
+dashed line represents the RQQ. The red vertical lines indicate the photon index values corresponding to each hardness ratio for Γ equal
+to (3, 2, 1, 0) from left to right. The left panel shows no difference between RLQ and RQQ distributions. The middle panel shows a slight
+difference. The right panel shows a significant difference between RLQ and RQQ with a higher tendency toward soft energy in the RQQ
+sample.
+We also mark the evolution of the photon index as a function of the hardness ratios. Using
+the fake pha function in Sherpa and the standard ACIS-S response files, we fix the photon
+index (Γ = 0, 1, 2, 3) to simulate the spectrum and calculate the corresponding hardness
+ratios for each Γ. Figures 3 show that the red marks of the photon index decrease (flat
+spectrum) as the hardness ratios increase (towards the hard band). RLQ and RQQ samples
+have a similar HRh/m distributions (see Figure 3) confirmed by the Kuiper-two test Dk =
+0.16, Fk = 0.37.
+The HRm/s distribution shown in Figure 3, shows a slight shift towards the soft energy
+band for RQQ in comparison to the RLQ sample, also indicated by the Kuiper-two test
+Dk = 0.21 and Fk = 0.05.
+Finally, the HRh/s distribution shows the most significant difference between RLQ and
+RQQ samples (see Figure 3) with the Kuiper-two test results of Dk = 0.25 and Fk = 0.01,
+the test accuracy is 99.8% (see Table 2). The HRh/s distribution indicates that the X-ray
+spectra of RQQ quasars are softer than the spectra of RLQ quasars in our samples.
+We investigate the X-ray properties of the CD and LD quasars separately in our compar-
+ison to RQQ by applying the Kuiper-two test on all the parameters. We find that LD and
+CD samples are consistent in all of the X-ray physical parameters except for hardness ratios,
+HRh/s and HRh/m with Fk = 0.16 and Fk = 0.10, respectively. However, HRh/s and HRh/m
+distributions for LD sample are similar to RQQ with Fk = 0.53 and Fk = 0.58, respectively.
+On the other hand, our LD sample is small (10 LD quasars). Future studies of CD and LD
+quasars with high-quality X-ray spectra are needed to confirm and investigate these results
+further.
+8
+
+3.0
+RQQ
+RLQ
+2.5
+2.0
+isity
+1.5
+1.0
+0.5
+0.0
+1.0
+0.5
+0.0
+0.5
+1.0
+HRh/m2.5
+RQQ
+RLQ
+2.0
+1.5
+Density
+1.0
+0.5
+0.0
+-1.00
+-0.75
+-0.50
+-0.25
+0.00
+0.25
+0.50
+0.75
+HRml/sRQQ
+2.00
+RLQ
+1.75
+1.50
+ensity
+1.25
+ 1.00
+0.75
+0.50
+0.25
+0.00
+-1.0
+0.5
+0.0
+0.5
+1.0
+HRh/sTable 1: The Statistical Analysis for X-ray Parameters
+Radio Loud Quasars
+Radio Quiet Quasars
+Parameter
+max
+min
+mean
+median
+SD
+max
+min
+mean
+median
+SD
+z
+4.7
+2.0
+2.88
+2.67
+0.76
+5.42
+2.08
+2.7
+2.45
+0.75
+fX
+10
+0.05
+1.4 +0.2
+−0.2
+0.7 +0.17
+−0.14
+2
+3
+0.004
+0.4 +0.10
+−0.09
+0.2 +0.89
+−0.68
+0.4
+LX
+47.07
+44.54
+45.66 +0.08
+−0.07
+45.9 +0.09
+−0.07
+0.56
+46.3
+43.68
+45.16 +0.11
+−0.01
+45.16 +0.10
+0.00
+0.45
+HRh/s
+0.90
+-0.99
+-0.10 +0.05
+−0.35
+-0.13 +0.06
+−0.33
+0.3
+0.99
+-0.99
+-0.21 −0.05
+−0.53
+-0.28 −0.06
+−0.52
+0.40
+HRm/s
+0.99
+-0.74
+-0.21 −0.02
+−0.39
+-0.21 +0.05
+−0.38
+0.29
+0.99
+-0.99
+-0.29 −0.02
+−0.49
+-0.3 −0.08
+−0.51
+0.3
+HRh/m
+0.6
+-0.99
+0.11 +0.23
+−0.16
+0.09 +0.27
+−0.11
+0.24
+0.99
+-0.99
+0.09 +0.30
+−0.26
+0.07 +0.28
+���0.26
+0.39
+Γ
+3.4
+-0.39
+1.8 +0.38
+−0.34
+1.76 +0.35
+−0.32
+0.50
+4.8
+-0.88
+2.14 +0.0.5
+−0.44
+2.06 +0.48
+−0.43
+0.65
+fX: The given X-ray flux (ergcm−2 s−1) must be multiplied by factor of 10−13.
+LX: The X-ray luminosity (ergs−1) is given in log scale.
+9
+
+3.3. X-ray Spectral Modeling and Photon Index
+CSC2 lists the photon index calculated by fitting a power law model multiplied by the
+photoelectric absorption. However, the CSC2 pipeline restricted the model fitting to spectra
+with at least 150 net counts (after subtracting the background) and applied the spectral bin-
+ning of 20 counts per energy bin to use the χ2 fit statistics (Evans et al., 2019; McCollough
+et al., 2020). The CSC2 fitting criteria mean that the majority of quasars in our study do not
+have a photon index available in the CSC2 catalog. We only found 13 RLQ and 26 RQQ.
+On the other hand, CSC2 provides X-ray spectra and response files for all the sources
+in the catalog. We obtained these spectral files and fit the absorbed power law model to all
+the quasars in our sample. In order to fit a larger number of quasar’s spectra, we put less
+restrictive criteria. We accept measurements with total counts greater than or equal to 10
+and use the wstat-statistics appropriate for low counts data fitting (Freeman et al., 2001).
+Furthermore, we reject any calculated photon index with an error greater than or equal to
+one. With these criteria, we increased the number of sources with the calculated photon
+index, for RQQ from 26 to 455, and RLQ from 13 to 63.
+We note that some quasars have multiple observations. In RQQ, there are 85 RQQ
+quasars with 243 observations. One of these quasars has 11 observations. In the RLQ sam-
+ple, we found 5 quasars with 12 observations. We checked for the variability between the
+multiple observations and confirmed that there is no variability as the measured flux is con-
+sistent for each quasar.
+In Figure 4, the left panel shows the distribution of our calculated photon index for the
+RQQ sample containing 445 quasars and the RLQ sample containing 63 quasars. The right
+panel shows the CSC2 photon index for the RQQ sample containing 26 quasars and the RLQ
+sample containing 13 quasars. The fitted photon index has a similar distribution to that of
+CSC2. We note that a range of the photon index values in our fitting is larger than in the
+CSC2. We discuss this in Section 3.4.
+The photon index distribution in the RQQ sample shows a steeper spectrum, with the
+mean value of Γ = 2.14 +0.05
+−0.44, while RLQ shows a flatter spectrum, the mean value of Γ =
+1.8 +0.38
+−0.34 (see Table 2). In addition, the Kuiper-two test for our calculated photon index
+shows that the difference between RLQ and RQQ samples is significant with Dk = 0.37 and
+Fk = 7.30×10−6. However, the Kuiper-two test gives an insignificant difference (Dk = 0.46
+and Fk = 0.18) for the CSC2 photon index, which may be due to the small sample size (only
+13 RLQ and 26 RQQ).
+3.4. Extreme cases in RLQ and RQQ
+Figure 4 shows some extreme values of the photon index in the distributions (13 RQQ and
+3 RLQ). The three RLQ quasars belong to CD class but they show extreme soft spectrum of
+Γ > 3: i.e. Γ = 3.14 +0.58
+−0.56 , 3.00 +0.58
+−0.56 and 3.4 +0.8
+−0.7 , a total counts = 33, 22 and 13 counts, and
+10
+
+Figure 4: The left panel shows our best-fit photon index using wstat-statistics. We fit 63 RLQ and 445 RQQ. The right panel shows the
+photon index we obtain from CSC2 for 13 RLQ and 26 RQQ. The RLQ distributions are represented by the solid green histogram and
+green KDE curve. The RQQ distributions are represented by the dashed-blue histogram and KDE curve.
+the background counts of 0.87, 0.25 and 0.86 counts, at z = 2.23, 2.11 and 3.7, respectively.
+Due to the high uncertainty of Γ and the low number of counts in their spectra we were not
+able to investigate their properties in more detail. These are interesting outliers identified in
+our RLQ distribution, which need to be observed in the future.
+On the other side, we identify 13 RQQ with extremely hard spectra, Γ < 1 , with a range
+of the photon index [-0.08 - 0.97]. The total number of counts for these sources range [16 -
+827] counts with background counts in the source region [0.12 - 655.8] counts. We selected
+three quasars with a relatively good signal-to-noise for detail modeling, with a total number
+of counts 133, 88, and 156 and a small number of background counts 0.48, 0.27, and 0.88, at
+z = 2.5, 2.1, and 3.2. These are RQQ with hard spectra potentially indicating a presence of
+the intrinsic absorption resulting in ”flattening” of the intrinsically soft spectrum (Zickgraf
+et al., 1997; de Kool et al., 2002; Page et al., 2005).
+We fit these three spectra of the RQQ assuming a power law model with additional
+multiplicative absorption components (Sherpa has built-in models for the intrinsic absorp-
+tion at the quasar redshift (xszphabs), and the photoelectric Galactic absorption compo-
+nent (xsphabs)). The best-fit Γ changes from (0.80 +0.14
+−0.14, 0.82 +0.16
+−0.16, 0.93 +0.12
+−0.13) to (1.69 +0.31
+−0.31,
+1.39 +0.32
+−0.32, 1.16 +0.21
+−0.21), bringing the photon index values closer to the bulk of the distribution
+(see Figure 4). Figure 5 shows the confidence contours for the best-fit Γ and the intrinsic
+absorption NH showing a high uncertainty in both the NH and Γ values. We need higher
+quality spectra for these quasars to confirm that they are intrinsically absorbed.
+After eliminating the extreme cases, the RLQ Γmean changes from 1.8 +0.38
+−0.34 to 1.70 +0.36
+−0.33
+and from 2.14 +0.0.5
+−0.44 to 2.19 +0.46
+−0.44 for RQQ. Consequently, the Kuiper-two test value between
+RLQ and RQQ increased to Dk = 0.38 and its corresponding probability decreased to Fk =
+10−7, which confirms a strong difference between RLQ and RQQ samples. Since these
+extreme cases are a small percentage, 4% RLQ and 2% RQQ for our sample sets, they are
+not changing the primary trend of RLQ (hard spectrum) and RQQ (soft spectrum).
+11
+
+RQQ(445)
+1.2
+RLQ(63)
+1.0
+0.8
+Density
+0.6
+0.4 
+0.2
+0.0
+-2
+-1
+0
+1
+2
+3
+4
+Photonindex2.00
+RQQ_CSC(26)
+RLQ_CSC(13)
+1.75
+1.50
+ensity
+1.25
+Der
+1.00
+0.75
+0.50
+0.25
+0.00
+-2
+-1
+0
+1
+2
+3
+4
+PhotonindexTable 2: The Kuiper-two sample test between RLQ and RQQ for all the parameters of interest
+Samples
+RLQ, RQQ
+CD, RQQ
+LD, RQQ
+CD, LD
+Parameters
+Dk
+Fk
+Dk
+Fk
+Dk
+Fk
+Dk
+Fk
+z
+0.19
+0.08
+0.24
+0.02
+0.39
+0.31
+0.50
+0.09
+LX
+0.42
+2.18x10−9
+0.42
+2.41x10−8
+0.48
+0.09
+0.22
+0.99
+Γ
+0.37
+7.30x10−6
+0.39
+2.56x10−5
+0.50
+0.04
+0.24
+0.98
+HRh/s
+0.25
+0.01
+0.31
+9.80x10−4
+0.37
+0.53
+0.49
+0.16
+HRm/s
+0.21
+0.05
+0.21
+0.10
+0.52
+0.07
+0.41
+0.37
+HRh/m
+0.16
+0.37
+0.18
+0.30
+0.34
+0.58
+0.52
+0.10
+Dk: is the maximum absolute difference between the two cumulative distribution functions.
+Fk: is a probability (P-value) of the hypothesis that the two samples come from the same population
+and therefore have the same CDF.
+Bolded values: are highlighting the highest difference distributions.
+4. Discussion
+We studied a sample of high redshift (z > 2) quasars selected from CSC2. The samples
+have similar redshift distribution, but the RQQ sample has (472) a higher number of quasars
+than the RLQ sample (81). We calculate the X-ray luminosity and the X-ray photon index.
+All the properties of the two samples are summarized in Table 1. The Kuiper-two test shows
+a significant difference between RLQ and RQQ for both LX and Γ indicating that the RLQ
+spectra were flatter than the spectra of RQQ. The Kuiper-two test values for all the X-ray
+parameters are given in Table 2.
+4.1. Comparing our parameterized results with literature
+Our studies indicate that the X-ray luminosity of RLQ is significantly higher than the X-
+ray luminosity of RQQ (Dk = 0.42, Fk = 2.18 × 10−9), see Table 1) in the sample of z > 2
+quasars in CSC2. This result agrees with the earlier studies (Scott et al., 2011), and suggests
+an additional X-ray radiation component present in RLQ (Bechtold et al., 1994; Zhu et al.,
+2020).
+12
+
+(a)
+(b)
+(c)
+Figure 5: The confidence regions of Γ and NH for the three quasars: (a)2CXO J011513.1+002013, (b)2CXO J123540.1+123620,
+(c)2CXO J095858.6+020139 with a number of counts (88, 156, 133), respectively. We fit Γ and NH with a power law model multiplied
+by the intrinsic absorption at a given redshift (and including the Galactic absorption). The cross marks the best fit value and the contours
+show 1σ (purple), 2σ (blue) and 3σ (yellow) levels. The NH values is in log scale.
+This additional component may also cause RLQ’s X-ray spectra to be flatter than the
+spectra of RQQ (Reeves and Turner, 2000; Piconcelli et al., 2005). Our studies cover a
+relatively high rest frame energies, exceeding 30 keV, in this high redshift sample. These
+energies are less sensitive to the intrinsic absorption, thus the flattening of the RLQ is less
+likely related to the absorption (e.g. high absorption columns, NH > [1022 − 1026] cm−2,
+are required to modify the high energy spectra), but more likely due to the differences in the
+radiation processes between the two classes (i.e. RLQ and RQQ).
+For our sample, the column density in the direction of the source ranges within [0.57-
+12.58]×1020 cm−2, with a mean of 2.49×1020 cm−2. The nuclear obscuration is parameter-
+ized by the hydrogen column density NH and the maximum value of NH in our sample is
+1.26 × 1021 cm−2, which does not affect the AGN X-ray continuum (Hickox and Alexander,
+2018). The obscuration due to the Compton-thick absorption requires a strong reflection
+component at E > 10 keV, and a prominent Fe-Kα emission line at 6.4 keV (Ricci et al.,
+2015). In our sample spectra, we did not find any Fe-Kα emission line.
+In addition to the photon index we studied the X-ray hardness ratios for the quasars in the
+two samples. Our analysis shows, no difference in HRh/m between RLQ and RQQ samples,
+a small difference in HRm/s, and a moderate difference in HRh/s with the RLQ having a
+harder spectra (see Table 1 and Table 2).
+The soft X-ray radiation might be produced anywhere in the vicinity of a SMBH in both
+RLQ and RQQ (Shen et al., 2006). However, we find that the peaks of the HRh/s and HRm/s
+distributions (see Figures 3) are shifted towards the soft energy band in RQQ but not in RLQ.
+Our result indicates that for RQQ, the soft X-ray radiation dominates over the radiation in the
+hard and medium energy bands. However, for RLQ, the radiation in the hard and medium
+energy X-ray bands dominates over the soft energy band.
+Page et al. (2005) considered a small sample of 7 RQQ and 16 RLQ at (z > 2) observed
+with XMM-Newton. They used the broad energy band [0.3 - 10] keV. They found 9 intrinsi-
+13
+
+18
+16
+14
+Column density (Nn)
+12
+10
+8
+6
+4
+2
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Photon index ()35
+30
+Column density (Nn)
+25
+20
+15 
+10
+5
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Photon index ()35
+30
+Column density (Nn)
+25
+20
+15 
+10
+5
+0
+0.5
+1.0
+1.5 
+2.0
+2.5
+3.0
+Photon index ()Figure 6: The comparison between our calculated photon index and the CSC2 photon index for the same set of quasars (13 RLQ and 26
+RQQ). Dark blue dashed lines and dark green solid lines show our photon index, and light blue dashed lines and light green solid lines
+show the CSC2 photon index.
+cally absorbed quasars with NH between [1 - 2]×1022 cm−2 in the rest frame of the objects.
+Using the absorbed power law model, they found that RLQ have flatter spectra than the RQQ
+counterparts (RLQ ≈ 1.55 and RQQ ≈ 1.98). Some studies compare RLQ and RQQ in a
+specific part of the X-rays (hard band) to specify the corresponding mechanism (Gupta et al.,
+2018; Zhu et al., 2020). In our study, the RLQ is flatter than RQQ by 0.49 +0.10
+−0.11, which is a
+bigger difference than that found by Page et al. (2005), due to our larger sample size. We do
+not see any clear intrinsically absorbed quasars, which could be due to lower signal-to-noise
+spectra in our sample. Furthermore, the extreme cases in our samples did not show strong
+evidence for intrinsic absorption.
+4.2. Calibrating our calculated photon index with CSC2
+We compare the photon index calculated by our spectral modeling to the photon index
+given in CSC2 for the same quasars (13 RLQ and 26 RQQ). Figure 6 shows the RLQ and
+RQQ distributions for the calculated Γ and the one given in CSC2. The distributions show
+a rough agreement between the two methods, with our modeled values indicating a slightly
+wider range.
+CSC2 uses the χ2 statistics with background subtraction and binning, while we use wstat-
+statistics and no background subtraction appropriate for low counts spectra. van Dyk et al.
+(2001) have explained the χ2 statistical bias at low counts spectra, see also (Protassov et al.,
+14
+
+2.00
+-
+RQQ_CSC
+RQQ
+1.75
+RLQ CSC
+RLQ
+1.50
+1.25
+Density
+1.00
+0.75
+0.50
+0.25
+0.00
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+PhotonindexFigure 7: The two panels show the 100 trials of the Kuiper test DK (left panel) and FK (right panel) for the HRh/s parameter, between
+RLQ and RQQ. The red color for HRh/s at 2 < z < 2.5, and the blue color for HRh/s at z > 2.5. The solid vertical lines are for the
+mean and the dashed vertical lines are for the median.
+2002). Humphrey et al. (2009) found that even high counts give an inherent bias in the
+χ2 fitting. These studies show that χ2 methods should not routinely be used for fitting an
+arbitrary, parameterized model to Poisson-distributed data, irrespective of the number of
+counts (Mighell, 1999), and instead, the Cash statistic should be adopted (Humphrey et al.,
+2009). We used the wstat, which is based on the Poisson likelihood and accounts for the
+background6.
+We applied the Kuiper-two test to evaluate the difference between the two photon-index
+distributions, Γfit and ΓCSC2. The test returns high values of Fk, for RQQ Fk = 0.33 and
+Fk = 0.77 for RLQ, which implies that the distributions of Γ resulting from our modeling
+are consistent with the CSC2 distributions for these small sub-samples.
+4.3. Redshift Dependence of the Hardness Ratio
+Our results on the hardness ratio parameter HRh/s indicate that the RQQ spectra are
+softer than the spectra of RLQ (see Sec.3.2). We perform simulations to confirm that the
+effect is an inherent physical property of RQQ and is not affected by the redshift. Because
+the rest frame energy range is shifted towards the lower energy in the observed frame we
+check the distributions of the hardness ratio parameter in the two redshift ranges. There are
+261 RQQ and 33 RLQ at 2 < z < 2.5 and 211 RQQ and 48 RLQ at 2.5 < z < 5.5 redshift.
+Thus the fraction of RQQ is higher at z < 2.5 than at z > 2.5, which may bias the RQQ’s
+HRh/s parameter in the full redshift range.
+6https://cxc.harvard.edu/sherpa/ahelp/wstat.html
+15
+
+Z<2.5
+Z>2.5
+6
+2
+0
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+D_K8
+Z<2.5
+Z>2.5
+6
+5 
+ISU
+3
+2
+1
+0.2
+0.0
+0.2
+0.4
+0.6 
+0.8
+1.0
+F_KWe apply the Kuiper test to the HRh/s at 2 < z < 2.5 sample and get the Kuiper param-
+eter values of DK = 0.33, and FK = 0.04. Then, we select the same sample size of 33 RQQ
+and RLQ by randomly selecting 33 quasars from the 261 RQQ sample and using all 33 RLQ
+in this low redshift bin. In the first random selection of the 33 RQQ we get DK = 0.47, and
+Fk = 0.02. Afterward, we perform the test for the hardness ratio difference by looking at
+the distribution of the Kuiper parameters, Dk and Fk, in 100 random samples (see Fig. 7).
+We selected the 100 random samples of 33 RQQ and used the existing 33 RLQ. The median
+values for the 100 Kuiper test parameters in this step are Dk = 0.36, and FK = 0.05.
+For the z > 2.5 sample, the Kuiper test results for HRh/s difference between RLQ and
+RQQ are DK = 0.30, FK = 0.04. We then performed the same simulation steps as described
+above for the z > 2.5 sample using the 48 RLQ and a random sample of 48 RQQ selected
+from the 211 RQQ. The median value of the Kuiper test distribution was Dk = 0.32, and
+FK = 0.06 (see Fig.7).
+The simulation results show that the difference in the HRh/s parameter between RLQ
+and RQQ samples is slightly more significant at 2 < z < 2.5 than z > 2.5. According to
+Peca et al. (2021), our sample selection is the least affected by the absorption dependence
+with redshift and the Chandra detector contamination. At low redshift (z < 2), the difference
+in the flux between hard and soft bands is larger for quasars with NH < 1022 cm−2 because
+the soft X-ray emission is present in the observed energy band. At high redshift (z > 2)
+the hardness ratio of the quasars with low NH is not affected by the hard band shift to the
+lower observed energies and only the quasars with high absorption, NH > 1023 cm−2, will
+show the impact on the HRh/s parameter. We conclude that the observed difference in the
+hardness ratio between RQQ and RLQ at z > 2 is not affected by redshift.
+5. Summary and Conclusions
+We studied the X-ray properties of high redshift quasars observed by Chandra. We found
+a total of 2,561 DR7 quasars in the CSC2 database. After applying redshift and radio-
+loudness filters we obtained two samples, one with 472 RQQ and the second with 81 RLQ.
+The two samples have a similar redshift range, 2 < z < 5, with the RLQ sample being
+one of the largest samples of RLQ within that redshift range to date. Our main results are
+summarized below.
+• We found that an average X-ray luminosity of RLQ at high redshift is higher than the
+average X-ray luminosity of RQQ, consistent with the previous studies.
+• We calculated the mean photon index of ΓRLQ = 1.70 +0.36
+−0.33 and ΓRQQ = 2.19 +0.46
+−0.44
+for the RLQ and RQQ samples, respectively. This result confirms that RLQ spectra
+16
+
+are flatter than the spectra of RQQ. We identified a few extremely soft RLQ and ex-
+tremely hard RQQ, but these sources have low signal-to-noise data and require further
+observations to understand their X-ray properties.
+• We found that the LD and RQQ have similar distributions of hardness ratios, HRh/m
+and HRh/s. In comparison, LD and CD have similar photon index and X-ray luminos-
+ity distributions. However, our sample has only 10 LD quasars and more LD observa-
+tions are needed to confirm this result.
+• The peaks of HRh/s and HRm/s distributions are shifted towards negative values (soft
+energy band) in RQQ compared to RLQ, which confirms that the X-ray luminosity in
+the RQQ is dominated by soft X-rays in comparison to RLQ.
+Our study shows potential directions for further investigation. The quasars of extreme cases
+need longer observation. The CD and LD comparison needs larger samples for statistically
+meaningful results. The current samples can be extended to include quasars at higher red-
+shifts, z > 5, with the future releases of the Chandra Source Catalog. Additionally, the
+available quasar catalogs can be used to study the early universe population of quasars using
+high redshift infrared observations which will become available with the JWST (Gardner et
+al., 2006).
+Software: Sherpa (Freeman et al., 2001), Topcat (Taylor, 2017), Python packages: As-
+tropy (Astropy Collaboration et al., 2018), Seaborn (Waskom, 2021), Numpy (Harris et al.,
+2020), and Matplotlib (Hunter, 2007).
+6. Acknowledgement
+This research has made use of data obtained from the Chandra Data Archive and the
+Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the
+application packages CIAO and Sherpa. F.S. thanks CXC Helpdesk and Nick Lee for the
+support in the analysis of Chandra data. A.S. was supported by NASA contract NAS8-03060
+(Chandra X-ray Center). We are very grateful to the referee for helpful and constructive
+comments that helped to improve the paper.
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@@ -0,0 +1,795 @@
+arXiv:2301.03054v1  [cond-mat.str-el]  8 Jan 2023
+Time-crystalline spin ice and Dirac strings in a driven magnet
+Mingxi Yue1 and Zi Cai1, 2, ∗
+1Wilczek Quantum Center and Key Laboratory of Artificial Structures and Quantum Control,
+School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
+2Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
+Studies on systems far from equilibrium open up new avenues for investigating exotic phases of
+matter. A driven-dissipative frustrated spin system is examined in this study, and we suggest an
+out-of-equilibrium non-magnetic phase where the spins do not order but adhere to the ice rule in
+space and establish a long-range crystalline order in time. It is shown that this time-crystalline
+spin ice phase is distinct from the equilibrium spin ice and conventional discrete time crystal. The
+dynamics of monopoles, Dirac strings and other space-time topological defects have been examined
+in the context of this far-from-equilibrium system, and the possible experimental realization of our
+model has been discussed.
+Introduction – Spin ice (SI) is an unusual magnet that
+does not order even as the temperature tends towards
+zero[1]. Here, geometrical frustration results in ground
+states with extensive degeneracy yet local constraints
+known as ice rule.
+For example, in the rare-earth ti-
+tanates such as Dy2Ti2O7 and Ho2Ti2O7, the energy is
+minimized for those configurations satisfying two spins
+pointing in and two out in each tetrahedra of the py-
+rochlore lattice[2–4]. Despite its simplicity, the ice rule is
+responsible for a wealth of interesting phenomena includ-
+ing the zero point entropy[5, 6], fractionalization[7, 8],
+and the emergent gauge field[9, 10].
+Locally break-
+ing the ice rule produces a pair of point-like defects -
+condensed matter analogs of monopoles[11]- that can be
+separated to a large distance with a finite energy cost.
+Most studies on this topic focused on the equilibrium or
+near-equilibrium (relaxation[12, 13] or transport[14–16]
+of monopoles) properties, while the spin ice physics in
+far-from-equilibrium systems is elusive. Because the ice
+rule is rooted in the energy minimization principle, while
+non-equilibrium systems, especially driven systems, are
+usually far from ground states.
+The scope of nonequilibrium physics is considerable.
+Nevertheless, far from equilibrium less is known in gen-
+eral. However, nonequilibrium systems present fresh op-
+portunities for investigating novel phases of matters ab-
+sent in thermal equilibrium. A prototypical example is
+the time crystal phase, which spontaneously breaks the
+temporal translational symmetry[17–26]. Further incor-
+porating spatial degrees of freedom might lead to more
+complex non-equilibrium phases with intriguing space-
+time structures[27–30]. As for magnetic systems, the role
+of frustration in a nonequilibrium magnet is still unclear
+despite great efforts[31–35]. For example, one may won-
+der whether a magnet driven far from the ground state
+can host an out-of-equilibrium analog of the SI phase,
+which does not order yet still obeys the ice rule that is
+typically assumed to be held only for the ground state.
+If exists, how does such a non-equilibrium SI differ from
+its equilibrium counterpart? Is it possible to define and
+characterize “excitations” above such a non-equilibrium
+state that has already been highly excited?
+In this study, we attempt to answer these questions by
+investigating a periodically driven classical spin system in
+a checkerboard lattice. Dynamical simulations of classi-
+cal spin systems, unlike quantum many-body systems, do
+not suffer from the notorious exponential wall problem,
+thus allowing us to simulate 2D systems up to very high
+system sizes. On the other hand, it has been realized that
+certain intriguing features of non-equilibrium physics do
+not crucially depend on the quantum or classical nature
+of the systems[36], and discrete time crystal (DTC) or
+other exotic orders have been investigated in classical pe-
+riodically driven systems[35–40]. In terms of SI physics,
+typically, a periodical driving will pump energy into the
+system thus is detrimental to the SI phase[41]. Here, we
+demonstrate that the interplay between periodic driving
+and frustration can lead to a non-equilibrium phase that
+displays oscillating SI patterns in space, accompanied by
+a DTC order in time. Furthermore, the dynamics of the
+space-time defects (monopoles, Dirac string and instan-
+ton) emerged from such a time-crystalline spin ice (TC-
+SI) phase have also been discussed.
+FIG. 1:
+(Color online)(a)Schematic of a perfect spin ice
+configuration (monopole vacuum) in a checkerboard lat-
+tice(blue/red dots indicate spin up/down); (b)Flipping one
+spin creates two monopole excitations (dark ⊠) above the vac-
+uum; (c)Two monopoles are separated in space by flipping all
+the spins in the associated Dirac string (the red line).
+The model – To examine the spin ice phase, we em-
+ploy a classical transverse Ising model in a checkerboard
+
+(a)
+(b)
+c2
+-2
+0
+2
+4
+-1
+0
+1
+0.0
+0.2
+0.4
+0.6
+s
+x
+(t)
+ 
+ 
+ 
+s
+z
+(t)
+(b)
+t
+3
+t
+4
+t
+2
+ 
+(t)
+t
+1
+548
+546
+544
+542
+ 
+t
+[J
+-1
+]
+540
+0
+5
+10
+-0.5
+0.0
+0.5
+1.0
+1
+10
+0.1
+1
+ 
+ 
+S(r
+i
+)
+i+1
+(c)
+ 
+ 
+|S(r
+i
+)|
+i+1
+|S(r)|~r
+-
+FIG. 2: (Color online) (a)the snapshot of {sz
+i } at three typical
+time slices; (b)dynamics of the x and z components of spin
+on site i; (c)equal-time correlation function at a AF time slice
+t = t1 (left panel), which exhibits an algebraic decay (right
+panel). The parameters are chosen as J′ = 4J, ω = 2πJ,
+Γ = 1.5J, γ = J, D = 0.01J and L = 30 (a 10 × 10 section is
+plotted in (a)).
+lattice, whose Hamiltonian reads:
+Hice =
+�
+⊠
+�
+ij∈⊠
+Θ(t)sz
+i sz
+j − Γ
+�
+i
+sx
+i ,
+(1)
+where ⊠ indicates the plaquette in the checkerboard
+lattice with the next nearest neighboring (NNN) cou-
+pling (the grey plaquette in Fig.1 a).
+si = [sx
+i , sy
+i , sz
+i ]
+is a classical vector with a fixed length |si| = 1.
+Γ
+is the strength of a time-independent transverse field,
+and Θ(t) = J + J′ cos ωt is a periodically varying inter-
+action strength, where Θ(t) being positive/negative in-
+dicates anti-ferromagnetic(AF)/ferromagnetic(FM) cou-
+pling, whereas J′ and ω represent the amplitude and
+frequency of the driving.
+Throughout this paper, we
+fix these Hamiltonian parameters.
+However, we shall
+demonstrate in the supplementary material(SM)that the
+key results of this work do not crucially depend on this
+specific choice of parameters[42] .
+Typically, periodic driving will heat closed interact-
+ing systems towards an infinite temperature state. We
+incorporate dissipation into our model by coupling each
+spin to a thermal bath, which can be phenomenologically
+described using stochastic methods, to avoid this feature-
+less asymptotic state. In the presence of a thermal bath,
+the dynamics of spin i can be described by a stochastic
+Landau-Lifshitz-Gilbert equation[43]:
+˙si = hi(t) × si − γsi × (si × hi(t))
+(2)
+where γ is the dissipation strength, which is fixed as
+γ = J for the numerical convenience.
+Although this
+value is larger than that in conventional magnet, the
+long-time asymptotic state does not importantly depends
+on γ[42]. hi(t) = h0
+i (t) + ξi(t), where h0
+i = −∇siHice =
+[Γ, 0, −Θ(t)¯sz
+i ] is the effective magnetic field on site i
+and ¯sz
+i = �
+j sz
+j where the summation is over all the six
+neighboring spins of site i. ξi(t) is a 3D zero-mean ran-
+dom field representing thermal fluctuations. The local
+bath satisfies: ⟨ξα
+i (t)ξβ
+j (t′)⟩ξ = D2δαβδijδ(t − t′) where
+α, β = x, y, z and D is the strength of the noise. If the
+bath is in thermal equilibrium, γ and D should satisfy
+D2 = 2T γ, where T is the temperature of the bath. The
+stochastic differential equation can be numerically solved
+by the standard Heun method with a Stratonovich’s dis-
+cretization formula[44], in which we select the discrete
+time step ∆t = 10−3J−1 (the convergence with smaller
+∆t has been verified). The simulation is performed over a
+L×L checkerboard lattice with periodic boundary condi-
+tion. In our simulations, we choose random initial states
+whose effect has also been analyzed in SM[42]. In the fol-
+lowing, we will focus on the long-time asymptotic dynam-
+ics of this model. The dynamical phase diagram of this
+model is extremely rich as shown in the SM[42]. Here, we
+only consider the scenario when the system concurrently
+displays SI patterns in space and DTC order in time, as
+opposed to listing all the dynamical phases.
+Time-crystalline spin ice – We consider the case where
+Θ(t) oscillates between the AF and FM couplings (this
+condition, however, is not necessary for the TC-SI phase
+as illustrated in the SM[42]), and the spin configuration
+accordingly varies. The snapshots of {sz
+i } at three typical
+time slices have been plotted in Fig.2 (a). At a time slice
+t1 = 541.2T0 with AF coupling (T0 = 1/J is the period
+of Θ(t), the magnetization has a 0.2T0 phase lag with
+respect to Θ(t)), each sz
+i reaches its maximum (|sz
+i | =
+0.9994), and {sz
+i } obeys the ice rule (�
+ij∈⊠ sz
+i vanishes
+for all ⊠). The {sz
+i } snapshot at the next time slice t2 =
+t1 + 0.5T0 with FM coupling (Θ(t2) < 0) shows neither
+spin ice pattern, nor FM long-range order, rather, it is
+a paramagnetic phase (PM) with magnetization along
+the x-direction (see Fig.2 b).
+At the time slices t3 =
+t1+T0, the system Hamiltonian return to the original one
+(Hice(t3) = Hice(t1)), but {sz
+i } does not. Instead, all of
+them are simultaneously reversed {sz
+i (t3)} = {−sz
+i (t1)},
+thus the ice rule is still preserved.
+{sz
+i } return to its
+original values after two periods of driving at t4 = t1+2T0
+({sz
+i (t4) = sz
+i (t1)}), which indicates a spontaneous Z2
+
+(a)
+0.8
+0.6
+0.4
+0.2
+0.2
+-0.4
+0.6
+-0.8
+t,=541.2T。
+t=541.7To
+t,=542.2To3
+time translational symmetry breaking (TTSB).
+The origin of the DTC order can be understood as a
+consequence of the periodically driven interaction. For a
+pair of adjacent sites ij, if sz
+i (t) and sz
+j(t) synchronize as
+sz
+i = sz
+j ∼ cos[ω′t + φ], the instantaneous interacting en-
+ergy HI(t) ∼ Θ(t) cos[2ω′t + 2φ] with Θ(t) ∼ cos ωt can
+be expressed as HI(t) ∼ cos[δωt − 2φ] + cos[Ωt + 2φ]
+with δω = ω − 2ω′ and Ω = ω + 2ω′.
+HI(t) oscil-
+lates around zero except for the period doubling case
+(ω′ = ω/2), where H(t) ∼ cos 2φ (the fast oscillating
+term cos[2ωt + 2φ] is omitted). Therefore HI(t) becomes
+approximately time-independent and takes its minimum
+value at two degenerate points φ1 =
+π
+2 and φ2 =
+3π
+2 ,
+which is responsible for the spontaneous Z2 TTSB in the
+DTC. This intuitive picture also explains the fact only
+{sz
+i } exhibit period doubling, while {sx
+i } do not, as shown
+in Fig.2 (b).
+The equilibrium SI supports a Coulomb phase char-
+acterized by an algebraic decay of the spatial corre-
+lation function, one may query whether this property
+holds for the non-equilibrium TC-SI phase. To answer
+this question, we select an AF time slices (t = t1),
+and calculate the equal-time correlation function S(r) =
+1
+L2
+�
+i⟨sz
+i (t1)sz
+i+r(t1)⟩, where the average ⟨⟩ is performed
+over the trajectories starting from different random ini-
+tial states.
+As shown in Fig.2 (c), along the diagonal
+direction r =
+1
+√
+2(r, r) with r = |r|, S(r) decays al-
+gebraically in distance S(r) ∼ r−α, with α = 1.9(2)
+agreeing very well with the exponent predicted by the
+Coulomb phase[10] (α = d with d = 2 the dimension of
+the lattice).
+However, this agreement does not indicate that the
+asymptotic state in our model adiabatically follows the
+ground state of the Hice. First, the ice rule only hold at
+the time slices with AF coupling. For example, at a time
+slice with FM coupling (e.g. t = t2), the ground state
+of Hice(t2) is supposed to be an FM state along the z-
+direction, while the system shows a PM state in our case.
+Furthermore, the spontaneous TTSB can is forbidden in
+thermal equilibrium due to the no-go theorem[45, 46].
+Therefore, the asymptotic state in our model is a gen-
+uine non-equilibrium state with alternating SI and PM
+configurations in space and DTC order in time.
+Dynamics of monopoles after a local spin flip – In a
+conventional SI, the elementary excitations can be intro-
+duced by flipping one spin in a perfect SI configuration,
+which violates the ice rule in the two adjacent ⊠. For a
+monopole “vacuum” (a perfect SI configuration), flipping
+a spin equals to create of a pair of monopoles, which can
+be separated by properly identifying a chain of spins with
+alternating spin up and down and flipping them simul-
+taneously, as shown in Fig.1 (c). The energy required to
+separate two monopoles in a SI model with short-range
+coupling is independent of their distance, and the string
+composed of the flipped spins is a condensed matter ana-
+100
+1000
+0.0
+0.5
+1.0
+4
+5
+6
+7
+8
+9
+10
+100
+<
+>
+l
+ 
+ 
+(t
+n
+)
+t
+n
+ l=3
+ l=5
+ l=7
+(c)
+[J
+-1
+]
+FIG. 3: (Color online) The spin difference configuration {δsz
+i }
+at the time slices t = t1 (initial state) and t = tN (final
+states). At t = t1, (a) only one spin is flipped and (b) a string
+of spins are flipped (Dirac string); (c) stroboscopic dynamics
+of the excess energy ∆E(tn) at the AF time slices with tn =
+t0 + 2T0(n − 1) starting from different initial states, each of
+which contains one Dirac string with different length l. The
+inset indicates the average relaxation time ⟨τ⟩ξ as a function
+of the length of the Dirac string in the initial state. Other
+parameters are chosen the same as in Fig.2.
+log of the Dirac string[47].
+In general, the definition of “excitation” above an out-
+of-equilibrium state is elusive. Nevertheless, for the TC-
+SI phase in our model, we adopt a similar procedure of
+perturbing the state by flipping one spin, and monitor-
+ing the subsequential dynamics.
+For this purpose, we
+first choose an AF time slice t0 when all sz
+i reach their
+maximum and the corresponding spin configuration {s0
+j}
+obeys the ice rule. Then we randomly pick a site (say, site
+i), flip its spin then study the evolution from such a con-
+figuration {s1
+j} (s1
+j = s0
+j except j = i where s1
+i = −s0
+i ).
+We only focus on the stroboscopic dynamics at the AF
+time slice with tn = t0 + 2T0(n − 1).
+At the time slice tn, we defined {δsn
+j } (δsn
+j = sn
+j − s0
+j)
+to measure the change of the spin configuration with re-
+spect to the initial SI configuration before the spin flip
+{s0
+j}. At t = t1, only one spin is flipped, and thus δs1
+j = 0
+except j = i. Due to the dissipative nature of the dynam-
+ics, after sufficiently long time (tn > tN), the system will
+approach a new SI configuration, which differs from the
+initial state as shown in {δsN
+j } in Fig.3 (a). By com-
+paring the final and initial state, we can find that the
+spins which have been flipped during this process form
+a closed ring, along which the δsN
+j exhibits an alternat-
+ing + and − structure. Flipping one spin produces two
+monopoles, each of which can propagate from one ⊠ to
+another by flipping the spin between them. The motion
+of the monopoles resembles a random walk under certain
+
+(a)
+(OsN)
+(0s,)
+(SsN
+14
+constraint. If these two monopoles contact and their tra-
+jectories form a closed ring, they could annihilate with
+each other, leaving behind a new SI configuration that
+differs from the original one by flipping all of the spins
+along the closed ring that the monopoles went through.
+Topology protected relaxation dynamics – A more in-
+triguing dynamics occur if we start from an initial state
+with a pair of well-separated monopoles attached by a
+Dirac string, as shown in Fig.3 (b).
+A monopole is a
+topological fractionalized object that can not be created
+or annihilated by itself. As an alternative, monopoles can
+only be annihilated in pairs when they intersect. There-
+fore, for a configuration with only two well-separated
+monopoles, despite the dissipative nature of the dynam-
+ics, the excess energy ∆En = ⟨Hice(tn)⟩ξ − ⟨Hice(t0)⟩ξ
+(⟨⟩ξ indicates the ensemble average over the trajectories
+of the thermal noise) can be protected for a sufficiently
+long time before these two monopoles collide.
+Conse-
+quently, the relaxation is supposed to be slower from an
+initial state with a pair of monopoles with larger separa-
+tion (see Fig.3 c). The inset of Fig.3 (c) shows that the
+average relaxation time ⟨τ⟩ξ exponentially diverges with
+the length of the Dirac string l.
+-4
+0
+4
+95
+100
+105
+110
+-1
+0
+1
+ 
+ 
+(t)
+(a)
+ 
+s
+z
+i
+t
+ D=0.01J
+ D=0.1J
+[J
+-1
+]
+-phase shift
+0
+10
+20
+30
+40
+50
+60
+-1
+0
+1
+0
+20
+40
+0.4
+0.6
+0.8
+1
+|<s
+z
+i
+>
+|
+t
+ 
+ 
+<s
+z
+i
+>
+t
+(b)
+[J
+-1
+]
+FIG. 4: (Color online) (a) The dynamics of sz
+i on site i in
+a single noise trajectory with D = 0.01J and D = 0.1J, the
+former demonstrates a perfect DTC order, while in the latter,
+thermal fluctuations activate a π-phase shift; (b) The dynam-
+ics of the average ⟨sz
+i ⟩ξ starting from a perfect SI state after
+ensemble average over 103 noise trajectories. The envelope
+of ⟨sz
+i ⟩ξ exhibits an exponential decay as shown in the inset.
+Other parameters except D are chosen the same as in Fig.2.
+Instanton activated by the thermal fluctuation – Al-
+though the stroboscopic dynamics of monopoles resemble
+the relaxation dynamics in the conventional SI phase, the
+proposed TC-SI phase is distinct from the equilibrium SI
+because of the spontaneous TTSB. A natural question
+thus arises: what’s the effect of the monopoles on the
+temporal order of the TC-SI phase. The answer to this
+question is directly related to the stability of TC-SI phase
+against thermal fluctuations, which excite monopole with
+a finite density. The Coulomb phase in equilibrium SI
+does not breaks any symmetry, and is not robust at finite
+temperature. However, the TC-SI phase is characterized
+by a spontaneous Z2 TTSB, while a discrete symmetry
+breaking phase is typically assumed to be robust against
+weak thermal fluctuations in 2D systems. For example,
+in a similar model without frustration, the corresponding
+DTC phase is indeed stable at low temperature[48]. The
+impact of thermal fluctuation on the TC-SI phase will
+then be discussed.
+Unlike the conventional SI phase, once a spin in our
+TC-SI phase is suddenly flipped at a typical AF time
+slice, it does not only produce a pair of monopoles in
+space, but also results in a π−phase shift on top of the
+periodic dynamics of this flipped spin, which corresponds
+to tunneling from one “degenerate” DTC phase (φ = π
+2 )
+to the other (φ = 3π
+2 ). Such a fluctuation-activated tun-
+neling between the two Z2 symmetry breaking states (see
+Fig.4 a) resembles the instanton excitation in the field
+theory[49], and is a topological defect in the temporal do-
+main. These instanton excitations, no matter how rare
+they are, are detrimental to the DTC long-range order
+in the time domain and result in an exponential decay
+of the DTC order at any finite temperature, as shown in
+Fig.4 (b). However, the life-time of TC-SI phase can be
+extraordinarily long at a temperatures much lower than
+the activated temperature of monopoles.
+Discussion – The proposed model is classical, while
+a quantum generalization might provide a new perspec-
+tive, although it is extremely challenging, if not impos-
+sible, to simulate its real-time evolution. For a quantum
+transverse Ising model in a checkerboard lattice, quan-
+tum fluctuation lifts the extensive classical degeneracy
+and leads to ordered ground states[50]. These magnetic
+orderings could be suppressed by increasing temperature,
+which leads to non-magnetic phases that resemble the
+Coulomb phase. In terms of a quantum generalization of
+our driven-dissipative model, we hypothesize that there
+may be a regime of intermediate temperature where the
+temperature could overwhelm the quantum fluctuation
+while remaining significantly lower than the activated
+temperature of monopoles, and the quantum system may
+exhibit dynamics similar to those in our classical model.
+Experimental realizations of dynamically modulated
+interactions– One of the primary obstacles to the experi-
+mental realization of our model is that it requires a peri-
+odical driving imposed on the interaction rather than on
+the external field, which seems unrealistic for solid-state
+magnets. This dynamical modulated interaction can be
+achieved using magnetophononics, in which the electric
+
+5
+field of a laser is coupled to the optical phonon, and the
+consequent periodic atomic displacements could dynami-
+cally modulate the magnetic exchange couplings between
+the spins[51, 52].
+This proposal has been realized ex-
+perimentally in the AF semiconductor α-MnTe[53]. Al-
+though the tunable coupling regime is small and it is
+impossible to change the sign of the interaction, we show
+in the SM[42] that, for a slower driving (e.g. ω = 0.5πJ)
+the TC-SI can exist even when the coupling is always
+AF (J′ < J). This periodically modulated interaction is
+accessible in synthetic quantum systems such as trapped
+ions[54] and cavity QED systems[55]. For instance, in the
+latter, by applying a periodic driving to the cavity pho-
+tons, the magnetic interaction mediated by cavity can be
+dynamically controlled.
+Conclusion and outlook – In summary, we examined
+a driven-dissipative frustrated magnetic system, and our
+results demonstrate that the interplay between the pe-
+riodic driving and frustration can give rise to a non-
+equilibrium TC-SI phase.
+Unlike earlier studies, the
+aim of this work is to investigate SI physics in the con-
+text of far-from-equilibrium systems rather than the out-
+of-equilibrium features of a conventional SI phase.
+In
+frustrated quantum magnetism, a similar phase without
+spontaneous symmetry breaking is the quantum spin liq-
+uid.
+One thus may wonder whether it is possible to
+realize similar exotic quantum phases of matter out of
+equilibrium[56], which can simultaneously show spatial
+topological order and non-trivial temporal (long-range or
+quasi-long-range) orders. Another important direction is
+to classify the topological space-time defects that emerge
+from non-equilibrium phases of matter and identify their
+properties, which are directly relevant to the physical ob-
+servable effect of these nonequilibrium phases.
+Acknowledgments.—This work is supported by the
+National Key Research and Development Program of
+China (Grant No.
+2020YFA0309000), NSFC of China
+(Grant No.12174251), Natural Science Foundation of
+Shanghai (Grant No.22ZR142830), Shanghai Munic-
+ipal Science and Technology Major Project (Grant
+No.2019SHZDZX01).
+∗ Electronic address: zcai@sjtu.edu.cn
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+Draft version January 16, 2023
+Typeset using LATEX preprint2 style in AASTeX63
+Globular Clusters in NGC 4839 Falling into Coma: Evidence for the Second Infall?
+Seong-A Oh,1 Myung Gyoon Lee,1 and In Sung Jang2
+1Astronomy Program, Department of Physics and Astronomy, SNUARC, Seoul National University, 1 Gwanak-ro,
+Gwanak-gu, Seoul 08826, Republic of Korea
+2Department of Astronomy & Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
+ABSTRACT
+NGC 4839 is the brightest galaxy (cD) of the NGC 4839 group at R ≈ 1 Mpc in the
+south-west of the Coma cluster, which is known to be falling into Coma. However, it
+has been controversial whether it is in the first phase of infall or in the second phase of
+infall after passing the Coma center. We present a wide field study of globular clusters
+(GCs) in NGC 4839 and its environment based on Hyper Suprime-Cam gr images in
+the Subaru archive. We compare the GC system of NGC 4839 with that of NGC 4816,
+which is the brightest member (S0) of the nearby group and lies at a similar distance
+in the west from the Coma center. Interestingly the spatial distribution of the GCs
+in NGC 4839 is significantly more compact than that of the GCs in NGC 4816.
+In
+addition, the radial number density profile of the GCs in NGC 4839 shows an abrupt
+drop at RN4839 ≈ 80 kpc, while that of the GCs in NGC 4816 shows a continuous slow
+decline even in the outer region at 80 < RN4816 < 500 kpc. The effective radius of
+the NGC 4839 GC system is about three times smaller than that of the NGC 4816 GC
+system. This striking difference can be explained if NGC 4839 lost a significant fraction
+of the GCs in its outskirt when it passed through Coma. This supports strongly the
+second infall scenario where the NGC 4839 passed the Coma center about 1.6 Gyr ago,
+and began the second infall after reaching the apocenter in the south-west recently.
+1. INTRODUCTION
+1.1. The NGC 4839 Group and the Main
+Cluster in Coma
+Coma is the most massive galaxy cluster in
+the local universe.
+It is connected with fil-
+aments from neighboring galaxy clusters and
+hosts various substructures indicating that it
+is a complex merger system (Colless & Dunn
+1996; Malavasi et al. 2020; Healy et al. 2021).
+Thus, Coma is one of the best targets to study
+how large scale substructures are assembled and
+Corresponding author: Myung Gyoon Lee
+sao@astro.snu.ac.kr,mglee@astro.snu.ac.kr
+evolve, and has been a focus of many cluster
+studies in various aspects (see Biviano (1998);
+Churazov et al. (2021) and references therein).
+Two most prominent substructures in Coma are
+the main cluster core in the center and the
+NGC 4839 group in the south-west, as shown by
+galaxy number density maps (Colless & Dunn
+1996; Healy et al. 2021), X-ray images of hot
+gas (White et al. 1993; Neumann et al. 2001;
+Lyskova et al. 2019; Churazov et al. 2021), and
+radio images of synchrotron emission (Bonafede
+et al. 2021, 2022; Lal et al. 2022). The main
+cluster core hosts two giant galaxies (NGC 4874
+(cD) and NGC 4889 (D)), which are merging
+now. The NGC 4839 group is at R ≈ 1 Mpc in
+arXiv:2301.05269v1  [astro-ph.GA]  12 Jan 2023
+
+2
+Oh et al.
+the south-west of Coma, and it is much smaller
+and less massive than the main cluster core
+(Colless & Dunn 1996; Lyskova et al. 2019). The
+NGC 4839 group is considered to be falling into
+Coma and that the two systems will merge to
+form a more massive system in the future (Bi-
+viano (1998) and references therein).
+1.2. Merger Scenarios for the NGC 4839
+Group: A Pre-merger or a Post-merger?
+It is generally accepted that the NGC 4839
+group is merging with the main cluster. How-
+ever, whether it is a pre-merger where the
+NGC 4839 group is in the first phase of infall
+(Briel et al. 1992; White et al. 1993; Colless &
+Dunn 1996; Neumann et al. 2001; Akamatsu et
+al. 2013) or a post-merger (Burns et al. 1994;
+Lyskova et al. 2019; Churazov et al. 2021) has
+been controversial (Sanders et al. 2020; Healy
+et al. 2021).
+We summarize the observational features re-
+lated with the merging of the NGC 4839 group
+in the previous studies in Table 1. These fea-
+tures include several substructures seen in X-
+ray and radio images, an excess of E+A galax-
+ies in the SW region of the cluster, and sub-
+structures found in the spatial distribution and
+kinematics of galaxies. Each feature can be ex-
+plained with either the pre-merger scenario or
+the post-merger scenario.
+Recently the post-
+merger scenario, which can better explain the
+existence of X-ray/radio substructures (in par-
+ticular, bridges and streams), appears to be
+more supported (Lyskova et al. 2019; Churazov
+et al. 2021, 2022; Bonafede et al. 2021). How-
+ever, even in the recent discussions of both sce-
+narios based on various observations, Healy et
+al. (2021) state that Nevertheless, the question
+whether the NGC 4839 group is on its first in-
+fall or has already passed through the cluster,
+remains open.
+1.3. Globular Clusters as a Probe
+The halos of massive galaxies in galaxy clus-
+ters grow via numerous mergers of less massive
+galaxies and host a large number of globular
+clusters (GCs).
+Thus, GCs are an excellent
+probe for investigating the structure of the outer
+halos in massive galaxies in the local universe,
+and they provide a critical clue for revealing the
+assembly history of galaxy halos.
+In this study, we present a wide field sur-
+vey of GCs covering the NGC 4839 group
+and its environment, based on the archival
+Subaru/Hypersuprime-Cam (HSC) gr images.
+The primary goals of this study are to derive
+wide field number density maps of GCs and to
+use them to constrain the merger scenarios of
+the NGC 4839 group. We adopt the distance to
+Coma as 100 Mpc (de Grijs & Bono 2020).
+1.4. Previous Studies of NGC 4839 GCs
+The main host of the NGC 4839 group is
+NGC 4839 (MV
+= −23.1 mag, vh = 7338
+km s−1), which is an elongated cD galaxy
+(Schombert 1988; Ali et al. 2014).
+There
+are only two previous studies of the GCs in
+NGC 4839. Mar´ın-Franch & Aparicio (2002) ap-
+plied the surface brightness fluctuation (SBF)
+method to estimate indirectly the total number
+of GCs in several bright Coma galaxies from
+r-band images obtained at the 2.5m Issac New-
+ton Telescope. They found that NGC 4839 is
+the second-most GC-rich in Coma, following
+NGC 4874 in their sample.
+Later Jord´an et
+al. (2004) presented a F450W(B)/F814W(I)
+photometry of GCs in NGC 4839 based on
+HST/WFPC2 images, deriving the total num-
+ber of GCs to be Ntot(GC) = 3060±850, which
+is three times smaller than the value given by
+Mar´ın-Franch & Aparicio (2002). These previ-
+ous studies either covered only the small field
+of NGC 4839 or used only one band, so lit-
+tle is known about the GCs in the outskirt
+of NGC 4839. Other previous HST surveys of
+GCs in Coma covered mainly the main cluster
+
+Globular Clusters in NGC 4839
+3
+core, and did not cover the NGC 4839 region
+(Peng et al. 2011; Madrid et al. 2018).
+2. DATA
+Utilizing the Subaru/Hyper Suprime-Cam
+(HSC) archival gr images from the Subaru Mi-
+taka Okayama Kiso Archive system (SMOKA)
+(Aihara et al. 2019), Oh et al. (2023) provided
+a wide field survey of GCs in the entire Coma
+cluster. At the distance of Coma (100 Mpc), one
+arcsec (arcmin) corresponds to a linear scale of
+484.8 pc (29.1 kpc). Thus GCs at the distance
+of Coma appear as point sources in the HSC im-
+ages. Oh et al. (2023) obtained photometry of
+the point sources in the seven HSC fields cover-
+ing the entire Coma cluster, using DAOPHOT
+(Stetson 1987). We adopt the AB magnitudes in
+the SDSS system. The limiting magnitude with
+50% completeness of detection derived from ar-
+tificial star experiments is r ≈ 27.1 mag. De-
+tailed description of the detection and photom-
+etry of the point sources is given in Oh et al.
+(2023), of which we used the data for NGC 4839
+and its environment in this study.
+We apply
+the foreground extinction correction using the
+extinction maps for Coma given in Schlegel et
+al. (1998); Schlafly & Finkbeiner (2011).
+3. RESULTS
+3.1. NGC 4839 in Comparison with NGC 4816
+In Figure 1 we show a gray scale map of the
+r-band SDSS image of the Coma cluster region
+including the NGC 4839 group.
+The zoom-in
+images (10′ × 10′) of NGC 4839 and NGC 4816
+show that the two galaxies are similar in their
+luminosity and size. In the following analysis of
+NGC 4839 we chose NGC 4816, a nearby bright
+S0 galaxy, as a comparison galaxy.
+NGC 4839 and NGC 4816 are at similar pro-
+jected distances from the Coma center.
+The
+projected separation between the two galaxies
+in the sky is 21.′8 (0.63 Mpc at the distance of
+Coma). Healy et al. (2021) found 15 groups us-
+ing the catalog of Coma member galaxies, and
+provided the number of members and velocity
+dispersion of each group. Groups S11 and S14
+in their study correspond to the NGC 4816 and
+NGC 4839 groups, respectively. We used these
+group data in the following analysis.
+NGC 4816 is the brightest member of the S14
+group (with N(member) = 17 and σv = 521 km
+s−1) at R = 49′ in the west of Coma (see Fig. 12
+in Healy et al. (2021)). Similarly, NGC 4839 is
+the brightest cD/SA0 member of the NGC 4839
+group (the S11 group with N(member) = 24
+and σv = 462 km s−1) at R = 43′, but in the
+south-west of Coma.
+Thus both galaxies are very bright, and the
+V
+magnitude of NGC 4816 is only 0.9 mag
+fainter than that of NGC 4839.
+While the
+NGC 4839 group shows a strong X-ray emission,
+the NGC 4816 group shows little detected X-
+ray emission even in the recent X-ray images
+(Lyskova et al. 2019; Sanders et al. 2020; Mi-
+rakhor & Walker 2020; Churazov et al. 2021,
+2022).
+Table 2 lists the basic parameters of the
+NGC 4839 and NGC 4816 groups in compari-
+son with the main cluster. We calculated the
+virial mass from the velocity dispersion of the
+two groups (Healy et al. 2021) using the group
+virial mass equation: Mvir/M⊙ = 1.5×106h−1σ3
+v
+in Tully (2015) (adopting h = 0.7), as listed
+in Table 2:
+Mvir
+= 2.1 × 1014M⊙ for the
+NGC 4839 group, and Mvir = 3.0 × 1014M⊙ for
+the NGC 4816 group. We also list the mass for
+the subhalo 2 corresponding to the NGC 4816
+group (Mvir = 1.3 × 1013M⊙), and the sub-
+halo 9 corresponding to the NGC 4839 group
+(Mvir = 1.7 × 1013M⊙) derived from the weak
+lensing analysis in Okabe et al. (2014).
+Ok-
+abe et al. (2014) derived the mass within the
+truncation radius of each subhalo. The trunca-
+tion radius of the subhalo 9 is 98 kpc, which is
+much smaller than the virial radius of the typi-
+cal galaxy groups (the truncation radius of the
+subhalo 2 is not given in Okabe et al. (2014)).
+
+4
+Oh et al.
+Thus weak-lensing masses of the two groups are
+significantly smaller than the dynamical masses.
+These results show that the NGC 4839 and
+NGC 4816 groups have comparable high masses.
+This indicates that the NGC 4816 group should
+show strong X-ray emission like the NGC 4839
+group, but it is not yet detected in any previous
+X-ray observations. Not all galaxy groups are
+detected in X-ray observations. About a half
+of the nearby galaxy groups show X-ray emis-
+sion (Mulchaey 2000). It is not clear why the
+NGC 4816 group does not show any strong X-
+ray emission, unlike the NGC 4839 group. It
+may need a study to investigate this issue fur-
+ther.
+3.2. CMDs of the GCs
+In Figure 2 we plot the color-magnitude dia-
+grams (CMDs) of the point sources in the cen-
+tral regions (Rgal < 3′ (< 87 kpc)) of NGC 4839
+and NGC 4816 as well as a nearby background
+region with the same area as the galaxy region.
+We also plot the color histograms of the bright
+sources with r0 < 26.5 mag of each region. To
+show the net color histograms we display the
+contribution of the background sources in the
+galaxy regions using the background histogram.
+The color histograms of the sources in the
+two galaxies clearly show an excess (open his-
+tograms) with respect to the background re-
+gion (hatched histograms) in the color range of
+(0.0 < (g − r)0 < 1.3). In the CMDs the verti-
+cal structure seen inside the red box represents
+mainly GCs in NGC 4839 and NGC 4816. We
+select GC candidates from the entire field using
+the color-magnitude criteria marked by the red
+box (0.35 < (g − r)0 < 1.0, and 22.5 < r0 <
+26.5 mag) in the CMDs for the following anal-
+ysis.
+3.3. Spatial Distribution of the GCs
+In Figure 3 we display the spatial number
+density contour map of the selected GC can-
+didates in NGC 4839 and its environment. The
+region covers out to the virial radius of Coma
+(96′ = 2.8 Mpc), and NGC 4839 and NGC 4816
+are located approximately at the half virial ra-
+dius.
+The strongest peak of the GC number
+density is seen at the position of NGC 4874,
+which is adopted as the Coma center in this
+study. Two other strong peaks in the main clus-
+ter core are visible at the position of NGC 4889
+and IC 4051. The main cluster core shows also a
+large extended distribution of intracluster GCs,
+the details of which are presented in Oh et al.
+(2023); Lee et al. (2022). Note that in the south-
+west outskirt two more strong peaks are found
+at the positions of NGC 4839 and NGC 4816,
+similar to those of NGC 4889 and IC 4051 in the
+main cluster core.
+One striking feature seen in this figure is a
+clear difference in the spatial distribution of
+GCs between NGC 4839 and NGC 4816:
+the
+spatial extent of the GC system in NGC 4839
+is very compact and that of the GC system in
+NGC 4816 is much more extended, despite both
+galaxies showing a similarly strong peak at their
+centers.
+NGC 4839 shows a weak excess tail
+of GCs in the east. There is no corresponding
+galaxies around the center of this excess. This
+may be due to stripped GCs from NGC 4839.
+Sasaki et al. (2016) noted that the center of the
+massive subhalo 9 in Okabe et al. (2014) is 1′
+east of NGC 4839 that is located at the X-ray
+peak in the XMM-Newton and Suzaku images.
+This offset of the subhalo 9 might have also pro-
+duced the east tail of the GCs in NGC 4839.
+No bright galaxies are found in the outer
+region of NGC 4816, which might have con-
+tributed to the extended distribution of GCs.
+The strong central concentration of the GCs
+around NGC 4816 (R < 1200′′) in the radial
+number density profiles, as shown in the fol-
+lowing section, indicates that a majority of the
+GCs in this region are bound to the NGC 4816
+group. In the GC number density map of Fig-
+ure 3 there are several weak GC clumps in the
+
+Globular Clusters in NGC 4839
+5
+outskirts of the NGC 4816 group, some of which
+can be due to some non-group member galaxies,
+but their contribution to the group GC system
+is negligible.
+3.4. Radial Number Density Profiles of the
+GCs
+We derive the radial number density profiles of
+the GCs in NGC 4839 and NGC 4816. We esti-
+mate the background levels from the surround-
+ing regions (at Rgal = 9.2′ for NGC 4839 and
+Rgal = 26.4′ for NGC 4816), and subtract them
+from the original counts for the galaxy regions.
+GC colors such as (g−i) are a useful proxy for
+metallicity. The (g−r) color in this study is less
+sensitive than the (g − i) color, but is still use-
+ful. We divide the GC sample into two subsam-
+ples according to their color: blue (metal-poor)
+GCs with 0.35 < (g − i) < 0.655, and the red
+(metal-rich) GCs with 0.655 < (g − r) < 1.0, as
+described in Oh et al. (2023). We derive the ra-
+dial number density profiles of the blue GCs and
+red GCs in NGC 4839 and NGC ,4816, display-
+ing them as well as that of all GCs in Figure 4.
+This figure shows that the blue GC system is
+slightly more extended than the red GC system
+in both NGC 4839 and NGC 4816.
+In Figure 5, we compare the radial profiles of
+the GC number density and surface brightness
+of galaxy light in NGC 4839 and NG 4816 For
+comparison with galaxy light, we derive the ra-
+dial surface brightness profiles of the two galax-
+ies from the HSC r-band images. First we mask
+out several bright sources except for the two
+galaxies in the images. Then we obtain surface
+brightness profiles of the galaxies using annu-
+lar aperture photometry, and plot them in the
+same figure.
+Several interesting features are noted in Fig-
+ure 5. First, the radial number density profiles
+of the GCs in the two galaxies show a striking
+difference in the outer region, while they are
+similar in the inner region. The decline in the
+central region at Rgal < 20′′ (< 10 kpc) is due
+to incompleteness of our photometry, so we use
+only the data for the outer region at Rgal > 20′′.
+We note only the difference in the outer regions
+between the two galaxies. The radial number
+density profile of the NGC 4839 GCs shows a
+sudden drop at RN4839 ≈ 80 kpc, and few GCs
+are found at RN4839 > 100 kpc. On the other
+hand, the radial number density profile of the
+NGC 4816 GCs shows a slow decline even in the
+outer region at RN4816 > 100 kpc, and some
+GCs are found even out to RN4816 ≈ 500 kpc.
+Second, the surface brightness profiles of the
+two galaxies are similar in the inner region at
+1 < Rgal < 20 kpc, and show a slight difference
+in the outer region at 20 < Rgal < 50 kpc. The
+shapes of these profiles are also similar to that
+of the GC number density profile of NGC 4816,
+but showing a clear difference against that of
+the GC number density profile of NGC 4839.
+Third, we fit the surface brightness profiles of
+the galaxies (3′′ < Rgal < 30′′) with a S´ersic
+law for n = 4 (i.e., a de Vaucouleurs law), as
+shown by the dot-dashed lines.
+The surface
+brightness profiles of the galaxy light in the in-
+ner regions of the two galaxies are reasonably
+fit by the S´ersic law.
+The effective radius of
+the NGC 4839 galaxy light, Reff,N4839 = 23.′′5 ±
+0.′′7 = 11.4 ± 0.3 kpc, is similar to that of the
+NGC 4816 galaxy light, Reff,N4816 = 23.′′9±1.′′1 =
+11.6 ± 0.5 kpc. The surface brightness profile of
+NGC 4839 shows a slight excess over the fitting
+line at R > 1′, which is a cD envelope, con-
+sistent with the previous results in Schombert
+(1988); Ali et al. (2014). On the other hand, this
+excess is much weaker in the case of NGC 4816.
+Fourth, we fit the radial number density pro-
+files of GCs at 50′′ < Rgal < 1260′′ in NGC 4816
+with a S´ersic law for n = 4, as shown by the
+dotted line in the figure.
+The radial number
+density profile of the NGC 4816 GCs is approxi-
+mately fit by the S´ersic law. The effective radius
+of the NGC 4816 GC system derived from this
+fitting, is Reff,GCS = 124±37 kpc. In the case of
+
+6
+Oh et al.
+NGC 4839, the radial number density profile of
+the GCs in the inner region (50′′ < Rgal < 200′′)
+is roughly fit by the S´ersic law, but the number
+density is significantly lower than the fitted line
+in the outer region at Rgal > 200′′. We derive
+the GC system effective radius of the two galax-
+ies, from the cumulative radial distribution of
+GCs. We assume that the number density pro-
+file is flat in the central region (Rgal < 25′′)
+where our data is incomplete (see Lee et al.
+(2008) for the radial number density profile of
+M60 GCs)). The effective radius of the GC sys-
+tem derived from this, is Reff,GCS = 101.′′5 ± 3.′′1
+= 49.1 ± 1.5 kpc for NGC 4839. We resample
+the radial density profiles from the data 1000
+times, and repeat the same procedure to de-
+rive an effective radius from each profile. From
+this we obtain a standard deviation of resam-
+pled Reff,GCS as a measuring error. Note that
+the true error must be larger than this error.
+Similarly we obtain Reff,GCS = 331.′′2 ± 10.′′8 =
+160.1 ± 5.2 kpc for NGC 4816, which is larger
+than, but consistent, within the error, with the
+value based on the fitting. Thus the effective ra-
+dius of the NGC 4839 GC system is about three
+times smaller than that of the NGC 4816 GC
+system.
+4. DISCUSSION AND CONCLUSION
+Coma is an ideal target for investigating not
+only the general assembly process of galaxy clus-
+ters but also the details of the merging pro-
+cess including the infall phase of substructures.
+Various substructures related with the merging
+process in Coma were discovered in previous X-
+ray images (see Briel et al. (1992); White et al.
+(1993); Neumann et al. (2001); Sanders et al.
+(2020); Mirakhor & Walker (2020); Churazov et
+al. (2021) and references therein). Early stud-
+ies based on X-ray observations suggested that
+the NGC 4839 group is in the first phase of in-
+fall (Briel et al. 1992; White et al. 1993). Then
+Burns et al. (1994) presented a new scenario,
+based on hydro/N-body simulations, that Coma
+already had a lunch (the NGC 4839 group) and
+the NGC 4839 group is in the second infall,
+which can explain the optical, radio, and X-
+ray properties of Coma. Later Colless & Dunn
+(1996) pointed out the shortcomings of the ar-
+guments in Burns et al. (1994), and argued that
+NGC 4839 is in the first phase of infall, based on
+dynamics of a large number of Coma galaxies.
+Most of these substructures could be explained
+either in pre-merger scenarios or in post-merger
+scenarios, as summarized in Table 1.
+Later Lyskova et al. (2019) noted two promi-
+nent features seen in the XMM-Newton and
+Chandra images of the NGC 4839 group: a long
+(600 kpc) bent tail of cool gas of NGC 4839, and
+a sheath of enhanced X-ray surface brightness
+due to hotter gas in the southwest, and tried
+SPH simulations to test both pre-merger and
+post-merger scenarios. They concluded that the
+post-merger scenario can explain better the ob-
+servational results (X-ray brightness and tem-
+peratures) than the pre-merger scenario.
+Ac-
+cording to this scenario (see their Fig. 8), the
+NGC 4839 group began falling to the main clus-
+ter from the northeast about 2 Gyr ago, passed
+the center about 1.6 Gyr ago, and began the
+second infall after reaching the apocenter in the
+southwest recently.
+Recently from the X-ray images obtained
+with the SRG/eROSITA, Churazov et al. (2021,
+2022) found a faint X-ray bridge connecting the
+NGC 4839 group with the main cluster. This
+bridge may be a remnant of stripped gas while
+NGC 4839 moves outward from the main clus-
+ter to the current position, showing that it is
+strong evidence that NGC 4839 already passed
+the main cluster core (see their Fig. 11). Chura-
+zov et al. (2021, 2022) also pointed out that the
+existence of the bow shock at R ≈ 33′ (960 kpc)
+in the west and the radio relic at R ≈ 2.1 Mpc
+in the southwest (Bonafede et al. 2021) may cor-
+respond, respectively, to the secondary shock
+(produced when crossing the apocenter) and
+
+Globular Clusters in NGC 4839
+7
+the primary shock (produced when crossing the
+main cluster core) caused by the merging event
+with NGC 4839.
+In Figure 6 we compare the GC number den-
+sity map (pseudocolor map) with the XMM-
+Newton X-ray contour map of hot gas ob-
+tained after β model subtraction (showing sub-
+structures better, Neumann et al. (2001, 2003))
+(based on Fig. 3 in Adami et al. (2005)).
+In
+this figure, the X-ray contours around the NGC
+4839 region show a slight offset from the cen-
+ter of the NGC 4839 GC clump. This offset is
+not seen in the recent X-ray data (Lyskova et
+al. 2019; Churazov et al. 2021). This offset is
+due to the outdated X-ray data (Neumann et
+al. 2003) used in Adami et al. (2005). The X-
+ray map shows three prominent substructures:
+(a) the NGC 4839 group where a strong con-
+centration of GCs is seen only at the position of
+NGC 4839, (b) a large arc-like western substruc-
+ture where few GCs are found, and (c) a smaller
+substructure associated with NGC 4911/4921 in
+the southeast where only a small population of
+GCs are seen.
+Note that the X-ray emission
+substructure is seen in the NGC 4839 group, but
+not in the NGC 4816 group.
+The center of the NGC 4839 group (G2 in
+Adami et al. (2005)) was close to NGC 4839 in
+the old study by Adami et al. (2005). However,
+the recent study based on a much larger sam-
+ple of Coma members by Healy et al. (2021)
+shows that the center of the NGC 4839 group
+(S11) is significantly offset to the southwest
+from NGC 4839 (see their Fig. 11).
+On the
+other hand, the recent SRG/eROSITA X-ray
+data with higher spatial resolution (Churazov
+et al. 2021) (as well as XMM-Newton data)
+shows clearly an X-ray peak at the position of
+NGC 4839 which is embedded in a much more
+diffuse X-ray emission. This diffuse component
+is significantly overlapped with the galaxy dis-
+tribution of the S11 group (see Fig. 11 in Healy
+et al. (2021)).
+In the figure we also add the trajectory (red
+dashed line) of the NGC 4839 group suggested
+for the second-infall scenario(Lyskova et al.
+2019; Churazov et al. 2021) (from Figure 11 in
+Churazov et al. (2021)), as well as other known
+substructures. The very compact spatial extent
+of the GC system in NGC 4839, much smaller
+than the GC system in NGC 4816, can be ex-
+plained if NGC 4839 lost a significant number of
+GCs in the outskirt of NGC 4839 when it passed
+the main cluster.
+On the other hand, the more extended GC
+system in NGC 4816 indicates that it may be in
+the first phase of infall, as described below. The
+radial velocity of the NGC 4839 group is 768 km
+s−1
+larger than that of the main cluster (6853
+km s−1). Colless & Dunn (1996) suggested that
+the angle between the observer and the velocity
+vector of the NGC 4839 group is about 74 deg
+so the merger is happening with ∆v = 1700 km
+s−1 almost in the projected sky plane. Which of
+the main cluster and NGC 4839 is closer to us is
+not yet known. On the other hand, the relative
+velocity of the NGC 4816 group with respect to
+the main cluster is only +35 km s−1
+and the
+NGC 4816 group is located along the large scale
+filament connecting with Abell 1367. Consider-
+ing these we infer that the NGC 4816 group is
+infalling to the cluster center in the sky plane.
+In addition, the GC system of the NGC 4816
+shows an extended structure with a continu-
+ously declining radial number density profile.
+These results indicate that the NGC 4816 is in
+its first infall. If it is in its second infall, its ra-
+dial profile of the GC system would have shown
+a significant drop in the outer region like the
+one in the NGC 4839 group.
+If NGC 4839 is in the first phase of infall,
+it should show a similar distribution to that
+of NGC 4816, and it would be difficult to ex-
+plain the observed difference between NGC 4839
+and NGC 4816.
+When NGC 4839 crosses the
+
+8
+Oh et al.
+main cluster core again, it would lose more GCs,
+which will become part of the intracluster GCs.
+In conclusion, the spatial distribution of GCs
+in NGC 4839 and its environment supports
+the second infall scenario where the NGC 4839
+passed the Coma center about 1.6 Gyr ago, and
+began the second infall after reaching the apoc-
+enter in the southwest. Previous simulations on
+GCs in galaxy clusters (e.g., Ramos-Almendares
+et al. (2018, 2020)) are useful to understand
+the spatial distribution and kinematics of the
+GCs in large scales. However none of them pro-
+vide any results on how the motion of individual
+groups in galaxy clusters affects the size of the
+GC systems in individual galaxies, which could
+be compared with the results in this study. We
+expect that our results motivate future simula-
+tions to address this issue.
+ACKNOWLEDGMENTS
+This work was supported by the National Re-
+search Foundation grant funded by the Korean
+Government (NRF-2019R1A2C2084019).
+We
+thank Brian S. Cho for his help in improving
+the English in the manuscript.
+The authors
+are grateful to the anonymous referee for use-
+ful comments.
+Facilities: Subaru(Hyper Suprime-Cam)
+
+Globular Clusters in NGC 4839
+9
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+doi:10.1088/0004-6256/149/5/171
+
+10
+Oh et al.
+White, S. D. M., Briel, U. G., & Henry, J. P. 1993,
+MNRAS, 261, L8. doi:10.1093/mnras/261.1.L8
+
+Globular Clusters in NGC 4839
+11
+Table 1. Merging-related Features for the NGC 4839 Group
+Band
+Features
+Infall Phasea Referenceb
+X-ray
+NGC 4839 inner tail (SW), SW main tail (sheath)
+1,2
+1,2
+SW bridge connecting NGC 4839 and the main cluster 2
+W sharp edge, E contact discontinuity
+1,2
+Radio (cont) Coma radio halo, W halo front
+2
+3,4
+SW bridge, SW streams, SW relic (R = 2.1 Mpc)
+2
+Radio (HI)
+HI deficiency and old galaxies in NGC 4839 group
+1,2
+5
+Optical
+E+A galaxies in SWc
+2
+6
+galaxy distribution and kinematics
+1,2
+7
+Compact globular cluster system in NGC 4839
+2
+This study
+a1 for the first infall phase (a pre-merger scenario), and 2 for the second infall phase (a post-merger
+scenario).
+b1.Lyskova et al. (2019);2.Churazov et al. (2021, 2022); 3. Kim et al. (1989); 4.Bonafede et al.
+(2021, 2022);5.Healy et al. (2021);6.Caldwell et al. (1993); 7.Colless & Dunn (1996).
+cColless & Dunn (1996) pointed out that a few of the E+A galaxies that are the members of the
+NGC 4839 group, and that these E+A galaxies may be falling recently into Coma like NGC 4839.
+
+12
+Oh et al.
+Table 2. Basic Parameters for the Main Cluster, NGC 4839 and NGC 4816 Group in Coma
+Parameter
+Main cluster
+NGC 4839 group NGC 4816 group Referencea
+Heliocentric galaxy velocity, vh
+7167 km s−1
+7338 km s−1
+6915 km s−1
+1,2,3
+Heliocentric group velocity, vh
+6853 km s−1
+7621 km s−1
+6898 km s−1
+1,2,3
+Velocity dispersion, σv
+1082 km s−1
+462 km s−1
+521 km s−1
+2,3
+Virial Mass (dynamics)b, Mvir
+2.7 × 1015M⊙
+2.1 × 1014M⊙
+3.0 × 1014M⊙
+4
+Weak Lensing Massc, MWL
+1.2 × 1015M⊙
+1.7 × 1013M⊙
+1.3 × 1013M⊙
+5
+a1: NED; 2: Colless & Dunn (1996); 3: Healy et al. (2021); 4: This study; 5: Okabe et al. (2014).
+bCalculated for the velocity dispersion (Healy et al. 2021) using the group virial mass equation:
+Mvir/M⊙ = 1.5 × 106h−1σ3
+v in Tully (2015). Note that Colless & Dunn (1996) presented Mvir =
+1.3×1015M⊙ for the main cluster, and Mvir = 8.6×1012M⊙ for the NGC 4839 group from galaxy
+dynamics.
+c Projected masses (M2D) within the truncation radius for the subhalo 2 for the NGC 4816 group,
+and the subhalo 9 for the NGC 4839 group derived from the weak lensing analysis in Okabe et al.
+(2014), given for h = 0.7.
+
+Globular Clusters in NGC 4839
+13
+Coma
+N
+E
+1 Mpc
+NGC 4839
+NGC 4816
+Figure 1. A gray scale map (4◦ × 4◦) of the r-band SDSS image of NGC 4839 and its environment in the
+Coma cluster. Zoom-in fields for NGC 4839 and NGC 4816 (red boxes) are 10′ × 10′.
+
+:
+.
+.
+:.14
+Oh et al.
+200
+400
+Number
+NGC 4839(r0 < 26.5)
+Background
+NGC 4816(r0 < 26.5)
+Background
+Background
+0.5
+0.0
+0.5
+1.0
+1.5
+(g
+r)0
+21
+22
+23
+24
+25
+26
+27
+28
+r0 PSF mag
+NGC 4839
+0.5
+0.0
+0.5
+1.0
+1.5
+(g
+r)0
+NGC 4816
+0.5
+0.0
+0.5
+1.0
+1.5
+(g
+r)0
+Background
+Figure 2. Color-magnitude diagrams (lower panels) and color distributions (upper panels) of the point
+sources with 22.5 < r0 < 26.5 mag in the central regions (1′.2 < Rgal < 3′.3) of NGC 4839, NGC 4816, and
+the background region with the same area based on the HSC images. The hatched histograms in the upper
+panels for the galaxy regions represent the background region. The red boxes in the lower panels represent
+the boundary for GC selection.
+
+Globular Clusters in NGC 4839
+15
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+R.A. [deg]
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+Dec. [deg]
+Rvir
+0.5°
+1.0°
+NGC 4839
+NGC 4816
+NGC 4854
+NGC 4923
+NGC 4874
+NGC 4889
+IC 4051
+NGC 4798
+Coma GCs
+ETG (E+S0, Doi+1995)
+LTG (Sa+Im, Doi+1995)
+E+A galaxies (Caldwell+1993)
+50
+60
+70
+80
+90
+100
+110
+120
+130
+140
+150
+160
+170
+180
+Number density [arcmin
+2]
+Figure 3.
+Spatial number density contour map of GCs in the Coma field including NGC 4839 and
+NGC 4816 (see Oh et al. (2023) for details). Dotted line circles represent R = 0.5◦, 1.0◦, and Rvir(=2.8
+Mpc) from NGC 4874 at the Coma center. Red circles and green triangles mark early-type galaxy members,
+and late-type galaxy members (Doi et al. 1995). Black boxes mark E+A galaxies (Caldwell et al. 1993).
+The contour levels denote 2σbg and larger with an interval of one σbg where σbg denotes the background
+fluctuation. The contour maps were smoothed using a Gaussian filter with σG = 1′. The color bar represents
+the GC number density.
+
+16
+Oh et al.
+102
+103
+Angular distance from galaxy center [arcsec]
+4.5
+4.0
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+Log GC number density [arcsec
+2]
+NGC 4816
+All GC
+Blue GC
+Red GC
+101
+102
+Linear distance [kpc]
+102
+103
+Angular distance from galaxy center [arcsec]
+4.5
+4.0
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+Log GC number density [arcsec
+2]
+NGC 4839
+All GC
+Blue GC
+Red GC
+101
+102
+Linear distance [kpc]
+Figure 4.
+Radial number density profiles of the GCs in NGC 4839 (upper panel) and NGC 4816 (lower
+panel): all GCs (black solid line), blue (metal-poor) GCs (blue dashed line), and red (metal-rich) GCs (red
+dashed line).
+
+Globular Clusters in NGC 4839
+17
+100
+101
+102
+103
+Angular distance from galaxy center [arcsec]
+14
+16
+18
+20
+22
+24
+26
+r [mag/arcsec2]
+NGC 4839
+NGC 4816
+Galaxy
+light
+GCS
+100
+101
+102
+Linear distance [kpc]
+4.0
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+Log GC number density [arcsec
+2]
+Figure 5.
+Radial profiles for HSC r-band surface brightness (solid lines) and GC number density (dashed
+lines) for NGC 4839 (red lines) and NGC 4816 (blue lines). Dot-dashed lines and dotted lines denote the
+results of S´ersic law (n = 4) fitting for galaxy light and GC number density profiles, respectively. Thicker
+lines denote the fitting ranges.
+
+18
+Oh et al.
+Figure 6. Comparison of the GC number density map (pseudo color map) with XMM-Newton X-ray map
+(Neumann et al. 2001, 2003) after β model subtraction (white contours, based on Figure 3 in Adami et al.
+(2005)). Dotted black lines mark the direction of neighboring large scale structures. Green, purple, and
+yellow lines denote the primary shock, secondary shock, and contact discontinuity, respectively, in Churazov
+et al. (2021) (from their Fig. 11). The red dashed line shows the trajectory of the NGC 4839 group suggested
+for the second-infall scenario (Lyskova et al. 2019; Churazov et al. 2021). The color bar represents the GC
+number density.
+
+Adami+2005 X-ray map
+175
+29.0
+NGC4839 path
+Secondary shock
+155
+28.5
+135
+A2199
+Dec. [deg]
+115
+A779
+28.0
+95
+N4816
+14839
+C.D.
+136
+75
+27.5
+55
+IMpc
+27.0
+35
+Primary shock
+15
+196.0
+195.5
+195.0
+194.5
+194.0
+R.A. [deg]

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+Efficient Quantum Simulation of Electron-Phonon Systems by Variational Basis State
+Encoder
+Weitang Li,1 Jiajun Ren,2 Sainan Huai,1 Tianqi Cai,1 Zhigang Shuai,3, 4, ∗ and Shengyu Zhang1, †
+1Tencent Quantum Lab, Tencent, Shenzhen, China
+2College of Chemistry, Beijing Normal Univerisity, Beijing, China
+3Department of Chemistry, Tsinghua University, Beijing, China
+4School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China
+(Dated: January 5, 2023)
+Digital quantum simulation of electron-phonon systems requires truncating infinite phonon levels
+into N basis states and then encoding them with qubit computational basis. Unary encoding and
+the more compact binary/Gray encoding are the two most representative encoding schemes, which
+demand O(N) and O(log N) qubits as well as O(N) and O(N log N) quantum gates respectively.
+In this work, we propose a variational basis state encoding algorithm that reduces the scaling of the
+number of qubits and quantum gates to both O(1). The cost for the scaling reduction is a constant
+amount of additional measurement. The accuracy and efficiency of the approach are verified by both
+numerical simulation and realistic quantum hardware experiments. In particular, we find using 1
+or 2 qubits for each phonon mode is sufficient to produce quantitatively correct results across weak
+and strong coupling regimes.
+Our approach paves the way for practical quantum simulation of
+electron-phonon systems on both near-term hardware and error-corrected quantum computers.
+Introduction
+Electron-phonon couplings are per-
+vasive in quantum materials, governing phenomena such
+as charge transport in semiconductors [1], vibrational
+spectra [2], polaron formation [3], and superconductiv-
+ity [4].
+Classically, expensive numerical methods such
+as density matrix renormalization group (DMRG) and
+quantum Monte-Carlo (QMC) are required to accurately
+simulate electron-phonon systems due to the interior
+many-body interaction [5–9]. Quantum computers hold
+promise for the simulation of quantum systems with ex-
+ponential speedup over classical computers [10]. In the
+wake of the tremendous progress in the implementation of
+quantum computers [11, 12] and the dawning of the noisy
+intermediate-scale quantum (NISQ) era [13], how to solve
+electron-phonon coupling problems with quantum com-
+puters has attracted a lot of research interest [14–17].
+A prominent problem for the digital quantum simu-
+lation of electron-phonon systems is how to encode the
+infinite phonon states with finite quantum computational
+basis states. The first step is usually truncating the in-
+finite phonon states into N basis states {|m⟩} and then
+the second step is encoding {|m⟩} into quantum compu-
+tational basis {|n⟩}. The phonon basis states are usu-
+ally the N lowest harmonic oscillator eigenstates or N
+uniformly distributed grid basis states. There are two
+established strategies to perform the encoding {|m⟩} �→
+{|n⟩} [18, 19]. The first is unary encoding [20, 21], in
+which each |m⟩ is encoded to |00 . . . 1m . . . 00⟩, and the
+total number of qubits required scales as O(N). The sec-
+ond is binary encoding, in which each |m⟩ is encoded to
+�
+i |⌊ m
+2i ⌋ mod 2⟩ represented by O(log N) qubits [14, 15].
+In terms of two-qubit gates required to simulate quantum
+operators such as ˆb† ± ˆb and ˆb†ˆb, unary encoding scales
+as O(N) and binary encoding scales as O(N log N) [19].
+The features of unary encoding and binary encoding are
+summarized in Table 1. Compared to the simulation of
+electrons, the simulation of phonons consumes quantum
+resources in a much faster manner, which becomes the
+bottleneck for efficient quantum simulation of electron-
+phonon systems.
+TABLE I. Comparison of traditional encoding schemes and
+the proposed variational encoding in terms of encoding for-
+mula, the number of qubits Nqubit required and the number
+of quantum gates Ngate required to simulate common phonon
+operators such as ˆb† ± ˆb and ˆb†ˆb .
+Scheme
+Formula
+Nqubit
+Ngate
+Unary
+|m⟩ �→ |00 . . . 1m . . . 00⟩
+O(N)
+O(N)
+Binary
+|m⟩ �→ �
+i |⌊ m
+2i ⌋ mod 2⟩ O(log N) O(N log N)
+Variational
+�
+m Cmn |m⟩ �→ |n⟩
+O(1)
+O(1)
+In this work, we propose a new basis encoding scheme
+called variational encoding. Variational encoding maps
+linear combinations of |m⟩ that are most entangled to
+the simulated system into the computational basis, i.e.
+�
+m Cmn |m⟩ �→ |n⟩, where Cmn is determined by varia-
+tional principle. The advantage of our approach is that,
+by encoding only the most entangled states and discard-
+ing the ones with little entanglement, the size of {|n⟩}
+can be made irrelevant to the size of {|m⟩}.
+In other
+words, the number of qubits required scales as O(1).
+Consequently, the scaling for the number of gates is also
+O(1).
+Variational encoding is best suited to work in
+combination with variational quantum algorithms such
+as variational quantum eigensolver (VQE) [22, 23] and
+variational quantum dynamics (VQD) [24, 25]. Besides,
+the variational encoding is also compatible with Trot-
+terized time evolution and quantum phase estimation
+(QPE) [10, 26, 27].
+Numerical simulation and experi-
+ments on realistic quantum hardware based on the Hol-
+arXiv:2301.01442v1  [quant-ph]  4 Jan 2023
+
+2
+stein model and spin-boson model shows that using 1
+or 2 qubits for each phonon mode is typically sufficient
+for highly accurate results even in the strong coupling
+regime.
+Variational Basis State Encoder
+Encoder coeffi-
+cients C are determined by variational principle for both
+static and dynamic cases. We start the derivation us-
+ing parameterized quantum circuit (PQC) and discuss
+how to incorporate the variational encoder in Trotterized
+time evolution and QPE later on. We use atomic units
+throughout the paper.
+More details for the derivation
+can be found in the Appendix.
+For each phonon mode l, encoded by Nl qubits, define
+the variational encoder ˆB[l]
+ˆB[l] =
+�
+m
+2Nl
+�
+n=1
+C[l]mn |n⟩l ⟨m|
+l
+,
+(1)
+with orthonormal constraint ˆB[l] ˆB[l]† = ˆI. The original
+Hamiltonian in |m⟩ basis can then be encoded to |n⟩ basis
+ˆ˜H = �
+l ˆB[l] ˆH �
+l ˆB[l]†.
+Suppose the quantum circuit
+is parameterized by |φ⟩ = �
+k eiθk ˆ
+Rk |φ0⟩, and then the
+ground state Lagrangian with multipliers λlnn′ is
+L = ⟨φ| ˆ˜H|φ⟩ +
+�
+lnn′
+λlnn′(
+�
+m
+C[l]mnC[l]mn′ − δnn′) . (2)
+Taking the derivative with respect to θk leads to tradi-
+tional VQE with encoded Hamiltonian ˆ˜H. Taking the
+derivative with respect to C[l] and setting it to 0 yields
+(1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ = 0 ,
+(3)
+with projector ˆP[l] =
+ˆB[l]† ˆB[l] and the half encoded
+Hamiltonian ˆ˜H′[l] = �
+k̸=l ˆB[k] ˆH �
+k ˆB[k]†. In practice,
+θk and C[l] are solved iteratively until convergence. In
+the following, this iteration is termed macro-iteration to
+avoid confusion with VQE iteration.
+Next, we discuss the measurement required to solve
+C[l] from Eq. 3. Suppose the Hamiltonian can be written
+as ˆH = �M
+x ˆhx and ˆhx = �
+k ˆh[k]x, where M is the total
+number of terms in the Hamiltonian and ˆh[k]x acts on the
+kth degree of freedom. The measurement of ⟨φ| ˆ˜H′[l]|φ⟩
+boils down to that of ⟨φ|n⟩l ⟨n′|
+l
+�
+k̸=l
+ˆ˜h[k]x |φ⟩, where
+ˆ˜h[k]x = ˆB[k]ˆh[k]x ˆB[k]† is the encoded local operator. For
+electron degree of freedom a dummy encoder ˆB[k] = ˆI is
+used for notational simplicity. The number of additional
+measurements for the update of C[l] is thus O
+�
+2NlM
+�
+,
+which is polynomial to the system size and does not in-
+crease with N. If the number of phonon modes is as-
+sumed to be linear with M and each C[l] is updated
+independently, then the total number of measurements
+for all C[l] is O
+�
+2NlM 2�
+.
+The measurement overhead
+increases exponentially with Nl. Due to arguments pre-
+sented later, Nl is usually small and does not increase
+with system size. From numerical experiments, we find
+Nl ≤ 2 is sufficient to produce excellent results.
+For time-dependent problems, it is convenient to define
+|ψ⟩ = �
+l ˆB[l]† |φ⟩ and use ΘK denote both θk and C[l].
+The Lagrangian with multipliers λlnn′ and γlnn′ is then
+L = |i
+�
+K
+∂ |ψ⟩
+∂ΘK
+˙ΘK − ˆH |ψ⟩ |2
++
+�
+lnn′
+λlnn′ Re
+��
+m
+C[l]∗
+mn ˙C[l]mn′
+�
++
+�
+lnn′
+γlnn′ Im
+��
+m
+C[l]∗
+mn ˙C[l]mn′
+�
+.
+(4)
+The constraints ensure that C[l]mn remains orthonormal
+during time evolution. Similar to the ground state prob-
+lem, the equation of motion for θk is the same as vanilla
+VQD with encoded Hamiltonian ˆ˜H
+�
+j
+Re
+�∂ ⟨φ|
+∂θk
+∂ |φ⟩
+∂θj
+�
+˙θj = Im
+�∂ ⟨φ|
+∂θk
+ˆ˜H |φ⟩
+�
+.
+(5)
+The equation of motion for C[l] reads
+iρ[l] ˙C[l]∗ = (1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ ,
+(6)
+where ρ[l]nn′ = Tr{⟨φ|n⟩ ⟨n′|φ⟩} is the reduced density
+matrix for the Nl qubits of |φ⟩.
+Eq. 3 represents a
+˙C[l] = 0 stationary point during real and imaginary
+time evolution.
+The measurement cost is the same as
+the ground state algorithm.
+The VQD step described by Eq. 5 can be natu-
+rally replaced by a Suzuki-Trotter time evolution step
+e−i ˆ˜
+Hτ ≈ �M
+x e−iˆ˜hxτ on a digital quantum simulator, so
+that Hamiltonian simulation is performed via Trotterized
+time evolution instead of VQD. To update C[l] based on
+Eq. 6, measurements on the circuit should be performed
+for every or every several Trotter steps. The variationally
+encoded ground state can then be prepared by adiabatic
+state preparation, whose energy is accessible by QPE us-
+ing ˆ˜H.
+It is instructive to observe that if the variational ba-
+sis encoder is viewed as a wavefunction ansatz |ψ⟩, then
+the algorithm proposed in this work can be viewed as a
+generalization for the local basis optimization method
+for DMRG [28, 29], or a special case of the recently
+proposed quantum-classical hybrid tensor network [30].
+Thus, ˆB[l] captures the 2Nl phonon states that are most
+entangled with the rest of the system. For local Hamil-
+tonian obeying the area law, the entanglement entropy
+between one phonon mode and the rest of the system S
+is a constant [31]. Consequently, | ⟨ψ|Ψ⟩ |2, the fidelity
+between the approximated encoded state and the target
+state has a lower bound of 2Nl
+eS , which lays the theoretical
+
+3
+0
+1
+2
+3
+Coupling strength g
+−8
+−6
+−4
+−2
+E/V
+(a)
+Exact (DMRG)
+Gray encoding
+Variational encoding
+0
+5
+10
+15
+Iteration
+−8
+−6
+−4
+−2
+E/V
+(b)
+g = 0.3
+g = 1.5
+g = 3.0
+8
+16
+24
+32
+Number of levels N
+10−3
+10−2
+10−1
+100
+101
+(E − Eexact)/V
+(c)
+g = 1.5, 1 qubit
+g = 3.0, 1 qubit
+g = 1.5, 2 qubits
+g = 3.0, 2 qubits
+0
+5
+10
+15
+Index
+10−14
+10−11
+10−8
+10−5
+10−2
+Singular values
+(d)
+g = 0.3
+g = 1.5
+g = 3.0
+FIG. 1.
+Numerical simulation results for the ground state of
+the Holstein model. (a) Ground state energy by numerically
+exact DMRG, binary encoding, and variational encoding with
+different coupling strength g; (b) Convergence of ground state
+energy with respect to the macro-iteration for variational en-
+coding; (c) Ground state energy error for the variational en-
+coding method at different numbers of phonon basis states
+N; (d) The singular values for the Schmidt decomposition
+between the last phonon mode and the rest of the system.
+foundation for the effectiveness of the variational encod-
+ing approach to ground state and low-lying excited state
+problems.
+Simulations
+We first show numerical simulation re-
+sults on a noiseless simulator and verify the algorithm
+on a superconducting quantum computer at the end of
+the section. The variational basis state encoder is first
+tested for VQE simulation of the one-dimensional Hol-
+stein model [32, 33]
+ˆH = −
+�
+⟨i,j⟩
+V ˆa†
+iˆaj +
+�
+i
+ωˆb†
+iˆbi +
+�
+i
+gωˆa†
+iˆai(ˆb†
+i +ˆbi) . (7)
+where V is the hopping coefficient, ⟨i, j⟩ denotes near-
+est neighbour pairs with periodic boundary condition, ω
+is the vibration frequency and g is dimensionless cou-
+pling constant. In the following, we assume V = ω = 1
+and adjust g for different coupling strengths. We con-
+sider a 3-site system corresponding to 3(Nl + 1) qubits.
+We use binary encoding to represent traditional encod-
+ing approaches. Unary encoding is expected to produce
+similar results with binary encoding only with different
+quantum resource budgets. The ansatz used and more
+details of the simulation are included in the Appendix.
+We first compare the accuracy of the variational encod-
+ing and the binary encoding with Nl = 1. It is clear from
+Fig. 1(a) that variational encoding is significantly more
+accurate than binary encoding, especially at the strong
+coupling regime. Within the setup, binary encoding uses
+only two phonon basis states to describe each phonon
+mode, yet the variational encoding is allowed to use up
+to 32 phonon basis states before combining them into the
+most entangled states. We note that the quantum circuit
+used for variational encoding and binary encoding is es-
+sentially the same. The number of macro-iterations to
+determine C[l] is found to be rather small, as shown in
+Fig. 1(b).
+Fully converged results are obtained within
+5 iterations.
+In Fig. 1(c) we show more details of the
+error for the variational approach.
+The simulation er-
+ror typically decreases exponentially with respect to the
+number of phonon levels N included in C[l]. It is worth
+noting that quantum computational resources, including
+the number of qubits, the number of gates in the circuit,
+and the number of measurements remain constant when
+N is increased from 2 to 32. Furthermore, by using 2
+qubits to encode each mode, it is possible to further re-
+duce the error at the N → ∞ limit. When g = 3.0, the
+error is not sensitive to Nl, which implies that the error
+is dominated by other sources such as limitations of the
+ansatz, instead of the small Nl. Fig. 1(d) shows the sin-
+gular values for the Schmidt decomposition between the
+last phonon mode and the rest of the system by DMRG.
+The exponential decay ensures the fast convergence of
+Nl. The von Neumann entropy S for the 3 systems is
+found to be 0.01, 0.25, and 0.65 respectively. We also
+note the g = 1.5 case has the largest 3rd singular value,
+which explains why setting Nl = 2 significantly reduces
+the g = 1.5 error in Fig. 1(c).
+We now turn to the spin-relaxation dynamics of the
+spin-boson model [34], described by the Hamiltonian
+ˆH = ϵ
+2 ˆσz + ∆ˆσx +
+�
+j
+gjωjˆσz(ˆb†
+j +ˆbj) +
+�
+j
+ωjˆb†
+jˆbj . (8)
+where ϵ is the eigenfrequency and ∆ is the tunnelling rate.
+The coupling term has a similar form with Eq. 7 and is
+more commonly written as �
+j cjˆσzˆxj. For systems in the
+condensed phase the coupling is usually characterized by
+the spectral density function J (ω) = π
+2
+�
+j
+c2
+j
+ωj δ(�� − ωk).
+In the following we assume ϵ = 0 and ∆ = 1. We first
+use VQD for the simulation and discuss Trotterized time
+evolution at last. The variational Hamiltonian ansatz [35]
+with 3 layers is used if not otherwise specified.
+The performance of variational encoding and binary
+encoding is first compared based on a 1-mode spin-boson
+model at the strong coupling (ω = 1 and g = 3) regime,
+shown in Fig. 2(a). Variational encoding with Nl = 1
+generates much more accurate dynamics than binary en-
+coding with fewer qubits and quantum gates. The sim-
+ulation of binary encoding with Nl > 4 is prohibited by
+the deep circuit depth in the ansatz. The variational en-
+coding scheme is exceptionally efficient for this 1-mode
+model because Schmidt decomposition guarantees that 2
+variational bases for the phonon mode are sufficient to ex-
+actly represent the system. In Fig. 2(b) a 2-mode model
+with ωj = 1
+2, 1 and gj = 1
+2, 1 is used. Variational encod-
+ing with Nl = 1 is accurate at t < 2 but as the entan-
+
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+⟨σz⟩
+(a)
+Exact
+Binary, N = 2
+Binary, N = 4
+Binary, N = 8
+Binary, N = 16
+Variational, N = 64
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+⟨σz⟩
+(b)
+Exact
+Variational, 1 qubit
+Variational, 2 qubits
+0
+1
+2
+3
+4
+5
+Time t
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+⟨σz⟩
+(c)
+Exact (DMRG)
+Variational
+Binary
+0
+1
+2
+3
+4
+5
+Time t
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+⟨σz⟩
+(d)
+Exact
+Trotter+Variational
+FIG. 2.
+Numerical simulation results for the spin-relaxation
+dynamics of the spin-boson model. (a) Comparison between
+binary encoding with different numbers of phonon basis states
+and variational encoding for a 1-mode spin-boson model;
+(b) Variational encoding with different numbers of encoding
+qubits Nl for a 2-mode spin-boson model; (c) Comparison
+between binary encoding and variational encoding for an 8-
+mode spin-boson model with sub-Ohmic spectral density; (d)
+Trotterized time evolution with variational encoding based on
+a 1-mode spin-boson model.
+glement builds up the dynamics deviate from the exact
+solution. Increasing Nl to 2 effectively eliminates the er-
+ror. Next, we move on to a more challenging model with
+8 modes, in which ω and g are determined by discretizing
+a sub-Ohmic spectral density J (ω) = π
+2 αωsω1−s
+c
+e−ω/ωc
+following the prescription in the literature [36]. The pa-
+rameters are s = 1
+4, ωc = 4 and α = 10. As illustrated in
+Fig. 2(c) variational encoding with Nl = 1 captures the
+localization behavior yet binary encoding with Nl = 1
+completely fails. The number of layers in the variational
+Hamiltonian ansatz is 8 and 32 for variational and binary
+encoding respectively. Fig. 2(d) demonstrates the possi-
+bility to incorporate variational basis state encoder into
+Trotterized time evolution with ω = g = 1 and Nl = 1.
+The measurement and the evolution of C[l] are performed
+at each Trotter step.
+Lastly, we verify the accuracy and efficiency of the vari-
+ational encoder approach on a superconducting quantum
+computer. We consider the ground state problem of a
+2-site Holstein model described by Eq. 7 with g = 3 and
+Nl = 1. The two electronic sites are represented by 1
+qubit and the total number of qubits for the system is
+thus 3.
+The quantum circuit for the simulation is de-
+picted in Fig. 3(a). The electronic degree of freedom is
+mapped to the second qubit, and the two phonon modes
+are mapped to the first and the third qubits respectively.
+There is one parameter to be determined by VQE in the
+circuit and the same ansatz is used for both binary en-
+coding and variational encoding. More simulation details
+(a)
+1
+2
+3
+Coupling strength g
+−8
+−6
+−4
+−2
+E/V
+(b)
+Binary
+Variational
+Simulation
+Exact
+1
+2
+3
+4
+5
+Macro-iteration
+(c)
+Binary
+Variational
+Simulation
+Exact
+FIG. 3.
+Quantum hardware experiments for the ground state
+energy of the Holstein model with variational basis state en-
+coder. (a) 3 qubits out of 9 qubits of a superconducting quan-
+tum computer and a one-parameter circuit are used for the
+simulation; (b) Ground state energy by binary encoding and
+variational encoding; (c) Convergence of ground state energy
+with respect to the macro-iteration for variational encoding.
+can be found in the Appendix. In Fig. 3(b) we show the
+ground state energy by variational encoding from weak
+to strong coupling, in analog to Fig. 1(a). The simulator
+result is based on the parameterized quantum circuit de-
+scribed in Fig. 3(a) without considering gate noise and
+measurement uncertainty. The results in Fig. 3(b) are
+consistent with that in Fig. 1(a). The residual error is
+dominated by the intrinsic gate noise in the quantum
+computer.
+In Fig. 3(c) we show the convergence with
+respect to the macro-iteration for variational encoding.
+The algorithm is resilient to the presence of quantum
+noise and measurement uncertainty. The convergent en-
+ergy is reached within 5 iterations.
+Conclusion
+We proposed variational basis state
+encoder to encode phonon basis states into quantum
+computational states for efficient quantum simulation
+of electron-phonon systems.
+The proposed variational
+encoding approach requires only O(1) qubits and O(1)
+quantum gates, which is significantly better than tradi-
+tional encoding schemes. The algorithm enables quan-
+tum simulation of electron-phonon systems with smaller
+quantum processors and shallower circuits.
+The addi-
+tional measurement required to implement the approach
+is found to be also O(1) with respect to the number of
+phonon basis states and it scales quadratically with the
+number of Pauli strings in the Hamiltonian. The accu-
+racy of the approach is ensured by the finite entangle-
+ment entropy between one phonon mode and the rest of
+the system in common electron-phonon systems. Vari-
+ational basis state encoder most naturally works with
+variational quantum algorithms and is compatible with
+Trotterized time evolution, adiabatic state preparation,
+
+Ry(-0)
+Ry(-0)
+Measurement
+Module
+Ry(0)
+Ry(-0)
+D5
+and QPE. Numerical simulation and quantum hardware
+experiments based on VQE of the Holstein model and
+dynamics of the spin-boson model indicates that varia-
+tional encoding is more accurate and resource-efficient
+than traditional encoding methods. In particular, using
+1 or 2 qubits to represent each phonon mode is suffi-
+cient for accurate simulation even at the strong coupling
+regime where N = 64 phonon basis states are involved.
+The approach could also be extended to other quantum
+simulation problems involving an infinite or large local
+Hilbert space.
+We thank Jinzhao Sun and Shixin Zhang for helpful
+discussions. This work is supported by the National Nat-
+ural Science Foundation of China through grand numbers
+22273005 and 21788102. This work is also supported by
+Shenzhen Science and Technology Program.
+∗ zgshuai@tsinghua.edu.cn
+† shengyzhang@tencent.com
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+I. Derivation of variational basis state encoder
+Time-independent equation
+We start with the Lagrangian defined in the main text, i.e. Eq. 2. Taking the derivative with respect to C[l]mn
+and setting it to 0 yields
+⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ +
+�
+n′
+λlnn′C[l]mn′ = 0 .
+(9)
+Multiply with C[l]mn′′
+�
+m
+C[l]mn′′ ⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ +
+�
+n′
+λlnn′
+�
+m
+C[l]mn′C[l]mn′′ = 0 ,
+(10)
+and use the C[l] orthonormal condition �
+m C[l]mnC[l]mn′ = δnn′ to get λ
+λlnn′ = −
+�
+m
+C[l]mn′ ⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ .
+(11)
+Substitute λ into Eq. 9 yields
+⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ −
+�
+n′m
+C[l]mn′ ⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ C[l]m′n′ = 0 .
+(12)
+Using
+ˆP = ˆB[l]†B[l] =
+�
+mm′
+�
+n
+|m⟩l C[l]mnC[l]m′n ⟨m′|
+l
+,
+(13)
+to simplify the notation of the second term
+⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ − ⟨φ|n⟩l ⟨m|
+l
+ˆP ˆ˜H′[l] |φ⟩ = 0 .
+(14)
+Rearranging and rewriting in matrix form, we get the equation for C[l]
+(1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ = 0 .
+(15)
+
+7
+Quantum circuit measurement
+In this section, we discuss the quantum circuit measurement required to solve C[l] from Eq. 3. The key quantity
+to be computed is matrix G[l]mn, defined as
+G[l]mn = ⟨φ|n⟩l ⟨m|
+l
+ˆ˜H′[l] |φ⟩ .
+(16)
+Express ˆH in sum-of-product form ˆH = �M
+x
+�
+k ˆh[k]x, using notations in the main text, and we get
+G[l]mn =
+M
+�
+x
+⟨φ|n⟩l ⟨m|
+l
+�
+k̸=l
+ˆB[k]
+�
+k
+ˆh[k]x
+�
+k
+ˆB[k]† |φ⟩
+=
+M
+�
+x
+⟨φ|n⟩l ⟨m|
+l
+ˆh[l]x ˆB[l]† �
+k̸=l
+ˆ˜h[k]x |φ⟩ .
+(17)
+Here we assume �
+k̸=l
+ˆ˜h[k]x can be expressed by a constant amount of Pauli strings. Represent ˆh[l]x in operator form
+ˆh[l]x =
+�
+mm′
+h[l]xm′m |m′⟩l ⟨m|
+l
+.
+(18)
+G[l]mn then becomes
+G[l]mn =
+M
+�
+x
+�
+m′n′
+h[l]xmm′C[l]m′n′ ⟨φ|n⟩l ⟨n′|
+l
+�
+k̸=l
+ˆ˜h[k]x |φ⟩ .
+(19)
+Thus to evaluate G[l]mn it is sufficient to measure ⟨φ|n⟩l ⟨n′|
+l
+�
+k̸=l
+ˆ˜h[k]x |φ⟩. |n⟩l ⟨n′|
+l
+in general is not Hermitian,
+and the real and imaginary parts can be measured by
+Re
+�
+�
+�⟨φ|n⟩l ⟨n′|
+l
+�
+k̸=l
+ˆ˜h[k]x |φ⟩
+�
+�
+� = 1
+2 ⟨φ| (|n⟩l ⟨n′|
+l
++ |n′⟩l ⟨n|
+l
+)
+�
+k̸=l
+ˆ˜h[k]x |φ⟩ ,
+Im
+�
+�
+�⟨φ|n⟩l ⟨n′|
+l
+�
+k̸=l
+ˆ˜h[k]x |φ⟩
+�
+�
+� = 1
+2 ⟨φ| i(|n′⟩l ⟨n|
+l
+− |n⟩l ⟨n′|
+l
+)
+�
+k̸=l
+ˆ˜h[k]x |φ⟩ .
+(20)
+To evaluate all matrix elements in G[l], the total number of measurements required scales as O
+�
+2NlM
+�
+.
+Time-dependent equation
+In this section, we derive the time-dependent equation for C[l]. For time-dependent problems, C[l] in general is
+complex
+C[l] = D[l] − iE[l] ,
+(21)
+where both D[l] and E[l] are real. The minus sign is for convenience expressing ˆB† |φ⟩. From the definition we have
+∂ |ψ⟩
+∂E[l]mn
+= i
+∂ |ψ⟩
+∂D[l]mn
+.
+(22)
+The starting point is the Lagrangian Eq. 4 defined in the main text. Taking the derivative with respect to ˙ΘK
+
+8
+yields
+∂L
+∂ ˙ΘK
+=
+�
+J
+∂ ⟨ψ|
+∂ΘJ
+∂ |ψ⟩
+∂ΘK
+˙ΘJ +
+�
+J
+∂ ⟨ψ|
+∂ΘK
+∂ |ψ⟩
+∂ΘJ
+˙ΘJ
++ i∂ ⟨ψ|
+∂ΘK
+ˆH |ψ⟩ − i ⟨ψ| ˆH ∂ |ψ⟩
+∂ΘK
++
+�
+lnn′
+λlnn′ Re
+��
+m
+C[l]∗
+mn
+∂ ˙C[l]mn′
+∂ ˙ΘK
+�
++
+�
+lnn′
+γlnn′ Im
+��
+m
+C[l]∗
+mn
+∂ ˙C[l]mn′
+∂ ˙ΘK
+�
+.
+(23)
+We first consider the case of ΘK = θk, and then
+∂L
+∂ ˙θk
+=
+�
+J
+∂ ⟨ψ|
+∂ΘJ
+∂ |ψ⟩
+∂θk
+˙ΘJ +
+�
+J
+∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂ΘJ
+˙ΘJ
++ i∂ ⟨ψ|
+∂θk
+ˆH |ψ⟩ − i ⟨ψ| ˆH ∂ |ψ⟩
+∂θk
+= 2
+�
+J
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂ΘJ
+�
+˙ΘJ − 2 Im
+�∂ ⟨ψ|
+∂θk
+ˆH |ψ⟩
+�
+,
+(24)
+which means at the
+∂L
+∂ ˙θk = 0 minimum, we have
+�
+J
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂ΘJ
+�
+˙ΘJ = Im
+�∂ ⟨ψ|
+∂θk
+ˆH |ψ⟩
+�
+.
+(25)
+Substitute ΘJ with θk, D[l]mn and E[l]mn
+�
+J
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂ΘJ
+�
+˙ΘJ =
+�
+j
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂θj
+�
+˙θj
++
+�
+lmn
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂D[l]mn
+�
+˙D[l]mn
++
+�
+lmn
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂E[l]mn
+�
+˙E[l]mn .
+(26)
+Using Eq. 22 the last two terms become
+�
+lmn
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂D[l]mn
+�
+˙D[l]mn +
+�
+lmn
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂E[l]mn
+�
+˙E[l]mn =
+�
+lmn
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂D[l]mn
+˙C[l]∗
+mn
+�
+,
+(27)
+which is zero because
+�
+mn
+∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂D[l]mn
+˙C[l]∗
+mn =
+�
+mn
+∂ ⟨φ|
+∂θk
+ˆB[l] |m⟩l ⟨n|
+l
+˙C[l]∗
+mn |φ⟩ = 0 ,
+(28)
+where the constraint �
+m C[l]mn ˙C[l]∗
+mn′ = 0 is used. Thus the simplified equation of motion reads
+�
+j
+Re
+�∂ ⟨ψ|
+∂θk
+∂ |ψ⟩
+∂θj
+�
+˙θj = Im
+�∂ ⟨ψ|
+∂θk
+ˆH |ψ⟩
+�
+,
+(29)
+or equivalently
+�
+j
+Re
+�∂ ⟨φ|
+∂θk
+∂ |φ⟩
+∂θj
+�
+˙θj = Im
+�∂ ⟨φ|
+∂θk
+ˆ˜H |φ⟩
+�
+.
+(30)
+
+9
+In short, the equation of motion for θk is the same as vanilla VQD with encoded Hamiltonian ˆ˜H .
+Next we consider the case of ΘK = D[l] and ΘK = E[l]. After some complex algebra, we have
+i
+�
+J
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂ΘJ
+˙ΘJ + i1
+2
+�
+n′
+λln′nC[l]∗
+mn′ − 1
+2
+�
+n′
+γln′nC[l]∗
+mn′ =
+∂ ⟨ψ|
+∂D[l]mn
+ˆH |ψ⟩ .
+(31)
+Similar to the case of ΘK = θk, substitute ΘJ with θk, D[l]mn and E[l]mn
+�
+J
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂ΘJ
+˙ΘJ =
+�
+k
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂θk
+˙θk +
+�
+km′n′
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂D[k]m′n′
+˙C[k]∗
+m′n′
+=
+�
+k
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂θk
+˙θk +
+�
+n′
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂D[l]mn′
+˙C[l]∗
+mn′ .
+(32)
+Here the orthonormal condition is again used. Substitute the equation back into Eq. 31.
+i
+�
+k
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂θk
+˙θk + i
+�
+n′
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂D[l]mn′
+˙C[l]∗
+mn′ + 1
+2
+�
+n′
+(iλln′n − γln′n)C[l]∗
+mn′ =
+∂ ⟨ψ|
+∂D[l]mn
+ˆH |ψ⟩ ,
+(33)
+Following the same strategy with the derivation of the time-independent equation, multiply Eq. 33 with C[l]mn
+i
+�
+k
+⟨φ|n⟩l ⟨n′|
+l
+∂ |φ⟩
+∂θk
+˙θk + 1
+2(iλln′n − γln′n) =
+�
+m
+C[l]mn′
+∂ ⟨ψ|
+∂D[l]mn
+ˆH |ψ⟩ ,
+(34)
+where �
+m C[l]∗
+mn′C[l]mn = δn′n and �
+m ˙C[l]∗
+mn′C[l]mn = 0 are used. Then, multiply again with C[l]∗
+mn
+i
+�
+k
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂θk
+˙θk + 1
+2
+�
+n′
+(iλln′n − γln′n)C[l]∗
+mn′ = ˆP[l]
+∂ ⟨ψ|
+∂D[l]mn
+ˆH |ψ⟩ .
+(35)
+Use this equation to eliminate λ and γ in Eq. 33, we get the equation of motion for C[l]
+i
+�
+n′
+∂ ⟨ψ|
+∂D[l]mn
+∂ |ψ⟩
+∂D[l]mn′
+˙C[l]∗
+mn′ = (1 − ˆP[l])
+∂ ⟨ψ|
+∂D[l]mn
+ˆH |ψ⟩ ,
+(36)
+which can be simplified to Eq. 6. The measurement required for time evolution is in the same order as the static VQE
+algorithm.
+In the end, we note that imaginary time evolution might be another approach to finding the ground state, in addition
+to the iterative method described in the main text. Imaginary time evolution might also be a feasible approach to
+determine C[l] as an alternative to solving Eq. 3.
+II. Numerical noiseless simulations
+All numerical quantum circuit simulation is performed using the TensorCircuit [37] package without considering
+noise. Classical DMRG simulation is performed using the Renormalizer package [38]. We use harmonic oscillator
+eigenstates for phonon basis states. Using positional states might affect the performance of traditional encodings
+because of the truncation, however, we expect variational encoding to be insensitive to the choice of phonon basis
+states at the N → ∞ limit. We use Gray code for binary encoding as an improvement to the standard approach [18].
+For both ground state simulation and dynamics simulation, C[l] is initialized as C[l]mn = δmn.
+For the VQE simulation of the Holstein model, the following ansatz is used
+|φ⟩ =
+L
+�
+l
+�
+�
+�
+�
+⟨j,k⟩
+eθljk(ˆa†
+j ˆak−ˆa†
+kˆaj) �
+j
+eθljˆa†
+j ˆaj(ˆb†
+j−ˆbj)
+�
+�
+� |φ0⟩ .
+(37)
+where L is the number of layers and L = 3 is adopted. The advantage of Eq. 37 is enforcing real-valued wavefunction.
+The circuit parameters ⃗θ are optimized by the L-BFGS-G method implemented in SciPy package [39]. The parameter
+gradient is calculated by auto-differentiation. The initial values for the parameters are set to zero at the first round
+
+10
+of the macro-iteration. In subsequent macro-iterations, the previously optimized parameters are used as the initial
+value for faster convergence. Eq. 3 is solved by the DF-SANE method implemented in SciPy [39]. Since this is a
+non-linear equation, we provide 3 initial guesses and adopt the one with the lowest energy. The solved C[l] sometimes
+does not satisfy the orthonormal condition due to numerical imprecision and the orthonormal condition is enforced
+by QR decomposition in each macro-iteration.
+For the VQD simulation of the spin-boson model, the variational Hamiltonian ansatz used is more complex than
+the VQE simulation. Because C[l] is complex, ˆB[l]ˆh[l]x ˆB[l]† spans the whole Hermitian matrix space. Thus for ˆh[l]x
+the whole Pauli matrix set {X, Y, Z, I}⊗Nl is added to the ansatz. To obtain the quantities required to calculate θk,
+the Jacobian of the wavefunction φ(⃗θ) is firstly calculated by auto-differentiation, and then the r.h.s and l.h.s of Eq. 5
+is calculated by matrix multiplication. How to measure the quantities in realistic quantum circuits is well described
+in the literature [16]. To calculate ˙C[l] it is necessary to take the inverse of ρ[l] which is sometimes ill-conditioned.
+We add 1 × 10−5 to the diagonal elements of ρ[l] for regularization. The time evolution of θk and C[l] is carried out
+using the RK45 method implemented in SciPy [39]. We observe that the gradient of θk is usually much larger than
+C[l]. Thus it is possible to evolve the two sets of parameters separately, which deserves further investigation. For
+Trotterized time evolution, N = 16 and a time step of 0.01 are used.
+III. Experiments on a superconducting quantum processor
+Device parameters
+The superconducting quantum processor, as shown in Fig. 3(a), is composed of nine computational transmon
+qubits with each pair of neighboring qubits mediated via a tunable coupler, forming a cross-shaped architecture.
+Each computational qubit has an independent readout cavity for state measurement and XY /Z control lines for state
+operation. High-fidelity simultaneous single-shot readout for all qubits are achieved with the help of the multistage
+amplification with the Josephson parametric amplifier (JPA) functioning as the first stage of the amplification. The
+fundamental device parameters including qubit parameters and gate parameters are outlined in Table. II and Table. III,
+where the parasitic ZZ interaction between qubits is suppressed by the coupler.
+TABLE II. Single qubit gate parameters. ωr is the resonant frequency of the readout cavity for each qubit. ωj,max (j = 1 ∼ 9)
+are the maximum resonant frequencies when qubits are biased at the sweet spot. ωj,idle (j = 1 ∼ 9) are the idle frequencies
+for implementing the single-qubit operations. αj (j = 1 ∼ 9) are the qubits’ anharmonicities. T1, T2,idle and T2E,idle are the
+corresponding energy relaxation time, Ramsey dephasing time and echoed dephasing time for the qubits measured at the idle
+frequency. The readout fidelities are typically characterized by detecting each qubit in |g⟩ (|e⟩) when it is prepared in |g⟩ (|e⟩),
+labeled by F0,j and F1,j. To mitigate the error coming from the readout infidelity, the outcomes are reconstructed with the
+calibration matrix through the Bayes’ rule. Single-qubit errors esq are measured with randomized benchmarking (RB).
+Q0
+Q1
+Q2
+Q3
+Q4
+Q5
+Q6
+Q7
+Q8
+ωr (GHz)
+6.874
+6.825
+6.931
+6.901
+6.845
+6.786
+6.991
+6.961
+6.806
+ωj,max (GHz)
+4.003
+4.215
+4.479
+4.689
+4.470
+4.479
+4.657
+4.512
+4.362
+ωj,idle (GHz)
+3.988
+4.187
+4.464
+4.668
+4.404
+4.359
+4.641
+4.498
+4.223
+αj/2π (MHz)
+−260
+−258
+−255
+−250
+−254
+−258
+−253
+−257
+−264
+T1 (µs)
+35.3
+31.6
+29.5
+27.7
+33.9
+34.3
+33.3
+22.1
+31.8
+T2,idle (µs)
+11.0
+10.2
+32.6
+38.2
+9.1
+5.6
+43.1
+24.1
+4.3
+T2E,idle (µs)
+48.2
+38.4
+47.8
+44.2
+31.6
+21.8
+56.8
+32.9
+18.6
+F0,j (%)
+96.9
+97.4
+98.6
+98.9
+98.7
+98.4
+96.3
+97.2
+94.1
+F1,j (%)
+93.7
+94.3
+92.5
+94.3
+94.5
+94.6
+92.7
+92.4
+90.9
+esq (%)
+0.07
+0.32
+0.06
+0.07
+0.08
+0.05
+0.06
+0.15
+0.08
+Experimental details
+We use 3 qubits out of the 9-qubit computer for the 2-site Holstein model
+ˆH = −V (a†
+1a2 + a†
+2a1) + ωb†
+1b1 + ωb†
+2b2 + gωa†
+1a1(b†
+1 + b1) + gωa†
+2a2(b†
+2 + b2) .
+(38)
+The electronic degree of freedom is mapped to the second qubit. Thus, a†
+1a1 is mapped to 1
+2(1 + Z1) and a†
+2a2 is
+mapped to 1
+2(1 − Z1). The phonon modes are mapped to the first and the third qubit. With binary encoding and
+
+11
+TABLE III. Two qubits gate parameters. ωc,idle are the idle frequencies for each coupler where the ZZ interaction between
+neighboring computational qubits are maximally suppressed. ξZZ is the residual ZZ interaction between each qubit pairs.
+Two-qubit gates are implemented with the controlled-Z (CZ) and the corresponding gate errors etq,CZ are characterized with
+RB.
+Q0 − Q1
+Q0 − Q2
+Q0 − Q3
+Q0 − Q4
+Q1 − Q5
+Q2 − Q6
+Q3 − Q7
+Q4 − Q8
+ωc,idle (GHz)
+5.020
+5.445
+5.570
+5.335
+5.325
+5.595
+5.695
+5.355
+|ξZZ| (kHz)
+18.0
+10.0
+5.0
+8.0
+2.0
+3.0
+5.0
+2.0
+etq,CZ (%)
+1.57
+2.22
+1.99
+2.47
+0.91
+1.04
+1.2
+0.96
+Nl = 1, the Hamiltonian in the Pauli string form reads
+ˆH = −V X1 + 1
+2ω(1 − Z0) + 1
+2ω(1 − Z2) + 1
+2gω(1 + Z1)X0 + 1
+2gω(1 − Z1)X2 .
+(39)
+For variational encoding, we assume C[l] = C. That is, the two modes share the same variational encoder. This is a
+reasonable assumption for translational invariant systems. Supposing ˆb†ˆb and ˆb† +ˆb are mapped to the following form
+ˆB(ˆb†ˆb) ˆB† = c1iI + c1xX + c1zZ
+ˆB(ˆb† + ˆb) ˆB† = c2iI + c2xX + c2zZ ,
+(40)
+the encoded Hamiltonian is then
+ˆH = −V X1 + ω(c1iI0 + c1xX0 + c1zZ0) + ω(c1iI2 + c1xX2 + c1zZ2)
++ 1
+2gω(1 + Z1)(c2iI0 + c2xX0 + c2zZ0) + 1
+2gω(1 − Z1)(c2iI2 + c2xX2 + c2zZ2) .
+(41)
+We use the following ansatz for the parameterized quantum circuit
+|φ⟩ =
+2
+�
+j=1
+eθja†
+jaj(b†
+j−bj) 1
+√
+2 (|000⟩ + |100⟩) ,
+(42)
+Because C[1] = C[2], the parameter space can be further simplified by setting θ1 = θ2. With binary encoding, the
+ansatz transforms to
+|φ⟩ = eiθY2e−iθZ1Y2eiθY0eiθZ1Y0H1 |0⟩ .
+(43)
+The ansatz is compiled into the following quantum circuit with 4 CNOT gates.
+q0 :
+RZ( −π
+2 )
+H
+RZ(−θ)
+RZ(−θ)
+H
+RZ( π
+2 )
+q1 :
+H
+•
+•
+•
+•
+q2 :
+RZ( −π
+2 )
+H
+RZ(θ)
+RZ(−θ)
+H
+RZ( π
+2 )
+Each energy term is measured by 8192 shots, and the uncertainty is obtained by repeating the measurement 5 times
+and taking the standard deviation. For the update of C[l], 4096 shots are performed for each term. Local readout
+error mitigation is applied for all results presented unless otherwise stated.
+In Fig. 4 we plot the energy landscape E(θ)/V in VQE with binary encoding. Both raw data and data with local
+readout error mitigation (EM) are presented for the energy expectation from quantum hardware. The mitigated
+landscape is in decent agreement with the perfect simulator. A minimum at around θ = 0.6 is clearly visible. We note
+that the perfect simulator is also based on the Nl = 1 ansatz and N is far smaller than what is physically demanded.
+Thus the minimum presented by the perfect simulator can not be recognized as the ground truth.
+
+12
+0.00
+0.25
+0.50
+0.75
+1.00
+θ
+−3
+−2
+−1
+0
+E/V
+Simulator
+Hardware (raw)
+Hardware (EM)
+FIG. 4.
+VQE energy landscape for the 2-site Holstein model with binary encoding. For the data from quantum hardware,
+both raw data and data with readout error mitigation are presented. The error bar indicates the measurement uncertainty.
+

1tFQT4oBgHgl3EQf1jZM/content/tmp_files/2301.13420v1.pdf.txt

@@ -0,0 +1,2265 @@
+Superhuman Fairness
+Omid Memarrast 1 Linh Vu 1 Brian Ziebart 1
+Abstract
+The fairness of machine learning-based decisions
+has become an increasingly important focus in
+the design of supervised machine learning meth-
+ods. Most fairness approaches optimize a spec-
+ified trade-off between performance measure(s)
+(e.g., accuracy, log loss, or AUC) and fairness met-
+ric(s) (e.g., demographic parity, equalized odds).
+This begs the question: are the right performance-
+fairness trade-offs being specified? We instead re-
+cast fair machine learning as an imitation learning
+task by introducing superhuman fairness, which
+seeks to simultaneously outperform human de-
+cisions on multiple predictive performance and
+fairness measures. We demonstrate the benefits
+of this approach given suboptimal decisions.
+1. Introduction
+The social impacts of algorithmic decisions based on ma-
+chine learning have motivated various group and individ-
+ual fairness properties that decisions should ideally satisfy
+(Calders et al., 2009; Hardt et al., 2016). Unfortunately, im-
+possibility results prevent multiple common group fairness
+properties from being simultaneously satisfied (Kleinberg
+et al., 2016). Thus, no set of decisions can be universally fair
+to all groups and individuals for all notions of fairness. In-
+stead, specified weightings, or trade-offs, of different criteria
+are often optimized (Liu & Vicente, 2022). Identifying an
+appropriate trade-off to prescribe to these fairness methods
+is a daunting task open to application-specific philosophical
+and ideological debate that could delay or completely derail
+the adoption of algorithmic methods.
+We consider the motivating scenario of a fairness-aware deci-
+sion task currently being performed by well-intentioned, but
+inherently error-prone human decision makers. Rather than
+seeking optimal decisions for specific performance-fairness
+trade-offs, which may be difficult to accurately elicit, we
+propose a more modest, yet more practical objective: out-
+perform human decisions across all performance and
+1Computer Science Department, University of Illinois Chicago.
+Correspondence to: O. Memarrast <omemar2@uic.edu>.
+Figure 1. Three sets of
+decisions (black dots)
+with different predictive
+performance and group
+disparity values defining
+the sets of 100%-, 67%-,
+and
+33%-superhuman
+fairness-performance
+values (red shades) based
+on Pareto dominance.
+fairness measures with maximal frequency. We implic-
+itly assume that available human decisions reflect desired
+performance-fairness trade-offs, but are often noisy and sub-
+optimal. This provides an opportunity for superhuman
+decisions that Pareto dominate human decisions across pre-
+dictive performance and fairness metrics (Figure 1) without
+identifying an explicit desired trade-off.
+To the best of our knowledge, this paper is the first to define
+fairness objectives for supervised machine learning with
+respect to noisy human decisions rather than using prescrip-
+tive trade-offs or hard constraints. We leverage and extend a
+recently-developed imitation learning method for subdomi-
+nance minimization (Ziebart et al., 2022). Instead of using
+the subdominance to identify a target trade-off, as previous
+work does in the inverse optimal control setting to estimate
+a cost function, we use it to directly optimize our fairness-
+aware classifier. We develop policy gradient optimization
+methods (Sutton & Barto, 2018) that allow flexible classes
+of probabilistic decision policies to be optimized for given
+sets of performance/fairness measures and demonstrations.
+We conduct extensive experiments on standard fairness
+datasets (Adult and COMPAS) using accuracy as a per-
+formance measure and three conflicting fairness definitions:
+Demographic Parity (Calders et al., 2009), Equalized Odds
+(Hardt et al., 2016), and Predictive Rate Parity (Choulde-
+chova, 2017)). Though our motivation is to outperform hu-
+man decisions, we employ a synthetic decision-maker with
+differing amounts of label and group membership noise to
+identify sufficient conditions for superhuman fairness of
+varying degrees. We find that our approach achieves high
+levels of superhuman performance that increase rapidly with
+reference decision noise and significantly outperform the
+superhumanness of other methods that are based on more
+arXiv:2301.13420v1  [cs.LG]  31 Jan 2023
+
+Superhuman Fairness
+narrow fairness-performance objectives.
+2. Fairness, Elicitation, and Imitation
+2.1. Group Fairness Measures
+Group fairness measures are primarily defined by confu-
+sion matrix statistics (based on labels yi ∈ {0, 1} and
+decisions/predictions ˆyi ∈ {0, 1} produced from inputs
+xi ∈ RM) for examples belonging to different protected
+groups (e.g., ai ∈ {0, 1}).
+We focus on three prevalent fairness properties in this paper:
+• Demographic Parity (DP) (Calders et al., 2009) requires
+equal positive rates across protected groups:
+P( ˆY = 1|A = 1) = P( ˆY = 1|A = 0);
+• Equalized Odds (EqOdds) (Hardt et al., 2016) requires
+equal true positive rates and false positive rates across
+groups, i.e.,
+P( ˆY =1|Y =y, A=1) = P( ˆY =1|Y =y, A=0), y ∈ {0, 1};
+• Predictive Rate Parity (PRP) (Chouldechova, 2017) re-
+quires equal positive predictive value (ˆy = 1) and negative
+predictive value (ˆy = 0) across groups:
+P(Y =1|A=1, ˆY = ˆy) = P(Y =1|A=0, ˆY = ˆy),
+ˆy ∈ {0, 1}.
+Violations of these fairness properties can be measured as
+differences:
+D.DP(ˆy, a) =
+�����
+�N
+i=1 I [ˆyi =1, ai =1]
+�N
+i=1 I [ai =1]
+(1)
+−
+�N
+i=1 I [ˆyi =1, ai =0]
+�N
+i=1 I [ai =0]
+�����;
+D.EqOdds(ˆy, y, a) = max
+y∈{0,1}
+�����
+�N
+i=1 I [ˆyi =1, yi =y, ai =1]
+�N
+i=1 I [ai =1, yi =y]
+−
+�N
+i=1 I [ˆyi =1, yi =y, ai =0]
+�N
+i=1 I [ai =0, yi =y]
+�����;
+(2)
+D.PRP(ˆy, y, a) = max
+y∈{0,1}
+�����
+�N
+i=1 I [yi =1, ˆyi =y, ai =1]
+�N
+i=1 I [ai =1, ˆyi =y]
+−
+�N
+i=1 I [yi =1, ˆyi =y, ai =0]
+�N
+i=1 I [ai =0, ˆyi =y]
+�����.
+(3)
+2.2. Performance-Fairness Trade-offs
+Numerous fair classification algorithms have been devel-
+oped over the past few years, with most targeting one fair-
+ness metric (Hardt et al., 2016). With some exceptions
+(Blum & Stangl, 2019), predictive performance and fairness
+are typically competing objectives in supervised machine
+learning approaches. Thus, though satisfying many fairness
+properties simultaneously may be na¨ıvely appealing, doing
+so often significantly degrades predictive performance or
+even creates infeasibility (Kleinberg et al., 2016).
+Given this, many approaches seek to choose parameters θ
+for (probabilistic) classifier Pθ that balance the competing
+predictive performance and fairness objectives (Kamishima
+et al., 2012; Hardt et al., 2016; Menon & Williamson, 2018;
+Celis et al., 2019; Martinez et al., 2020; Rezaei et al., 2020).
+Recently, Hsu et al. (2022) proposed a novel optimization
+framework to satisfy three conflicting fairness metrics (de-
+mographic parity, equalized odds, and predictive rate parity)
+to the best extent possible:
+min
+θ
+Eˆy∼Pθ
+�
+loss(ˆy, y) + αDPD.DP(ˆy, a)
+(4)
++ αOddsD.EqOdds(ˆy, y, a) + αPRPD.PRP(ˆy, y, a)
+�
+.
+2.3. Preference Elictation & Imitation Learning
+Preference elicitation (Chen & Pu, 2004) is one natural ap-
+proach to identifying desirable performance-fairness trade-
+offs. Preference elicitation methods typically query users
+for their pairwise preference on a sequence of pairs of op-
+tions to identify their utilities for different characteristics of
+the options. This approach has been extended and applied to
+fairness metric elicitation (Hiranandani et al., 2020), allow-
+ing efficient learning of linear (e.g., Eq. (4)) and non-linear
+metrics from finite and noisy preference feedback.
+Imitation learning (Osa et al., 2018) is a type of supervised
+machine learning that seeks to produce a general-use policy
+ˆπ based on demonstrated trajectories of states and actions,
+˜ξ = (˜s1, ˜a1, ˜s2, . . . , ˜sT ). Inverse reinforcement learning
+methods (Abbeel & Ng, 2004; Ziebart et al., 2008) seek
+to rationalize the demonstrated trajectories as the result
+of (near-) optimal policies on an estimated cost or reward
+function. Feature matching (Abbeel & Ng, 2004) plays a key
+role in these methods, guaranteeing if the expected feature
+counts match, the estimated policy ˆπ will have an expected
+cost under the demonstrator’s unknown fixed cost function
+weights ˜w ∈ RK equal to the average of the demonstrated
+trajectories:
+Eξ∼ˆπ [fk(ξ)] = 1
+N
+N
+�
+i=1
+fk
+�
+˜ξi
+�
+, ∀k
+(5)
+=⇒ Eξ∼ˆπ [cost ˜
+w(ξ)] = 1
+N
+N
+�
+i=1
+cost ˜
+w
+�
+˜ξi
+�
+,
+where fk(ξ) = �
+st∈ξ fk (st).
+Syed & Schapire (2007) seeks to outperform the set of
+demonstrations when the signs of the unknown cost function
+
+Superhuman Fairness
+are known, ˜wk ≥ 0, by making the inequality,
+Eξ∼π [fk(ξ)] ≤ 1
+N
+N
+�
+i=1
+fk
+�
+˜ξi
+�
+, ∀k,
+(6)
+strict for at least one feature. Subdominance minimization
+(Ziebart et al., 2022) seeks to produce trajectories that out-
+perform each demonstration by a margin:
+fk(ξ) + mk ≤ fk(˜ξi), ∀i, k,
+(7)
+under the same assumption of known cost weight signs.
+However, since this is often infeasible, the approach in-
+stead minimizes the subdominance, which measures the
+α-weighted violation of this inequality:
+subdomα(ξ, ˜ξ) ≜
+�
+k
+�
+αk
+�
+fk(ξ) − fk(˜ξ)
+�
++ 1
+�
++ , (8)
+where [f(x)]+ ≜ max(f(x), 0) is the hinge function and
+the per-feature margin has been reparameterized as α−1
+k .
+Previous work (Ziebart et al., 2022) has employed subdom-
+inance minimization in conjunction with inverse optimal
+control:
+min
+w min
+α
+N
+�
+i=1
+K
+�
+k=1
+subdomα(ξ∗(w), ˜ξi), where:
+ξ∗(w) = argmin
+ξ
+�
+k
+wkfk(ξ),
+learning the cost function parameters w for the optimal tra-
+jectory ξ∗(w) that minimizes subdominance. One contribu-
+tion of this paper is extending subdominance minimization
+to the more flexible prediction models needed for fairness-
+aware classification that are not directly conditioned on cost
+features or performance/fairness metrics.
+3. Subdominance Minimization for Improved
+Fairness-Aware Classification
+We approach fair classification from an imitation learning
+perspective. We assume vectors of (human-provided) ref-
+erence decisions are available that roughly reflect desired
+fairness-performance trade-offs, but are also noisy. Our
+goal is to construct a fairness-aware classifier that outper-
+forms reference decisions on all performance and fairness
+measures on withheld data as frequently as possible.
+3.1. Superhumanness and Subdominance
+We consider reference decisions ˜y = {˜yj}M
+j=1 that are
+drawn from a human decision-maker or baseline method ˜P,
+on a set of M items, XM×L = {xj}M
+j=1, where L is the num-
+ber of attributes in each of M items xj. Group membership
+Figure 2. A Pareto fron-
+tier for possible ˆPθ (blue)
+optimally trading off pre-
+dictive performance (e.g.,
+inaccuracy) and group
+unfairness. The model-
+produced decision (red
+point) defines dominance
+boundaries (solid red)
+and margin boundaries
+(dashed red), which in-
+cur subdominance (green
+lines) on three examples.
+attributes am from vector a indicate to which group item m
+belongs.
+The predictive performance and fairness of decisions ˆy for
+each item are assessed based on ground truth y and group
+membership a using a set of predictive loss and unfairness
+measures {fk(ˆy, y, a)}.
+Definition 3.1. A fairness-aware classifier is considered γ-
+superhuman for a given set of predictive loss and unfairness
+measures {fk} if its decisions ˆy satisfy:
+P (f (ˆy, y, a) ⪯ f (˜y, y, a)) ≥ γ.
+If strict Pareto dominance is required to be γ-superhuman,
+which is often effectively true for continuous domains, then
+by definition, at most (1 − γ)% of human decision makers
+could be γ-superhuman. However, far fewer than (1 − γ)
+may be γ−superhuman if pairs of human decisions do not
+Pareto dominate one another in either direction (i.e., neither
+f (˜y, y, a) ⪯ f (˜y′, y, a) nor f (˜y′, y, a) ⪯ f (˜y, y, a)
+for pairs of human decisions ˜y and ˜y′). From this perspec-
+tive, any decisions with γ−superhuman performance more
+than (1 − γ)% of the time exceed the performance limit for
+the distribution of human demonstrators.
+Unfortunately, directly maximizing γ is difficult in part
+due to the discontinuity of Pareto dominance (⪯). The
+subdominance (Ziebart et al., 2022) serves as a convex upper
+bound for non-dominance in each metric {fk} and on 1 − γ
+in aggregate:
+subdomk
+αk(ˆy, ˜y, y, a) ≜ [αk (fk(ˆy, y, a) − fk(˜y, y, a)) + 1]+ .
+subdomα(ˆy, ˜y, y, a) ≜
+�
+k
+subdomk
+αk(ˆy, ˜y, y, a).
+(9)
+Given N vectors of reference decisions as demonstrations,
+˜Y = {˜yi}N
+i=1, the subdominance for decision vector ˆy with
+respect to the set of demonstrations is1
+subdomα(ˆy, ˜Y, y, a) = 1
+N
+�
+˜y∈ ˜
+Y
+subdomα(ˆy, ˜y, y, a),
+1For notational simplicity, we assume all demonstrated deci-
+sions ˜y ∈ ˜Y correspond to the same M items represented in X.
+Generalization to unique X for each demonstration is straightfor-
+ward.
+
+Superhuman Fairness
+where ˆyi is the predictions produced by our model for the
+set of items Xi, and ˆY is the set of these prediction sets,
+ˆY = {ˆyi}N
+i=1. The subdominance is illustrated by Figure 2.
+Following concepts from support vector machines (Cortes &
+Vapnik, 1995), reference decisions ˜y that actively constrain
+the predictions ˆy in a particular feature dimension, k, are
+referred to as support vectors and denoted as:
+˜YSVk(ˆy, αk) =
+�
+˜y|αk(fk(ˆy) − fk(˜y)) + 1 ≥ 0
+�
+.
+3.2. Performance-Fairness Subdominance
+Minimization
+We consider probabilistic predictors Pθ : X M → ∆YM
+that make structured predictions over the set of items in
+the most general case, but can also be simplified to make
+conditionally independent decisions for each item.
+Definition 3.2. The minimally subdominant fairness-aware
+classifier ˆPθ has model parameters θ chosen by:
+argmin
+θ
+min
+α⪰0 Eˆy|X∼Pθ
+�
+subdomα,1
+�
+ˆy, ˜Y, y, a
+��
++ λ∥α∥1.
+Hinge loss slopes α ≜ {αk}K
+k=1 are also learned from the
+data during training. For subdominance of kth feature, αk
+indicates the degree of sensitivity to how much the algo-
+rithm fails to sufficiently outperform demonstrations in that
+feature. When αk value is higher, the algorithm chooses that
+feature to minimize subdominance. In our setting, features
+are loss/violation metrics defined to measure the perfor-
+mance/fairness of a set of reference decisions.
+We use the subgradient of subdominance with respect to θ
+and α to update these variables iteratively, and after con-
+vergence, the best learned weights θ∗ are used in the final
+model ˆPθ∗. A commonly used model like logistic regression
+can be used for ˆPθ.
+Theorem 3.3. The gradient of expected subdominance un-
+der Pθ with respect to the set of reference decisions {˜yi}N
+i=1
+is:
+∇θEˆy|X∼ ˆ
+Pθ
+�
+����
+�
+k
+Γk(ˆy, ˜
+Y,y,a)
+�
+��
+�
+min
+αk
+�
+subdomk
+αk
+�
+ˆy, ˜Y, y, a
+�
++ λkαk
+�
+�
+����
+= Eˆy|X∼ ˆ
+Pθ
+� ��
+k
+Γk(ˆy, ˜Y, y, a)
+�
+∇θ log ˆPθ(ˆy|X)
+�
+,
+where the optimal αk for each γk is obtained from:
+αk = argmin
+α(m)
+k
+m such that fk (ˆy) + λ ≤ 1
+m
+m
+�
+j=1
+fk
+�
+˜y(j)�
+,
+using α(j)
+k
+=
+1
+fk(ˆy(j))−fk(˜y(j)) to represent the αk value
+that would make the demonstration with the jth smallest fk
+feature, ˜y(j), a support vector with zero subdominance.
+Using gradient descent, we update the model weights θ
+using an approximation of the gradient based on a set of
+sampled predictions ˆy ∈ ˆY from the model ˆPθ:
+θ ← θ + η
+�
+��
+ˆy∈ ˆ
+Y
+��
+k
+Γk(ˆy, ˜Y, y, a)
+�
+∇θ log ˆPθ(ˆy|X)
+�
+� ,
+We show the steps required for the training of our model in
+Algorithm 1. Reference decisions {˜yi}N
+i=1 from a human
+decision-maker or baseline method ˜P are provided as input
+to the algorithm.
+θ is set to an initial value. In each
+iteration of the algorithm, we first sample a set of model
+predictions {ˆyi}N
+i=1 from ˆPθ(.|Xi) for the matching items
+used for reference decisions {˜yi}N
+i=1. We then find the new
+θ (and α) based on the algorithms discussed in Theorem
+3.3.
+Algorithm 1: Subdominance policy gradient opti-
+mization
+Draw N set of reference decisions {˜yi}N
+i=1 from a
+human decision-maker or baseline method ˜P.
+Initialize: θ ← θ0;
+while θ not converged do
+Sample model predictions {ˆyi}N
+i=1 from
+ˆPθ(.|Xi) for the matching items used in
+reference decisions {˜yi}N
+i=1;
+for k ∈ {1, ..., K} do
+Sort reference decisions {˜yi}N
+i=1 in
+ascending order based on their kth feature
+value fk(˜yi): {˜y(j)}N
+j=1;
+Compute α(j)
+k
+=
+1
+fk(˜y(j))−fk(ˆy(j));
+αk = argmin
+α(m)
+k
+m
+such that fk
+�ˆy(j)�
+≤ 1
+m
+�m
+j=1 fk
+�˜y(j)�
+;
+Compute Γk(ˆyi, ˜Y, y, a);
+θ ← θ +
+η
+N
+�
+i
+��
+k Γk(ˆyi, ˜Y, y, a)
+�
+∇θ log ˆPθ(ˆyi|Xi);
+3.3. Generalization Bounds
+With a small effort, we extend the generalization bounds
+based on support vectors developed for inverse optimal con-
+trol subdominance minimization (Ziebart et al., 2022).
+Theorem 3.4.
+A classifier
+ˆPθ
+trained to minimize
+subdomα (ˆy, ˜yi) on a set of N iid reference decisions
+has the support vector set
+��
+ˆy:Pθ(ˆy|X)>0 ˜YSVk (ˆy, αk)
+�
+defined by the union of support vectors for any decision
+with support under ˆPθ. Such a classifier is on average γ-
+superhuman on the population distribution with: γ = 1−
+1
+N
+����K
+k=1
+�
+ˆy:Pθ(ˆy|X)>0 ˜YS Vk (ˆy, αk)
+���.
+This generalization bound requires overfitting to the training
+
+Superhuman Fairness
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+DP
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+EqOdds
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+D.PRP
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+PRP
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+D.DP
+0.35
+0.40
+0.45
+0.50
+0.55
+Prediction error
+1/
+DP
+1/
+error
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+D.EqOdds
+0.35
+0.40
+0.45
+0.50
+0.55
+Prediction error
+1/
+EqOdds
+1/
+error
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+D.PRP
+0.35
+0.40
+0.45
+0.50
+0.55
+Prediction error
+1/
+PRP
+1/
+error
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+Figure 3. Prediction error versus difference of: Demographic Parity (D.DP), Equalized Odds (D.EqOdds) and Predictive Rate Parity
+(D.PR) on test data using noiseless training data (ϵ = 0) for Adult (top row) and COMPAS (bottom row) datasets.
+data so that the ˆPθ has restricted support (i.e., ˆPθ(ˆy|X) = 0
+for many ˆy) or becomes deterministic.
+4. Experiments
+The goal of our approach is to produce a fairness-aware
+prediction method that outperforms reference (human) de-
+cisions on multiple fairness/performance measures. In this
+section, we discuss our experimental design to synthesize
+reference decisions with varying levels of noise, evaluate
+our method, and provide comparison baselines.
+4.1. Training and Testing Dataset Construction
+To emulate human decision-making with various levels of
+noise, we add noise to the ground truth data of benchmark
+fairness datasets and apply fair learning methods over re-
+peated randomized dataset splits. We describe this process
+in detail in the following section.
+Datasets
+We perform experiments on two benchmark fair-
+ness datasets:
+• UCI Adult dataset (Dheeru & Karra Taniskidou, 2017)
+considers predicting whether a household’s income is
+higher than $50K/yr based on census data. Group mem-
+bership is based on gender. The dataset consists of 45,222
+items.
+• COMPAS dataset (Larson et al., 2016) considers predict-
+ing recidivism with group membership based on race. It
+consists of 6,172 examples.
+Partitioning the data
+We first split entire dataset
+randomly into a disjoint train (train-sh) and test
+(test-sh) set of equal size. The test set (test-sh) is
+entirely withheld from the training procedure and ultimately
+used solely for evaluation. To produce each demonstration
+(a vector of reference decisions), we split the (train-sh)
+set, randomly into a disjoint train (train-pp) and test
+(test-pp) set of equal size.
+Noise insertion
+We randomly flip ϵ% of the ground truth
+labels y and group membership attributes a to add noise to
+our demonstration-producing process.
+Fair classifier ˜P:
+Using the noisy data, we provide ex-
+isting fairness-aware methods with labeled train-pp
+data and unlabeled test-pp to produce decisions on the
+test-pp data as demonstrations ˜y. Specifically, we em-
+ploy:
+• The Post-processing method of Hardt et al. (2016), which
+aims to reduce both prediction error and {demographic
+parity or equalized odds} at the same time. We use de-
+mographic parity as the fairness constraint. We produce
+demonstrations using this method for Adult dataset.
+• Robust fairness for logloss-based classification (Rezaei
+et al., 2020) employs distributional robustness to match
+target fairness constraint(s) while robustly minimizing the
+log loss. We use equalized odds as the fairness constraint.
+
+Superhuman Fairness
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.DP
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+Prediction error
+1/
+DP
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+Prediction error
+1/
+EqOdds
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+D.PRP
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+Prediction error
+1/
+PRP
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+D.DP
+0.35
+0.40
+0.45
+0.50
+0.55
+Prediction error
+1/
+DP
+1/
+error
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+D.EqOdds
+0.35
+0.40
+0.45
+0.50
+0.55
+Prediction error
+1/
+EqOdds
+1/
+error
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+D.PRP
+0.35
+0.40
+0.45
+0.50
+0.55
+Prediction error
+1/
+PRP
+1/
+error
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+Figure 4. Experimental results on the Adult and COMPAS datasets with noisy demonstrations (ϵ = 0.2). Margin boundaries are shown
+with dotted red lines. Each plot shows the relationships between two features.
+We employ this method to produce demonstrations for
+COMPAS dataset.
+We repeat the process of partitioning train-sh N =
+50 times to create randomized partitions of train-pp
+and test-pp and then produce a set of demonstrations
+{˜y}50
+i=1.
+4.2. Evaluation Metrics and Baselines
+Predictive Performance and Fairness Measures
+Our
+focus for evaluation is on outperforming demonstrations in
+multiple fairness and performance measures. We use K = 4
+measures: inaccuracy (Prediction error), difference
+from demographic parity (D.DP), difference from equalized
+odds (D.EqOdds), difference from predictive rate parity
+(D.PRP).
+Baseline methods
+As baseline comparisons, we train five
+different models on the entire train set (train-sh) and
+then evaluate them on the withheld test data (test-sh):
+• The Post-processing model of (Hardt et al., 2016)
+with demographic parity as the fairness constraint
+(post proc dp).
+• The Post-processing model of (Hardt et al., 2016)
+with
+equalized
+odds
+as
+the
+fairness
+constraint
+(post proc eqodds).
+• The Robust Fair-logloss model of (Rezaei et al., 2020)
+with demographic parity as the fairness constraint
+(fair logloss dp).
+• The Robust Fair-logloss model of (Rezaei et al.,
+2020)
+equalized
+odds
+as
+the
+fairness
+constraint
+(fair logloss eqodds).
+• The Multiple Fairness Optimization framework of Hsu
+et al. (2022) which is designed to satisfy three conflict-
+ing fairness metrics (demographic parity, equalized odds
+and predictive rate parity) to the best extent possible
+(MFOpt).
+Hinge Loss Slopes
+As discussed previously, αk value cor-
+responds to the hinge loss slope, which defines by how far
+a produced decision does not sufficiently outperform the
+demonstrations on the kth feature. When the αk is large, the
+model chooses heavily weights support vector reference de-
+cisions for that particular k when minimizing subdominance.
+We report these values in our experiments.
+4.3. Superhuman Model Specification and Updates
+We use a logistic regression model Pθ0 with first-order mo-
+ments feature function, φ(y, x) = [x1y, x2y, . . . xmy]⊤,
+and weights θ applied independently on each item as our
+decision model. During the training process, we update the
+model parameter θ to reduce subdominance.
+Sample from Model ˆPθ
+In each iteration of the algorithm,
+we first sample prediction vectors {ˆyi}N
+i=1 from ˆPθ(.|Xi)
+
+Superhuman Fairness
+Table 1. Experimental results on noise-free datasets, along with the αk values learned for each feature in subdominance minimization.
+Method
+Dataset
+Adult
+COMPAS
+Prediction error
+DP diff
+EqOdds diff
+PRP diff
+Prediction error
+DP diff
+EqOdds diff
+PRP diff
+αk
+62.62
+35.93
+6.46
+4.98
+82.5
+4.27
+3.15
+12.72
+γ-superhuman
+98%
+94%
+100%
+100%
+100%
+100%
+100%
+100%
+MinSub-Fair (ours)
+0.210884
+0.025934
+0.006690
+0.183138
+0.366806
+0.040560
+0.124683
+0.171177
+MFOpt
+0.195696
+0.063152
+0.077549
+0.209199
+0.434743
+0.005830
+0.069519
+0.161629
+post proc dp
+0.212481
+0.030853
+0.220357
+0.398278
+0.345964
+0.010383
+0.077020
+0.173689
+post proc eqodds
+0.213873
+0.118802
+0.007238
+0.313458
+0.363395
+0.041243
+0.060244
+0.151040
+fair logloss dp
+0.281194
+0.004269
+0.047962
+0.124797
+0.467610
+0.000225
+0.071392
+0.172418
+fair logloss eqodds
+0.254060
+0.153543
+0.030141
+0.116579
+0.451496
+0.103093
+0.029085
+0.124447
+for the matching items used in demonstrations {˜yi}N
+i=1. In
+the implementation, to produce the ith sample, we look up
+the indices of the items used in ˜yi, which constructs item set
+Xi. Now we make predictions using our model on this item
+set ˆPθ(.|Xi). The model produces a probability distribution
+for each item which can be sampled and used as a prediction
+{ˆyi}N
+i=1.
+Update model parameters θ
+We update θ until conver-
+gence using Algorithm 1. For our logistic regression model,
+the gradient is:
+∇θ log ˆPθ(ˆy|X) = φ(ˆy, X) − Eˆy|X∼ ˆ
+Pθ [φ(ˆy, X)] ,
+where φ denotes the feature function and φ(ˆy, X) =
+�M
+m=1 φ(ˆym, xm) is the corresponding feature function
+for the ith set of reference decisions.
+4.4. Experimental Results
+After training each model, e.g., obtaining the best
+model weight θ∗ from the training data (train-sh)
+for superhuman, we evaluate each on unseen test data
+(test-sh). We employ hard predictions (i.e., the most
+probable label) using our approach at time time rather than
+randomly sampling.
+Noise-free reference decisions
+Our first set of experi-
+ments considers learning from reference decisions with no
+added noise. The results are shown in Figure 3. We ob-
+serve that our approach outperforms demonstrations in all
+fairness metrics and shows comparable performance in accu-
+racy. The (post proc dp) performs almost as an average
+of demonstrations in all dimensions, hence our approach
+can outperform it in all fairness metrics. In comparison
+to (post proc dp), our approach can outperform in all
+fairness metrics but is slightly worse in prediction error.
+We show the experiment results along with αk values in
+Table 1. Note that the margin boundaries (dotted red lines)
+in Figure 3 are equal to
+1
+αk for feature k, hence there is re-
+verse relation between αk and margin boundary for feature
+k. We observe larger values of αk for prediction error and
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Noise Ratio
+0.86
+0.88
+0.90
+0.92
+0.94
+0.96
+0.98
+1.00
+-Superhumn
+Adult
+Predictive value difference
+Equalized odds difference
+Demographic parity difference
+ZeroOne
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+Noise Ratio
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+-Superhumn
+Adult
+Predictive value difference
+Equalized odds difference
+Demographic parity difference
+ZeroOne
+Figure 5. The relationship between the ratio of augmented noise
+in the label and the protected attribute of reference decisions
+produced by post-processing (upper) and fair-logloss (lower)
+and achieving γ-superhuman performance in our approach.
+demographic parity difference. The reason is that these fea-
+tures are already optimized in demonstrations and our model
+has to increase αk values for those features to sufficiently
+outperform them.
+Noisy reference decisions
+In our second set of experi-
+ments, we introduce significant amounts of noise (ϵ = 0.2)
+into our reference decisions. The results for these experi-
+ments are shown in Figure 4. We observe that in the case of
+learning from noisy demonstrations, our approach still out-
+performs the reference decisions. The main difference here
+is that due to the noisy setting, demonstrations have worse
+prediction error but regardless of this issue, our approach
+
+Superhuman Fairness
+Table 2. Experimental results on datasets with noisy demonstrations, along with the αk values learned for each feature.
+Method
+Dataset
+Adult
+COMPAS
+Prediction error
+DP diff
+EqOdds diff
+PRP diff
+Prediction error
+DP diff
+EqOdds diff
+PRP diff
+αk
+29.63
+10.77
+5.83
+13.42
+29.33
+4.51
+3.34
+57.74
+γ-superhuman
+100%
+100%
+100%
+100%
+100%
+100%
+100%
+98%
+MinSub-Fair (ours)
+0.202735
+0.030961
+0.009263
+0.176004
+0.359985
+0.031962
+0.036680
+0.172286
+MFOpt
+0.195696
+0.063152
+0.077549
+0.209199
+0.459731
+0.091892
+0.039745
+0.153257
+post proc dp
+0.225462
+0.064232
+0.237852
+0.400427
+0.353164
+0.087889
+0.088414
+0.160538
+post proc eqodds
+0.224561
+0.103158
+0.010552
+0.310070
+0.351269
+0.144190
+0.158372
+0.148493
+fair logloss dp
+0.285549
+0.007576
+0.057659
+0.115751
+0.484620
+0.005309
+0.145502
+0.183193
+fair logloss eqodds
+0.254577
+0.147932
+0.012778
+0.118041
+0.487025
+0.127163
+0.011918
+0.153869
+Table 3. Percentage of reference demonstrations that each method outperforms in all prediction/fairness measures.
+Method
+Adult(ϵ = 0.0)
+Adult(ϵ = 0.2)
+COMPAS(ϵ = 0.0)
+COMPAS(ϵ = 0.2)
+MinSub-Fair (ours)
+96%
+100%
+100%
+98%
+MFOpt
+42%
+0%
+18%
+18%
+post proc dp
+16%
+86%
+100%
+80%
+post proc eqodds
+0%
+66%
+100%
+88%
+fair logloss dp
+0%
+0%
+0%
+0%
+fair logloss eqodds
+0%
+0%
+0%
+0%
+still can achieve a competitive prediction error. We show
+the experimental results along with αk values in Table 2.
+Relationship of noise to superhuman performance
+We
+also evaluate the relationship between the amount of aug-
+mented noise in the label and protected attribute of demon-
+strations, with achieving γ-superhuman performance in our
+approach. As shown in Figure 5, with slightly increasing the
+amount of noise in demonstrations, our approach can outper-
+form 100% of demonstrations and reach to 1-superhuman
+performance. In Table 3 we show the percentage of demon-
+strations that each method can outperform across all predic-
+tion/fairness measures (i.e., the γ−superhuman value).
+5. Conclusions
+In this paper, we introduce superhuman fairness, an ap-
+proach to fairness-aware classifier construction based on im-
+itation learning. Our approach avoids explicit performance-
+fairness trade-off specification or elicitation. Instead, it
+seeks to unambiguously outperform human decisions across
+multiple performance and fairness measures with maximal
+frequency. We develop a general framework for pursuing
+this based on subdominance minimization (Ziebart et al.,
+2022) and policy gradient optimization methods (Sutton
+& Barto, 2018) that enable a broad class of probabilistic
+fairness-aware classifiers to be learned. Our experimental
+results show the effectiveness of our approach in outper-
+forming synthetic decisions corrupted by small amounts of
+label and group-membership noise when evaluated using
+multiple fairness criteria combined with predictive accuracy.
+Societal impacts
+By design, our approach has the po-
+tential to identify fairness-aware decision-making tasks in
+which human decisions can frequently be outperformed by
+a learned classifier on a set of provided performance and
+fairness measures. This has the potential to facilitate a tran-
+sition from manual to automated decisions that are preferred
+by all interested stakeholders, so long as their interests are
+reflected in some of those measures. However, our approach
+has limitations. First, when performance-fairness tradeoffs
+can either be fully specified (e.g., based on first principles)
+or effectively elicited, fairness-aware classifiers optimized
+for those trade-offs should produce better results than our
+approach, which operates under greater uncertainty cast by
+the noisiness of human decisions. Second, if target fair-
+ness concepts lie outside the set of metrics we consider,
+our resulting fairness-aware classifier will be oblivious to
+them. Third, our approach assumes human-demonstrated
+decision are well-intentioned, noisy reflections of desired
+performance-fairness trade-offs. If this is not the case, then
+our methods could succeed in outperforming them across all
+fairness measures, but still not provide an adequate degree
+of fairness.
+Future directions
+We have conducted experiments with
+a relatively small number of performance/fairness measures
+using a simplistic logistic regression model. Scaling our ap-
+proach to much larger numbers of measures and classi���ers
+with more expressive representations are both of great inter-
+est. Additionally, we plan to pursue experimental validation
+using human-provided fairness-aware decisions in addition
+to the synthetically-produced decisions we consider in this
+paper.
+
+Superhuman Fairness
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+Superhuman Fairness
+A. Proofs of Theorems
+Proof of Theorem 3.3. The gradient of the training objective with respect to model parameters θ is:
+∇θEˆy|X∼ ˆ
+Pθ
+�
+����
+�
+k
+Γk(ˆy, ˜
+Y,y,a)
+�
+��
+�
+min
+αk
+�
+subdomk
+αk
+�
+ˆy, ˜Y, y, a
+�
++ λkαk
+�
+�
+���� = Eˆy|X∼ ˆ
+Pθ
+� ��
+k
+Γk(ˆy, ˜Y, y, a)
+�
+∇θ log ˆPθ(ˆy|X)
+�
+,
+which follows directly from a property of gradients of logs of function:
+∇θ log ˆP(ˆy|X) =
+1
+ˆP(ˆy|X)
+∇θˆP(ˆy|X) =⇒ ∇θˆPθ(ˆy|X) = ˆP(ˆy|X)∇θ log ˆP(ˆy|X).
+(10)
+We note that this is a well-known approach employed by policy-gradient methods in reinforcement learning (Sutton & Barto,
+2018).
+Next, we consider how to obtain the α−minimized subdominance for a particular tuple (ˆy, ˜Y,y,a), Γk
+�
+ˆy, ˜Y, y, a
+�
+=
+minαk
+�
+subdomk
+αk
+�
+ˆy, ˜Y, y, a
+�
++ λkαk
+�
+, analytically.
+First, we note that subdomk
+αk
+�
+ˆy, ˜Y, y, a
+�
++ λkαk is comprised of hinged linear functions of αk, making it a convex
+and piece-wise linear function of αk. This has two important implications: (1) any point of the function for which the
+subgradient includes 0 is a global minimum of the function (Boyd & Vandenberghe, 2004); (2) an optimum must exist at a
+corner of the function: αk = 0 or where one of the hinge functions becomes active:
+αk(fk(ˆyi) − fk(˜yi)) + 1 = 0 =⇒ αk =
+1
+fk(˜yi) − fk(ˆyi).
+(11)
+The subgradient for the jth of these points (ordered by fk value from smallest to largest and denoted fk(˜y(j)) for the
+demonstration) is:
+∂αk subdomk
+αk
+�
+ˆy, ˜Y, y, a
+� ���
+αk=(fk(ˆy)−fk(˜y(j)))−1 = ∂αk
+�
+1
+N
+j
+�
+i=1
+�
+αk
+�
+fk(ˆy) − fk(˜y(i))
+�
++ 1
+�
++
++ λαk
+�
+= λ + 1
+N
+j−1
+�
+i=1
+�
+fk(ˆy) − fk(˜y(i))
+�
++
+�
+0, fk(ˆy) − fk(˜y(j))
+�
+,
+where the final bracketed expression indicates the range of values added to the constant value preceding it.
+The smallest j for which the largest value in this range is positive must contain the 0 in its corresponding range, and is thus
+the provides the j value for the optimal αk value.
+Proof of Theorem 3.4. We extend the leave-one-out generalization bound of Ziebart et al. (2022) by considering the set of
+reference decisions that are support vectors for any learner decisions with non-zero probability. For the remaining reference
+decisions that are not part of this set, removing them from the training set would not change the optimal model choice
+and thus contribute zero error to the leave-one-out cross validation error, which is an almost unbiased estimate of the
+generalization error (Vapnik & Chapelle, 2000).
+B. Additional Results
+In the main paper, we only included plots that show the relationship of a fairness metric with prediction error. To show the
+relation between each pair of fairness metrics, in Figures 6 and 7 we show the remaining plots removed from Figures 3 and
+4 respectively.
+
+Superhuman Fairness
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+1/
+EqOdds
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+D.PRP
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+1/
+PRP
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+D.PRP
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+1/
+PRP
+1/
+EqOdds
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+D.EqOdds
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+D.DP
+1/
+EqOdds
+1/
+DP
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+D.PRP
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+D.DP
+1/
+PRP
+1/
+DP
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+D.PRP
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+D.EqOdds
+1/
+PRP
+1/
+EqOdds
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+Figure 6. The trade-off between each pair of: difference of Demographic Parity (D.DP), Equalized Odds (D.EqOdds) and Predictive
+Rate Parity (D.PR) on test data using noiseless training data (ϵ = 0) for Adult (top row) and COMPAS (bottom row) datasets.
+B.1. Experiment with more measures
+Since our approach is flexible enough to accept wide range of fairness/performance measures, we extend the experiment on
+Adult to K = 5 features. In this experiment we use Demographic Parity (D.DP), Equalized Odds (D.EqOdds), False
+Negative Rate (D.FNR), False Positive Rate (D.FPR) and Prediction Error as the features to outperform reference decisions
+on. The results are shown in Figure 8.
+
+Superhuman Fairness
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.DP
+1/
+EqOdds
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+D.PRP
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.DP
+1/
+PRP
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+D.PRP
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+1/
+PRP
+1/
+EqOdds
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+D.EqOdds
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+D.DP
+1/
+EqOdds
+1/
+DP
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+D.PRP
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+D.DP
+1/
+PRP
+1/
+DP
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.15
+0.20
+0.25
+0.30
+0.35
+D.PRP
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+D.EqOdds
+1/
+PRP
+1/
+EqOdds
+COMPAS
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+Figure 7. The trade-off between each pair of: difference of Demographic Parity (D.DP), Equalized Odds (D.EqOdds) and Predictive
+Rate Parity (D.PR) on test data using noiseless training data (ϵ = 0.2) for Adult (top row) and COMPAS (bottom row) datasets.
+
+Superhuman Fairness
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+DP
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+EqOdds
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.FNR
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+FNR
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.FPR
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Prediction error
+1/
+FPR
+1/
+error
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+1/
+EqOdds
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.FNR
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+1/
+FNR
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.FPR
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+D.DP
+1/
+FPR
+1/
+DP
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.FNR
+1/
+EqOdds
+1/
+FNR
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.FPR
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.FNR
+1/
+FPR
+1/
+FNR
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+D.EqOdds
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+D.FPR
+1/
+EqOdds
+1/
+FPR
+Adult
+fair_logloss_eqodds
+fair_logloss_dp
+post_proc_eqodds
+post_proc_dp
+MFOpt
+post_proc_demos
+superhuman_train
+superhuman_test
+Figure 8. The trade-off between each pair of: difference of Demographic Parity (D.DP), Equalized Odds (D.EqOdds), False Negative
+Rate (D.FNR), False Positive Rate (D.FPR) and Prediction Error on test data using noiseless training data (ϵ = 0) for Adult dataset.
+

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+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND
+BROWNIAN GRAPHON LIMITS
+TH´EO LENOIR
+Abstract. We consider large uniform labeled random graphs in different classes with
+few induced P4 (P4 is the graph consisting of a single line of 4 vertices), which generalize
+the case of cographs. Our main result is the convergence to a Brownian limit object in the
+space of graphons. We also obtain an equivalent of the number of graphs of size n in the
+different classes. Finally we estimate the expected number of induced graphs isomorphic
+to a fixed graph H for a large variety of graphs H.
+Our proofs rely on tree encoding of graphs. We then use mainly combinatorial argu-
+ments, including the symbolic method and singularity analysis.
+1. Introduction
+1.1. Motivation. Random graphs are one of the most studied objects in probability theory
+and in combinatorics. A natural question is to investigate the scaling limits of a uniformly
+chosen graph in a given family (an important example for this paper are the cographs).
+Cographs have been studied since the seventies by various authors, especially for their
+algorithmic properties: recognizing cographs can be solved in linear time [4, 6, 12], and
+many hard problems can be solved in polynomial time for cographs. Several equivalent
+definitions exists of the class of cographs exists, here are two important ones:
+• A graph is a cograph if and only if it has no induced P4 (a line of 4 vertices).
+• The class of cograph is the smallest class containing every graph reduced to a single
+vertex, and stable by union and by join1.
+Simultaneously in [1] and [21], the authors exhibit a Brownian limiting object for a
+uniform cograph, called the Brownian cographon, which can be explicitly constructed from
+the Brownian excursion and a parameter p ∈ [0, 1].
+The convergence holds in distribution in the sense of graphons. Introduced in [2], graphon
+is a well-established topic in graph theory but their probabilistic counterpart is more recent.
+Graphon convergence can be seen as the convergence of the renormalized adjacency matrix
+for the so-called cut metric (a good reference on graphon theory is [19]).
+One natural question to go further than the case of cographs is to study more complicated
+classes with, in some specific sense, few P4’s. A natural question is to study classes of graphs
+to which some algorithmic properties of cographs extend. Several classes characterized by
+properties of their induced P4’s have thus been considered in the graph theory literature.
+1the join of two graphs (G, H) is the graph obtained by adding an edge between every pair of vertices
+(g, h) ∈ G × H
+1
+arXiv:2301.13607v1  [math.PR]  31 Jan 2023
+
+2
+TH´EO LENOIR
+The classes we will focus on here are the following: P4-reducible graphs [15,18], P4-sparse
+graphs [13,17] P4-lite graphs [14], P4-extendible graphs [16] and P4-tidy graphs [10] which
+can all be seen as classes defined by some constraints on the induced P4’s. All these classes
+will be defined precisely in Section 3. The inclusion relations between these classes are
+sketched in Figure 1.
+P4-tidy
+P4-lite
+P4-sparse
+P4-extendible
+P4-reducible
+P4-free
+(=cographs)
+Figure 1. Inclusion relations between the different classes of graphs
+To our knowledge, these different classes have not been studied from a probabilistic point
+of view. The main aim of this paper is to prove a result of universality of the Brownian
+cographon: for every class previously mentioned, a random graph will converge towards the
+Brownian cographon of parameter 1
+2 (the rigorous construction is given by [1, Definition
+10]). An intermediate result is the asymptotic enumeration of each of these classes, which
+was unknown up to now.
+1.2. Main results. For a finite graph G, let WG be the embedding of the finite graph G
+in the set of graphon (the formal construction will be recalled in Definition 6.2). Our main
+result is:
+Theorem 1.1. Let G(n) be a uniform graph of size n taken uniformly at random in one
+of the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible or P4-reducible. The
+following convergence in distribution holds in the sense of graphons:
+WG(n)
+n→∞
+−→ W
+1
+2
+where W
+1
+2 is the Brownian cographon of parameter 1
+2.
+Graphon convergence is equivalent to the joint convergence of subgraphs density. Di-
+aconis and Janson extended this criterion in [7] to random graphs: the convergence of a
+family (H(n))n≥1 of random graphs is characterized by the convergence in distribution of
+OccH(n)(H)
+nk
+for every positive integer k and for every finite graph H of size k, where OccG(H)
+is the number of induced subgraphs of G isomorphic to H. All the necessary material on
+graphon will be recalled at the beginning of Section 6.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+3
+Figure 2 shows an example of the adjacency matrix of a random P4-extensible graph of
+size 200. This picture gives an idea of what a realization of the Brownian cographon could
+look like.
+Figure 2. The adjacency matrix of a random P4-extensible graph of size
+200, simulation by Micka¨el Maazoun
+In the course of proving Theorem 1.1, we get an equivalent of the number of graphs in
+the different classes.
+Theorem 1.2. The number of labeled P4-sparse, P4-tidy, P4-lite, P4-extendible, P4-reducible
+or the number of P4-free graphs of size n is asymptotically equivalent to
+C
+n!
+Rnn
+3
+2 ,
+for some R, C > 0, depending on the class.
+We can compute with arbitrary precision the numerical values of R and C (see Sec-
+tion 4.2). All the numerical values of R and C vary according to each class which confirms
+that all these classes are significantly different.
+Theorem 1.1 provides a precise estimation of OccH(G(n)) for every cograph H. But for
+every graph H which is not a cograph, the only information given by the convergence in
+the sense of graphon is that the number of induced H in G(n) is typically o(n|H|). Quite
+unexpectedly, thanks to the tools developed to prove Theorem 1.1, we are able to estimate
+the expected number of induced subgraphs isomorphic to a specific class of graphs H in
+G(n): the graphs that are called ”prime” for the modular decomposition (see Definition 2.8).
+Theorem 1.3. Let G(n) be a uniform graph of size n taken uniformly at random in one of
+the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible or P4-reducible. Let H be
+a prime graph, denote by OccH(G(n)) the number of labeled subgraphs of G(n) isomorphic
+to H.
+
+回4
+TH´EO LENOIR
+Then there exists KH ≥ 0 such that:
+E[OccH(G(n))] ∼
+�
+�
+�
+KHn
+3
+2
+if H verifies condition (A)
+KHn
+otherwise
+where (A) is defined p.38 and constant KH is given in Theorem 6.9.
+This results follows from Theorem 6.9 which is stated in a more general setting. The
+condition (A) depends on the class of graphs, checking if H verifies condition (A) and if
+KH is positive is quite straightforward.
+To make things more concrete, let us apply Theorem 1.3 to the example of H = P4. We
+can check that for each class P4 does not verify condition (A). Thus a uniform random
+graph contains in average a linear number of induced P4, while Theorem 1.1 only implies
+that this number is o(n4). The different numerical values of KP4 are explicitly computed
+p.41, and happen to take different values for each class. For each class, the graph called
+”bull” (see Fig. 7) does not verify condition (A). Thus a uniform random graph contains
+in average a number of induced bulls growing as n3/2, while Theorem 1.1 only implies that
+this number is o(n5). However, for non prime graphs H, the behavior of the expected
+value of induced subgraphs of G(n) isomorphic to H is not well-understood, which leads to
+interesting open questions.
+1.3. Proof strategy. The proof is essentially combinatorial and is based on modular de-
+composition, which allows to encode a graph with a decorated tree. Modular decomposition
+is a standard tool in graph theory (it was introduced in the 60’s by Gallai [9]) but to our
+knowledge it has been very little used in the context of random graphs. In this paper we
+introduce an enriched modular decomposition which enables us to obtain exact enumer-
+ations for a large family of graph classes. The five classes mentioned before fit in this
+framework. We exploit those enumerative results with tools from analytic combinatorics
+to get asymptotic estimates in order to prove Theorem 1.2.
+The more technical part of the proof is, for every finite graph H, to estimate the number
+of induced subgraphs of G(n) isomorphic to H. The enriched modular decomposition allows
+us to count the number of graphs with a specific induced subgraph H. Again asymptotics
+are derived with tools from combinatorics to prove Theorem 1.1 and Theorem 1.3.
+1.4. Outline of the paper.
+• In Section 2 we define the encoding of graphs with trees, the modular decomposition
+and the enriched modular decomposition which will be used throughout the different
+proofs.
+• Section 3 presents the necessary material on the different classes of graphs studied:
+results are already widely known, most of them are quoted from the litterature and
+reformulated to suit our enriched modular decomposition.
+• Sections 4 and 5 are about calculating generating series related to our graph classes:
+in Section 4 we prove Theorem 1.2 and Section 5 deals with the generating series
+of graphs with a given induced subgraph.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+5
+• Section 6 presents the necessary material on graphons, and the proofs of Theo-
+rem 1.1 and Theorem 1.3.
+2. Modular decomposition of graphs: old and new
+2.1. Labeled graphs. In the following all the graphs considered are simple and finite.
+Each time a graph G is defined, we denote by V its set of vertices and E its set of edges.
+Whenever there is an ambiguity, we denote by VG (resp. EG) the set of vertices (resp. edges)
+of G.
+Definition 2.1. We say that G = (V, E) is a weakly-labeled graph if every element of V
+has a distinct label in N and that G = (V, E) is a labeled graph if every element of V has
+a distinct label in {1, . . . , |V |}.
+The size of a graph G, denoted by |G|, is its number of vertices.
+The minimum of a graph G, denoted min(G), is the minimal label of its vertices.
+In the following, every graph will be labeled, otherwise we will mention explicitly that
+the graph is weakly-labeled.
+Remark. We do not identify a vertex with its label. A vertex of label i will be denoted vi.
+The label of a vertex v will be denoted ℓ(v).
+Definition 2.2. For any weakly-labeled object (graph or tree) of size n, we call reduction
+the operation that reduces its labels to the set {1, . . . , n} while preserving the relative order
+of the labels.
+For example if G labels 2, 4, 12, 63 then the reduced version of G is a copy of G in which
+2, 4, 12, 63 are respectively replaced by 1, 2, 3, 4.
+2.2. Encoding graphs with trees.
+Definition 2.3. Let G be a graph of size n and H1, . . . , Hn be weakly-labeled graphs such
+that no label is given to two distinct vertices of �n
+i=1 Hi. The graph G[H1, . . . , Hn] = (V, E)
+is the graph whose set of vertices is V = �n
+i=1 VHi and such that:
+• for every i ∈ {1, . . . , n} and every pair (v, v′) ∈ V 2
+Hi, {v, v′} ∈ E if and only if
+{v, v′} ∈ EHi;
+• For every (i, j) ∈ {1, . . . , n} with i ̸= j, and every pair (v, v′) ∈ VHi ×VHj, {v, v′} ∈
+E if and only if {vi, vj} ∈ EG.
+Notation. In Definition 2.3 we will use the shortcut ⊕ for the complete graph of size n.
+Thus ⊕[H1, . . . , Hn] is the graph obtained from copies of H1, . . . , Hn in which for every
+i ̸= j every vertex of Hi is connected to every vertex of Hj. This graph is called the join
+of H1, . . . , Hn
+We use the shortcut ⊖ for the empty graph of size n. Thus ⊖[H1, . . . , Hn] is the graph
+given by the disjoint union of H1, . . . , Hn This graph is called the union of H1, . . . , Hn.
+This construction allows us to transform non-plane labeled trees with internal nodes
+decorated with graphs, ⊕ and ⊖ into graphs.
+
+6
+TH´EO LENOIR
+Definition 2.4. Let T0 be the set of rooted non-plane trees whose leaves have distinct labels
+in N and whose internal nodes carry decorations satisfying the following constraints:
+• internal nodes are decorated with ⊕, ⊖ or a graph;
+• If a node is decorated with some graph G then |G| ≥ 2 and this node has |G|
+children. If a node is decorated with ⊕ or ⊖ then it has at least 2 children.
+A tree t ∈ T0 is called a substitution tree if the labels of its leaves are in {1, . . . , |t|}.
+We call linear the internal nodes decorated with ⊕ or ⊖ and non-linear the other ones.
+Notation. For a non-plane rooted tree t, and an internal node v of t, let tv be the multiset
+of trees attached to v and let t[v] be the non-plane tree rooted at v containing only the
+descendants of v in t.
+Convention. We only consider non-plane trees. However it is sometimes convenient to
+order the subtrees of a given node. The convention is that for some v in a tree t the trees
+of tv are ordered according to their minimal leaf labels.
+Definition 2.5. Let t be an element of T0, the weakly-labeled graph Graph(t) is inductively
+defined as follows:
+• if t is reduced to a single leaf labeled j, Graph(t) is the graph reduced to a single
+vertex labeled j;
+• otherwise, the root r of t is decorated with a graph H, and
+Graph(t) = H[Graph(t1), . . . , Graph(t|H|)]
+where ti is the i-th tree of tr.
+1
+2
+7
+5
+8
+3
+4
+6
+9
+Root
+1
+9
+6
+2
+3
+8
+7
+5
+4
+t0
+Graph(t0)
+Figure 3. A substitution tree t0 and the corresponding graph Graph(t0)
+Note that if t is a substitution tree then Graph(t) is a labeled graph.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+7
+The following simple Lemma is essential to the study of the enriched decomposition of
+graphs introduced in Section 2.4.
+Lemma 2.6. Let t be a substitution tree such that the decoration of the root of t (resp. its
+complementary) is connected. Then Graph(t) (resp. its complementary) is connected.
+Proof. Since both cases are similar, we only deal with the case of a connected decoration.
+Let r be the root of t, H its decoration and k the size of H. Let w1, . . . , wk be vertices of
+Graph(t) such that for each i ∈ {1, . . . , k} there is a leaf labeled ℓ(wi) in the i-th tree of tr.
+Since the unlabeled graph induced by {wi | 1 ≤ i ≤ k} is isomorphic to H, it is connected.
+Let C be the connected component of Graph(t) containing all wi’s. Note that for every
+vertex v of Graph(t), there exists p ∈ {1, . . . k} such that the leaf labeled ℓ(v) belongs to
+the p-th tree of tr. Since H is connected and of size at least 2, there exists q ̸= p such that
+the vertices of label q and p are connected by an edge in H. Thus v and wq are connected
+by an edge in Graph(t), which means that v ∈ C. This implies that C = V , thus Graph(t)
+is connected.
+□
+2.3. Modular decomposition. In this short section we gather the main definitions and
+properties of modular decomposition. The historical reference is [9], the interested reader
+may also look at [3] or [20].
+The next definitions and theorems allows to get a unique recursive decomposition of any
+graph in the sense of Definition 2.5, the modular decomposition, and to encode it by a
+tree.
+Definition 2.7. Let G be a graph (labeled or not). A module M of G is a subset of V
+such that for every (x, y) ∈ M 2, and every z ∈ V \M, {x, z} ∈ E if and only if {y, z} ∈ E.
+Remark. Note that ∅, V and {v} for v ∈ V are always modules of G. Those sets are called
+the trivial modules of G.
+Definition 2.8. A graph G is prime if it has at least 3 vertices and its only modules are
+the trivial ones.
+Definition 2.9. A graph is called ⊖-indecomposable (resp. ⊕-indecomposable) if it cannot
+be written as ⊖[G1, . . . , Gk] (resp. ⊕[G1, . . . , Gk]) for some k ≥ 2 and weakly-labeled graphs
+G1, . . . , Gk.
+Note that a graph is ⊖-indecomposable if and only if it is connected, and ⊕-indecomposable
+if and only if its complementary is connected.
+Theorem 2.10 (Modular decomposition, [9]). Let G be a graph with at least 2 vertices,
+there exists a unique partition M = {M1, . . . , Mk} for some k ≥ 2, where each Mi is a
+module of G and such that either
+• G = ⊕[M1, . . . , Mk] and the (Mi)1≤i≤k are ⊕-indecomposable;
+• G = ⊖[M1, . . . , Mk] and the (Mi)1≤i≤k are ⊖-indecomposable;
+• G = P[M1, . . . , Mk] for some prime graph P.
+Moreover, only one of the possibilities occurs.
+
+8
+TH´EO LENOIR
+This decomposition can be used to encode graphs by specific trees to get a one-to-one
+correspondence.
+Definition 2.11. Let t be a substitution tree. We say that t is a canonical tree if its
+internal nodes are either ⊕, ⊖ or prime graphs, and if there is no child of a node decorated
+with ⊕ (resp. ⊖) which is decorated with ⊕ (resp. ⊖).
+To a graph G we associate a canonical tree by recursively applying the decomposition
+of Theorem 2.10 to the modules (Mi)1≤i≤k, until they are of size 1. First of all, at each
+step, we order the different modules increasingly according to their minimal vertex labels.
+Doing so, a labeled graph G can be encoded by a canonical tree. The internal nodes are
+decorated with the different graphs that are encountered along the recursive decomposition
+process (⊕ if G = ⊕[M1, . . . , Mk], ⊖ if G = ⊖[M1, . . . , Mk], P if G = P[M1, . . . , Mk]).
+At the end, every module of size 1 is converted into a leaf labeled by the label of the vertex.
+This construction provides a one-to-one correspondence between labeled graphs and
+canonical trees that maps the size of a graph to the size of the corresponding tree.
+Proposition 2.12. Let G be a graph, and t its canonical tree, then t is the only canonical
+tree such that Graph(t) = G.
+Remark. It is crucial to consider canonical trees as non-plane: otherwise, since prime graphs
+can have several labelings, there would be several canonical trees associated with the same
+graph.
+2.4. Enriched modular decomposition. Unfortunately the modular decomposition alone
+does not provide usable decompositions for the graph classes that we consider. The aim of
+this section is to solve this issue: we will state and prove Proposition 2.18 which provides
+in a very general settings a one-to-one encoding of graphs with substitution trees with
+constraints. In Section 3 we will show that P4-reducible graphs, P4-sparse graphs, P4-lite
+graphs, P4-extendible graphs, P4-tidy graphs fit in the settings of Proposition 2.18.
+Definition 2.13. We say that G is a graph with blossoms if there exists k ∈ {0, . . . , |V |}
+such that exactly k vertices of G are labeled ∗, and the others ones have a distinct label in
+{1, . . . , |V | − k}.
+The vertices labeled ∗ are called the blossoms of G. Let BG the set of vertices that are
+blossoms of G and N(G) := |V | − |BG| the number of vertices that are not a blossom of G.
+Remark. In the above definition, we allow k = 0, then the definition reduces to the one of
+a labeled graph.
+Definition 2.14. Let G be a graph with blossoms and π be a permutation of {1, . . . , N(G)}.
+The π-relabeling of G is the graph G′ such that:
+• VG′ = VG and BG′ = BG;
+• for every vertex v in VG′\BG′, we replace the label of the leaf v by π(ℓ(v)).
+We write G ∼ G′ if there exists a permutation π of {1, . . . , N(G)} such that G is iso-
+morphic to the π-relabeling of G′.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+9
+Note that ∼ is an equivalence relation.
+Definition 2.15. Let G be a graph with blossoms, a permutation π of {1, . . . , N(G)} is an
+automorphism of G if the π-relabeling of G is G.
+Definition 2.16. A module of a graph with blossoms is called flowerless if it does not
+contain any blossom.
+Let G be a graph with blossoms and M a non-empty flowerless module of G. We define
+bloM(G) to be the labeled graph obtained after the following transformations:
+• M is replaced by a new vertex v, that is now labeled ∗;
+• for every vertex w ∈ G\M, {w, v} is an edge if and only if {w, m} is an edge of G
+for every m ∈ M;
+• the graph obtained is replaced by its reduction as defined in Definition 2.2.
+If G is a graph with one blossom and M is a non-empty flowerless module of G, we
+define bloM,0(G) (resp. bloM,1(G)) to be the graph bloM(G) where the label of the initial
+blossom of G is replaced by ∗0 (resp. ∗1) and the label of the new blossom is replaced by ∗1
+(resp. ∗0).
+1
+2
+3
+4
+5
+∗
+1
+2
+4
+5
+6
+3
+7
+8
+M = {v3, v7, v8}
+bloM(G)
+Figure 4. Illustration of Definition 2.16 Left: A graph G in which we have
+highlighted the module M
+= {v3, v7, v8}.
+Right:
+The corresponding
+bloM(G).
+In this paper, we only consider the construction bloM(G) for graphs with 0 or 1 blossom.
+We are now ready to precise the general framework of our study. One of the key ingredient
+is the following recursive definition of families of graphs.
+Definition 2.17. Let P be a set of graphs with no blossom and P• be a set of graphs with
+one blossom. A tree t ∈ T0 is called (P, P•)-consistent if one of the following conditions
+holds:
+(D1) The tree t is a single leaf.
+(D2) The root r of t is decorated with a graph H ∈ P and tr (the multiset of trees attached
+to r) is a union of leaves.
+(D3) The root r of t is decorated with ⊕ (resp. ⊖) and all the elements of tr are (P, P•)-
+consistent and their roots are not decorated with ⊕ (resp. ⊖).
+
+10
+TH´EO LENOIR
+(D4) The root r of t is decorated with a graph H /∈ {⊕, ⊖} and there exists at least
+one index i ∈ {1, . . . , |H|} such that the i-th tree of tr is (P, P•)-consistent, the
+remaining subtrees of tr are reduced to a single leaf and blo{vi}(H) ∈ P•.
+We define TP,P• to be the set of trees t that are (P, P•)-consistent and such that each leaf
+has a distinct label in {1, . . . , |t|}.
+1
+2
+5
+7
+4
+9 8
+3
+6
+10
+12
+11
+(D2)
+(D3)
+(D4)
+(D3)
+(D3)
+Figure 5. An example of tree in some TP,P•. The different colours illustrate
+the different cases of Definition 2.17. The subtree with leaves {5, 6} on the
+top-right is attached to the vertex which is circled in red inside the vertex of
+case (D4). This corresponds to the i-th subtree of case (D4)
+A graph G is called (P, P•)-consistent if there exists a (P, P•)-consistent tree t such
+that G = Graph(t). We let GP,P• be the set of Graph(t) for t ∈ TP,P•.
+The map t �→ Graph(t) from TP,P• to GP,P• is surjective, but without conditions on
+(P, P•) this map is not one-to-one. To solve this issue, we introduce the following additional
+constraints on the set P, P•:
+Condition (C).
+(C1) P and P• do not contain a graph of size 1.
+(C2) For every F ∈ P and every module M of F, either bloM(F) ̸∈ P• or the subgraph
+of F induced by M is not (P, P•)-consistent.
+(C3) For every F and F ′ in P•, and every flowerless modules M and M ′ of respectively
+F and F ′ one of the following conditions is verified:
+• bloM,0(F) ̸= bloM′,1(F ′)
+• The subgraph of F induced by M is not (P, P•)-consistent.
+• The subgraph of F ′ induced by M ′ is not (P, P•)-consistent.
+(C4) Every element of P and P• is ⊕-indecomposable and ⊖-indecomposable.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+11
+(C5) For every G ∈ P•, the only modules of G containing the blossom are {∗} and G.
+We say that (P, P•) verifies condition (C) if (C1) − (C5) hold.
+Remark. The last two constraints are not necessary to ensure that the map is bijective.
+However, giving necessary and sufficient conditions to have unicity that can be checked
+easily is quite complicated.
+Note that if condition (C) is satisfied for a pair of sets (P, P•) and Q ⊂ P and Q• ⊂ P•,
+it is also verified by (Q, Q•).
+Proposition 2.18. Let P be a set of graphs with no blossom and P• a set of graphs with
+one blossom. Assume that (P, P•) verifies condition (C). For any G ∈ GP,P•, there exists
+a unique t ∈ TP,P• such that G = Graph(t). Moreover, for any element of TP,P• satifying
+case (D4) in Definition 2.17, the index i such that case (D4) holds is unique.
+Proof. Existence is guaranted by definition of GP,P•.
+We proceed by contradiction to prove the uniqueness of t.
+Let t be a smallest tree
+in TP,P• such that there exists another t′ in TP,P• verifying Graph(t) = Graph(t′). Let
+G = Graph(t).
+The graph G cannot be reduced to a single vertex due to (C1), otherwise t and t′ would
+be a single leaf with label 1. Thus we can assume that t and t′ are not in case (D1).
+By Lemma 2.6 and (C4), G is ⊕-indecomposable (resp. ⊖-indecomposable) if and only
+if t is not in case (D3) with a root decorated with ⊕ (resp. ⊖). Thus either t and t′ are
+both in case (D3) and their roots are both decorated ⊕ or ⊖, or they are both in case
+(D2) or (D4).
+Case (i): t, t′ are both in case (D3) and their are both decorated ⊕ or ⊖.
+Let r and r′ be the roots of respectively t and t′. Assume that both decorations are
+⊖, the other case is similar. The elements of tr induce connected graphs by Lemma 2.6
+as their roots are either decorated with ⊕, or ⊖-indecomposable by (C4). Since the roots
+of t and t′ are decorated with ⊖, we have a one-to-one correspondence between trees
+of tr and connected components of G. The same is true for t′
+r′. Assume that two trees
+corresponding to the same connected component of G are different. Since their set of labels
+are the same (they correspond to the labels of the vertices in the connected component)
+after reduction, one would obtain two trees t1, t2 that are different, (P, P•)-consistent and
+such that Graph(t1) = Graph(t2) since both are equal to the reduction of the corresponding
+connected component of G. This contradicts the minimality of t. Therefore tr = t′
+r′ and
+t = t′.
+Case (ii): t, t′ are both in case (D2).
+The graph G is simply the decoration of the root of t so t = t′.
+Case (iii): t is in case (D4), t′ is in case (D2).
+Let r be the root of t and H its decoration. Let i be one of the elements of {1, . . . |VH|}
+such that (D4) holds for t, H and i. Let M be the set of vertices of G whose labels are
+labels of leaves that belong to the i-th tree of tr: M is a module of G. Then bloM(G) is
+equal to blo{vi}(H) and thus belongs to P•. Moreover the subgraph of G induced by M is
+(P, P•)-consistent as the i-th subtree of t is also (P, P•)-consistent. This contradicts (C2).
+
+12
+TH´EO LENOIR
+Case (iv): t, t′ are both in case (D4).
+Let r and r′ be the roots of respectively t and t′ and H and H′ be their decorations. Let
+i be an element of {1, . . . , |VH|} such that (D4) is true for t, H and i, and i′ be an element
+of {1, . . . , |VH′|} such that (D4) is true for t′, H′ and i′. Consider M (resp. M ′) the set of
+vertices of G whose labels are labels of leaves that belong to the i-th tree of tr (resp. i′-th
+tree of t′
+r′): M (resp. M ′) is a module of G. Since the i-th tree of tr (resp. the i′-th tree of
+t′
+r′) is (P, P•)-consistent the subgraph of G induced by M (resp. M ′) is (P, P•)-consistent.
+We now prove by contradiction that M = M ′.
+By symmetry we can assume that
+M ′ ̸⊂ M.
+First assume that M ∩ M ′ = ∅. Note that bloM,1(bloM′(G)) = bloM′,0(bloM(G)). Since
+bloM(G) = blo{vi}(H) and bloM′(G) = blo{vi′}(H′), we get that bloM′,0(blo{vi}(H)) =
+bloM,1(blo{vi′}(H′)) which contradicts (C3) as both subgraphs of G induced by M and M ′
+are (P, P•)-consistent.
+Now assume that M ∩ M ′ ̸= ∅. Let L be the subset of VH such that v ∈ L if and only if
+the ℓ(v)-th tree of tr contains a leaf labeled with the label of an element of M ′. Since M ′
+is a module of G and M ∩ M ′ ̸= ∅, L is a module of blo{vi}(H) containing the blossom.
+Since M ′ is not included in M, by (C5), L = H. Since M ′ ̸= G, there exists a vertex w
+in G such that w ̸∈ M ′. Let w′ be the vertex of H such that w is in the ℓ(w′)-th tree
+of tr. Since M ′ is a module, every vertex of M ′ is either connected or not to w, thus w′
+is connected to every vertex of H (except w′) or to none of them. This means that H is
+either ⊕-decomposable or ⊖-decomposable, which is a contradiction.
+Thus M = M ′ and blo{vi}(H) = bloM(G) = bloM′(G) = blo{vi′}(H′), and we get that
+H = H′, and that i = i′: thus i is unique.
+We know that the i-th tree of tr and the i-th tree of t′
+r′ are (P, P•)-consistent and the
+associated graph is the one induced by M. By taking the reduction of the trees and the
+graph, we get by minimality of t that the reductions of both trees are equal. Since M = M ′,
+it implies that both subtrees are the same: thus t = t′.
+□
+3. Zoology of graph classes with few P4’s
+Several classes have been defined as generalizations of the class of P4-free graphs, the
+cographs. Here the classes we will focus on are the following: P4-reducible graphs [15,18],
+P4-sparse graphs [13,17] P4-lite graphs [14], P4-extendible graphs [16], P4-tidy graphs [10].
+The aim of this section is to give explicit sets P and P• such that GP,P• is one of the
+previously mentioned classes.
+3.1. Basic definitions. The following results and definitions are from [3, Section 11.3].
+Definition 3.1. A graph G is a Pk if it is a path of k vertices, and a Ck if it is a cycle of
+k vertices.
+The two vertices of degree one of a P4 are called the endpoints, the two vertices of degree
+two are called the midpoints.
+Notation. For a graph G, we denote by G its complementary.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+13
+The modular decompositions of classes of graphs we consider are already well-known [10].
+To explain the different properties, we need the notion of spider and bull.
+Definition 3.2. A spider is a graph G, such that there exists a partition of VG in three
+parts, K, S, R, verifying:
+• |K| ≥ 2;
+• K induces a clique;
+• S induces a graph without edges;
+• every element of R is connected to every element of K but to none of S;
+• there exists a bijection f from K to S such that for every k ∈ K, k is only connected
+to f(k) in S, or such that for every k ∈ K, k is connected to every element of S
+except f(k). In the first case the spider is called thin, in the second one it is called
+fat.
+K
+S
+K
+S
+R
+R
+1
+1
+2
+2
+3
+3
+4
+4
+5
+5
+6
+6
+7
+7
+Figure 6. Left: a thin spider. Right: a fat spider. Both with |K| = 3.
+Remark. For every spider G, the partition (K, S, R) is uniquely determined by G. Moreover,
+the bijection f given by the definition is unique, except in the case |K| = 2. In this case,
+since there is no difference between a thin and a fat spider, a spider with |K| = 2 is called
+thin. A spider with |K| = 2 and |R| = 1 is called a bull, and a spider with |K| = 2 and
+|R| = 0 is simply a P4.
+1
+2
+3
+4
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+Figure 7. From left to right: a P4, a bull, a C5
+Proposition 3.3. A spider is prime if and only if |R| ≤ 1.
+
+14
+TH´EO LENOIR
+In the following, if |R| = 1, the vertex belonging to R will be a blossom of the spider,
+and it will be its only blossom: such spiders will be called blossomed spiders. If |R| = 0,
+the spider will have no blossom. This also applies for bulls and P4.
+Definition 3.4. We call a graph H a pseudo-spider if there exists a prime spider G such
+that, if we duplicate a vertex that is not a blossom of G (his label is the new number of
+vertices), and if either by adding or not an edge between the vertex and its duplicate, the
+graph obtained is a relabeling of H. If |K| = 2, we also call H a pseudo-P4.
+Moreover, we say that H is a blossomed pseudo-spider if G is a blossomed spider. If
+|K| = 2, we also call H a pseudo-bull.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+∗
+K
+S
+R
+1
+2
+∗
+3
+4
+5
+6
+Duplicate
+7
+Figure 8. A blossomed pseudo-spider, a pseudo-bull, a pseudo P4
+Lemma 3.5. A prime spider with 0 or 1 blossom has |K|! automorphisms (as there is a
+natural bijection between the automorphisms of the spider and the automorphisms of K).
+A pseudo-spider with 0 or 1 blossom has 2 × (|K| − 1)! automorphisms.
+3.2. P4-tidy graphs.
+Definition 3.6. A graph G is said to be a P4-tidy graph if, for every subgraph H of G
+inducing a P4, there exists at most one vertex y ∈ VG\VH such that y is connected to at
+least one element of H but not all, and y is not connected to exactly both midpoints of H.
+Theorem 3.7. Let Ptidy be the set containing all C5, P5, P5, all prime spiders without
+blossom and all pseudo-spiders without blossom.
+Let P•
+tidy be the set of all blossomed
+prime spiders and all blossomed pseudo-spiders. Then the set of graphs that are P4-tidy is
+GPtidy,P•
+tidy.
+Proof. It is simply a reformulation in our setting of [10, Theorem 3.3] that states that a
+graph G is P4-tidy if and only if its canonical tree t verifies the following conditions:
+• Every node in t is labeled with ⊕, ⊖, C5, P5, P5 or a prime spider.
+• If a node w in t is decorated with C5, P5 or P5, every element of tw is reduced to a
+single leaf.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+15
+• If a node w in t is decorated with a prime spider with |R| = 0, every element of tw
+is a tree of size at most two, and at most one is of size two.
+• If a node w in t is decorated with a prime spider H with |R| = 1, let v be the vertex
+of H in R, and t′ the ℓ(v)-th tree of tw. Every element of tw\{t′} is a tree of size
+at most two, and at most one is of size two.
+□
+Proposition 3.8. The pair (Ptidy, P•
+tidy) verifies (C)
+Proof. Note that all the graph in Ptidy or P•
+tidy are prime except the pseudo-spiders. The
+only modules of the pseudo-spiders are the trivial ones, and the module formed by the
+vertex that was duplicated and its duplicate, which implies (C5).
+(C2) is also verified with the previous observation, as the modules of every graph in Ptidy
+are trivial.
+(C1) is clearly verified and (C4) can be checked easily as all the graphs in P ∪ P• are
+connected, and their complementary is also connected.
+For (C3), assume that for (F, F ′)2 ∈ P•
+tidy and M, M ′ are respectively flowerless modules
+of F and F ′, bloM,0(G) = bloM′,1(G).
+By cardinality argument, F and F ′ are either
+both spiders, or both pseudo-spiders of same size. If both are spiders, as R is uniquely
+determined by the spiders, and the only element of R does not have the same label in
+bloM,0(G) and in bloM,1(G), we get a contradiction. If both are pseudo-spiders, note that
+the original node and its duplicate form the only module of size 2 of bloM,0(G). Thus the
+only element of R (in the original spiders) is uniquely determined by the pseudo-spiders,
+and the only element of R does not have the same label in bloM,0(G) and in bloM,1(G), we
+get a contradiction.
+□
+3.3. P4-lite graphs.
+Definition 3.9. A graph G is said to be a P4-lite graph if every subgraph of G of size at
+most 6 does not contain three induced P4.
+Theorem 3.10. Let Plite be the set containing all P5, P5, all prime spiders without blossom
+and all pseudo-spiders without blossom. Let P•
+lite to be the set containing all blossomed
+prime spiders and all blossomed pseudo-spiders. Then the set of graphs that are P4-lite is
+GPlite,P•
+lite.
+Proof. It is simply a reformulation in our setting of [10, Theorem 3.8] that states that a
+graph G is P4-lite if and only if its canonical tree t verifies the following conditions:
+• Every node in t is labeled with ⊕, ⊖, P5, P5 or a prime spider.
+• If a node w in t is decorated with P5 or P5, every element of tw is reduced to a
+single leaf.
+• If a node w in t is decorated with a prime spider with |R| = 0, every element of tw
+is a tree of size at most two, and at most one is of size two.
+• If a node w in t is decorated with a prime spider H with |R| = 1, let v be the vertex
+of H in R, and t′ the ℓ(v)-th tree of tw. Every element of tw\{t′} is a tree of size
+at most two, and at most one is of size two.
+□
+
+16
+TH´EO LENOIR
+By Proposition 3.8 since Plite ⊂ Ptidy, P•
+lite ⊂ P•
+tidy we get that the pair (Plite, P•
+lite)
+verifies (C).
+3.4. P4-extendible graphs.
+Definition 3.11. A graph G is said to be a P4-extendible graph if, for every subgraph H
+of G inducing a P4, there exists at most one vertex y ∈ VG\VH such that y belongs to an
+induced P4 sharing at least one vertex with H.
+Theorem 3.12. Let Pext be the set containing all C5, P5, P5, P4 and all pseudo-P4. Let
+P•
+ext be the set containing all bulls and all pseudo-bulls. Then the set of graphs that are
+P4-extendible is GPext,P•
+ext.
+Proof. It is simply a reformulation in our setting of [10, Theorem 3.7] that states that a
+graph G is P4-extendible if and only if its canonical tree t verifies the following conditions:
+• Every node in t is labeled with ⊕, ⊖, C5, P5, P5, P4 or a bull.
+• If a node w in t is decorated with C5, P5 or P5, every element of tw is reduced to a
+single leaf.
+• If a node w in t is decorated with P4, every element of tw is a tree of size at most
+two, and at most one is of size two.
+• If a node w in t is decorated with a bull G, let v be the vertex of G in R, and t′
+the ℓ(v)-th tree of tn. Every element of tw\{t′} is a tree of size at most two, and at
+most one is of size two.
+□
+By Proposition 3.8 since Pext ⊂ Ptidy, P•
+ext ⊂ P•
+tidy we get that the pair (Pext, P•
+ext)
+verifies (C).
+3.5. P4-sparse graphs.
+Definition 3.13. A graph G is said to be a P4-sparse graph if every subgraph of G of size
+5 does not contain two induced P4.
+Theorem 3.14. Let P be the set containing all prime spiders without blossom. Let P• be
+the set containing all blossomed prime spiders. Then the set of graphs that are P4-sparse
+is GP,P•.
+Proof. It is simply a reformulation in our setting of [11, Theorem 3.4] that states that a
+graph G is P4-sparse if and only if its canonical tree t verifies the following conditions:
+• Every node in t is labeled with ⊕, ⊖ or a prime spider.
+• If a node w in t is decorated with a prime spider with |R| = 0, every element of tw
+is reduced to a single leaf.
+• If a node w in t is decorated with a prime spider h with |R| = 1, let v be the vertex
+of H in R, and t′ the ℓ(v)-th tree of tw. Every element of tw\{t′} is reduced to a
+single leaf.
+□
+By Proposition 3.8 since Pspa ⊂ Ptidy, P•
+spa ⊂ P•
+tidy we get that the pair (Pspa, P•
+spa)
+verifies (C).
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+17
+3.6. P4-reducible graphs.
+Definition 3.15. A graph G is said to be a P4-reducible graph if every vertex of G belongs
+to at most one induced P4.
+Theorem 3.16. Let Pred be the set containing all P4. Let P•
+red be the set containing all
+bulls. Then the set of graphs that are P4-reducible is GPred,P•
+red.
+Proof. It is simply a reformulation in our setting of [11, Theorem 4.2] that states that a
+graph G is P4-reducible if and only if its canonical tree t verifies the following conditions:
+• Every node in t is labeled with ⊕, ⊖, P4 or a bull.
+• If a node w in t is decorated with a P4, every element of tw is reduced to a single
+leaf.
+• If a node w in t is decorated with a bull H, let v be the vertex of H in R, and t′
+the ℓ(v)-th tree of tn. Every element of tw\{t′} is reduced to a single leaf.
+□
+By Proposition 3.8 since Pred ⊂ Ptidy, P•
+red ⊂ P•
+tidy we get that the pair (Pred, P•
+red)
+verifies (C).
+3.7. P4-free graphs (cographs).
+Definition 3.17. A graph G is said to be a cograph if no subgraph of G induces a P4.
+Theorem 3.18. Set Pcog = ∅ and P•
+cog = ∅. Then the set of graphs that are cographs is
+GPcog,P•cog.
+Proof. It is simply a reformulation in our setting of [5, Theorem 7] that states that a graph
+G is a cograph if and only if its canonical tree t has no internal node decorated with a
+prime graph.
+□
+Clearly the pair (Pcog, P•
+cog) verifies (C).
+4. Enriched modular decomposition: enumerative results
+4.1. Exact enumeration. In the following, we establish combinatorial identities between
+formal power series involving subsets of P and P•.
+Throughout this section, we consider generic pairs (P, P•) where P (resp. P•) is a set
+of graphs with no blossom (resp. with one blossom) verifying condition (C) defined p.10.
+Recall that for a graph G with blossoms, N(G) is the number of vertices that are not
+a blossom: this will be the crucial parameter in the subsequent analysis. Let P •(z) :=
+�
+s∈P•
+zN(s)
+N(s)! and P(z) := �
+s∈P
+zN(s)
+N(s)!.
+For n ∈ N, let Pn (resp. P•
+n) be the set of graphs G in P (resp. P•) such that N(G) = n.
+Note that, if both P and P• are stable under relabeling (which is the case for the classes
+of graphs mentioned in Section 3), for each n ∈ N, there is a natural action Φn of the
+
+18
+TH´EO LENOIR
+permutations of {1, . . . , n} over Pn and P•
+n. Let RPn and RP•n be a system of representants
+of every orbit under this action, then
+P •(z) =
+�
+n∈N
+|P•
+n|zn
+n! =
+�
+n∈N
+�
+s∈RPn
+|RP•n|
+n!
+|Aut(s)|
+zn
+n! =
+�
+n∈N
+�
+s∈RPn
+|RP•n|
+zn
+|Aut(s)|
+Similarly, we have:
+P(z) =
+�
+n∈N
+�
+s∈RPn
+|RPn|
+zn
+|Aut(s)|
+Theorem 4.1. For each graph class introduced in Section 3, we have the following expres-
+sions for P and P •:
+P4-tidy
+P •
+tidy(z) = (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4
+2 − 2z5
+Ptidy(z) = P •
+tidy(z) + z5 + z5
+10
+P4-lite
+P •
+lite(z) = (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4
+2 − 2z5
+Plite(z) = P •
+lite(z) + z5
+P4-extendible
+P •
+ext(z) = z4
+2 + 2z5
+Pext(z) = P •
+ext(z) + z5 + z5
+10
+P4-sparse
+P •
+spa(z) = Pspa(z) = 2(exp(z2) − 1 − z2 − z4
+4 )
+P4-reducible
+P •
+red(z) = Pred(z) = z4
+2
+P4-free
+P •
+cog(z) = Pcog(z) = 0
+Proof. We only detail the computation of Ptidy and P •
+tidy for P4-tidy graphs as this is the
+most involved case. According to Theorem 3.7, Ptidy is composed of one C5 that has 10
+automorphisms and all its relabelings, one P5, and one P5 that both have 2 automorphisms
+and all their relabelings.
+For k ≥ 3 (resp. k = 2), there are thin and fat spiders corresponding to the 2 (resp. 1)
+different orbits of the action Φ2k over prime spiders of size 2k, each having k! automor-
+phisms.
+For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can
+come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4)
+cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over pseudo-spiders
+of size 2k + 1, each having 2(k − 1)! automorphisms.
+Thus we have
+Ptidy(z) = z5
+10 + 2z5
+2 + z4
+2 + 2
+�
+k≥3
+z2k
+k! + 4z5
+2 + 8
+�
+k≥3
+z2k+1
+2(k − 1)!
+Hence
+Ptidy(z) = z5 + z5
+10 + (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4
+2 − 2z5.
+Now let’s compute P •
+tidy. For k ≥ 3 (resp. k = 2), there are thin and fat spiders with
+blossom corresponding to the 2 (resp. 1) different orbits of the action Φ2k over blossomed
+prime spiders G with 2k non blossomed vertices, each having k! automorphisms.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+19
+For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can
+come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4)
+cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over blossomed
+pseudo-spiders with 2k + 1 non blossomed vertices, each having 2(k − 1)! automorphisms.
+Hence
+P •
+tidy(z) = z4
+2 +2
+�
+k≥3
+z2k
+k! +4z5
+2 +8
+�
+k≥3
+z2k+1
+2(k − 1)! = (2+4z3) exp(z2)−2−2z2−4z3− z4
+2 −2z5,
+which gives the announced result.
+□
+Let T be the exponential generating function of TP,P•, the set of trees defined in Def-
+inition 2.17 counted by their number of leaves. Denote by Tnot⊕ (resp. Tnot⊖) the set of
+all t ∈ TP,P• whose root is not decorated with ⊕ (resp. ⊖) and by Tnot⊕ (resp. Tnot⊖) the
+corresponding exponential generating function.
+Theorem 4.2. Together with Tnot⊕ = 0, the exponential generating function Tnot⊕ is de-
+termined (as a formal series) by the following equation:
+Tnot⊕ = z + P + (exp(Tnot⊕) − 1)P • + exp(Tnot⊕) − 1 − Tnot⊕,
+(1)
+and the series T and Tnot⊖ are simply given by the following equations:
+T = exp(Tnot⊕) − 1
+(2)
+Tnot⊖ = Tnot⊕
+(3)
+Moreover, Eq. (1) with Tnot⊕(0) = 0 determines uniquely the generating function Tnot⊕.
+Proof. Note that there is a natural involution on TP,P•: the decoration of every linear node
+can be changed to its opposite: ⊕ to ⊖, and ⊖ to ⊕. Therefore Tnot⊕ = Tnot⊖.
+First, we prove that
+T = z + T × P • + P + 2 × (exp(Tnot⊕) − 1 − Tnot⊕)
+(4)
+We split the enumeration of the trees t ∈ TP,P• according to the different cases of
+Definition 2.17.
+(D1) The tree t is a single leaf (which gives the z in Eq. (4)).
+(D2) The tree t has a root decorated with a graph H belonging to P. The exponential
+generating function for a fixed H is zN(H)
+N(H)!. Summing over all H and all n gives the
+term P in Eq. (4).
+(D3) The tree t has a root r decorated with ⊕ and having k children with k ≥ 2. In this
+case, the generating function of the set of the k subtrees of tr is
+T k
+not⊕
+k! . Summing
+over all k implies that the exponential generating function of all trees in case (D3)
+with a root labeled ⊕ is exp(Tnot⊕) − 1 − Tnot⊕.
+The tree t can also have a root r decorated with ⊖. Since Tnot⊕ = Tnot⊖, the
+exponential generating function of all trees in case (D3) with a root labeled ⊖ is
+exp(Tnot⊕) − 1 − Tnot⊕.
+
+20
+TH´EO LENOIR
+(D4) The tree t has a root r decorated with a graph H and there exists v ∈ VH such that
+blov(H) = W where W ∈ P•. Denote t′ the ℓ(v)-th tree of tr.
+The exponential generating function corresponding to the set of leaves in t\t′
+is
+zN(W )
+N(W)!, and the exponential generating function corresponding to t′ is T. Note
+that the tree t is uniquely determined by W, the labeled product of t′ and the
+set of leaves of t\t′. Thus the corresponding generating function for a fixed W is
+T × zN(W )
+N(W)!. Summing over all W and all n gives the term T × P • in Eq. (4).
+Summing all terms gives Eq. (4).
+Similarly, we get
+Tnot⊕ = z + T × P • + P + exp(Tnot⊕) − 1 − Tnot⊕.
+(5)
+Substracting Eq. (5) to Eq. (4) gives Eq. (2). Then Eq. (1) is an easy consequence from
+Eqs. (2) and (5).
+Note that Eq. (1) can be rewritten as:
+Tnot⊕ = z + P + (exp(Tnot⊕) − 1)P • +
+�
+k≥2
+T k
+not⊕
+k! .
+(6)
+For every n ≥ 1, the coefficient of degree n of Tnot⊕ only depends on coefficients of lower
+degree as P •(z) has no term of degree 0 or 1 and Tnot⊕(0) = 0. Thus Eq. (1) combined
+with Tnot⊕(0) = 0 determines uniquely Tnot⊕.
+□
+We are going to define the notions of trees with marked leaves, and of blossomed trees,
+which will be crucial in the next section. We insist on the fact that the size parameter will
+count the number of leaves including the marked ones but not the blossoms.
+Definition 4.3. A marked tree is a pair (t, I) where t is a tree and I a partial injection
+from the set of labels of leaves of t to N. The number of marked leaves is the size of the
+domain of I denoted by |(t, I)|, and a leaf is marked if its label j is in the domain, its mark
+being I(j).
+Remark. In the following, we will consider marked trees (t, I), and subtrees t′ of t. The
+marked tree (t′, I) will refer to the marked tree (t′, I′) where I′ is the restriction of I to
+the set of labels of leaves of t′.
+Remark. Let F ∈ {TP,P•, Tnot⊖, Tnot⊕}, and F be its generating exponential function. The
+exponential generating function of trees in F with a marked leaf is zF ′(z): if there are fn
+trees of size n in F, there are nfn trees with a marked leaf. Thus the generating exponential
+function is �
+n≥1
+nfn
+n! zn = zF ′(z).
+Blossoming transformation. Let t be a tree not reduced to a leaf in TP,P•, ℓ a leaf of t
+and n the parent of ℓ. If n is a linear node, we replace the label of ℓ by ∗, and do the
+reduction on t. If v is a non-linear node, and ℓ is in the i-th tree of tv (where i is the
+element such that (D4) holds in Definition 2.17), we replace the label of ℓ by ∗ and i by
+∗ in the decoration of v, and do the reduction on both t and the decoration of v. If t is
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+21
+reduced to a leaf, we replace the leaf by a blossom. We call such this transformation the
+blossoming of (t, ℓ).
+We extend this operation to internal node: if n is a internal node, we replace t[n] by its
+leaf of smallest label, and do the blossoming operation on the tree obtained. The resulting
+tree is still called the blossoming of (t, n).
+Definition 4.4 (Blossomed tree). A blossomed tree is a tree that can be obtained by the
+blossoming of a tree in TP,P•. Its size is its number of leaves without blossom.
+A blossom is ⊕-replaceable (resp. ⊖-replaceable) if its parent is not decorated with ⊕
+(resp. ⊖).
+Remark. Similarly to a tree, a blossomed tree can be marked by a partial injection I.
+We will denote T b and T b
+a with a = not⊕, not⊖, and b = ⊕, ⊖ or blo the set of trees
+whose root is not ⊕ (resp. ⊖) if a = not⊕ (resp. a = not⊖), and with one blossom that is
+b-replaceable if b = ⊕ or ⊖, or just with one blossom if b = blo.
+We define T b
+a to be the corresponding exponential generating function of trees, counted
+by the number of non blossomed leaves.
+However, we take the convention that T ⊕
+not⊕(0) = 0 = T ⊖
+not⊖. In other words, a single leaf
+is neither in T ⊕
+not⊕ nor in T ⊖
+not⊖. The other series have constant coefficient 1.
+Remark. From the previously defined involution, it follows that T ⊖
+not⊕ = T ⊕
+not⊖, T ⊕
+not⊕ =
+T ⊖
+not⊖ et T ⊕ = T ⊖ and T blo
+not⊕ = T blo
+not⊖.
+Theorem 4.5. The functions T ⊕, T ⊕
+not⊕, T ⊕
+not⊖ are given by the following equations:
+T ⊕ =
+1
+2 − exp(Tnot⊕) − P • exp(Tnot⊕)
+(7)
+T ⊖
+not⊕ =
+T ⊕
+exp(Tnot⊕)
+(8)
+T ⊕
+not⊕ =
+T ⊕ − 1
+exp(Tnot⊕)
+(9)
+Proof. Let t be a tree in T ⊕
+not⊕. Note that it cannot be reduced to a single leaf, have a root
+decorated with ⊕ or be in case (D2) of Definition 2.17.
+(D3) The tree t can have a root r decorated with ⊖ and having k children with k ≥ 2.
+There are k−1 subtrees without blossom, and 1 with a blossom. Thus the generating
+function of the set of the k subtrees of tr is
+T k−1
+not⊖
+(k−1)!T ⊖
+not⊕. Summing over all k gives
+that the exponential generating function of all trees in case (D3) with a root labeled
+⊖ is
+�
+k≥2
+T k−1
+not⊖
+(k − 1)!T ⊕
+not⊖ = (exp(Tnot⊖) − 1)T ⊕
+not⊖
+
+22
+TH´EO LENOIR
+⊕ or (D4)
+⊖ or (D4)
+⊖ or (D4)
+not (D2)
+(D4)
+2
+9
+6
+or
+At least one tree, each does not
+have a root decorated with ⊕
+If the previous node is in (D4) the marked leaf must be in the i-th tree
+Figure 9. Illustration of both cases in the proof of Theorem 4.5
+(D4) The tree t can have a root r decorated with H and v ∈ VH such that blov(H) = W
+with W ∈ P•. Then the blossom must be in the ℓ(v)-th tree of tr that will be
+denoted t′.
+The exponential generating function corresponding to the set of leaves in t\t′
+is zN(W )
+N(W)!, and the exponential generating function corresponding to t′ is T ⊕. Note
+that the tree t is uniquely determined by W, the labeled product of t′ and the
+set of leaves of t\t′. Thus the corresponding generating function for a fixed W
+is T ⊕ × zN(W )
+N(W)!. Summing over all W and all n gives the exponential generating
+function T ⊕ × P •.
+This implies the following equation:
+T ⊕
+not⊕ = (exp(Tnot⊖) − 1)T ⊕
+not⊖ + P •T ⊕ = (exp(Tnot⊕) − 1)T ⊖
+not⊕ + P •T ⊕
+(10)
+We have similarly:
+T ⊖
+not⊕ = 1 + (exp(Tnot⊖) − 1)T ⊖
+not⊖ + P •T ⊖ = 1 + (exp(Tnot⊕) − 1)T ⊕
+not⊕ + P •T ⊕
+(11)
+T ⊕ = 1 + (exp(Tnot⊖) − 1)T ⊕
+not⊖ + (exp(Tnot⊕) − 1)T ⊕
+not⊕ + P •T ⊕
+(12)
+Thus:
+T ⊕ = 1 + (exp(Tnot⊕) − 1)(T ⊕
+not⊕ + T ⊖
+not⊕) + P •T ⊕
+(13)
+By substracting Eq. (11) to Eq. (13), we get T ⊕ − T ⊖
+not⊕ = (exp(Tnot⊕) − 1)T ⊖
+not⊕ which
+implies Eq. (8).
+Using Eqs. (10) and (13), we get
+T ⊕ = 1 + (exp(Tnot⊕) − 1)T ⊕
+not⊕ + T ⊕
+not⊕ = 1 + exp(Tnot⊕)T ⊕
+not⊕
+which implies Eq. (9).
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+23
+Substituting T ⊕
+not⊕ and T ⊖
+not⊕ with Eqs. (9) and (10) ins Eq. (8), it follows that:
+T ⊕ − 1 = (exp(Tnot⊕) − 1)T ⊕ + exp(Tnot⊕)P •T ⊕
+and T ⊕(2 − exp(Tnot⊕) − P • exp(Tnot⊕)) = 1 which implies Eq. (7).
+□
+Theorem 4.6. We also have the following equations:
+T blo =
+exp(Tnot⊕)
+2 − exp(Tnot⊕) − P • exp(Tnot⊕)
+(14)
+T blo
+not⊕ =
+1
+exp(Tnot⊕)T blo
+(15)
+Proof. By the same techniques used as those of the previous proof, we establish that:
+T blo = 1 + 2(exp(Tnot⊕) − 1)T blo
+not⊕ + P •T blo;
+(16)
+T blo
+not⊕ = 1 + (exp(Tnot⊕) − 1)T blo
+not⊕ + P •T blo.
+(17)
+By substracting Eq. (17) to Eq. (16), we get that:
+T blo − T blo
+not⊕ = (exp(Tnot⊕) − 1)T blo
+not⊕
+which implies Eq. (15).
+By multiplying Eq. (17) by exp(Tnot⊕) and using Eq. (15) we get that:
+T blo (2 − P • exp(Tnot⊕) − exp(Tnot⊕)) = exp(Tnot⊕)
+which implies Eq. (14).
+□
+Combining Theorem 4.5 and Theorem 4.6 we obtain:
+Corollary 4.7. We have the following equations:
+T blo = exp(Tnot⊕)T ⊕;
+(18)
+T blo
+not⊕ = T ⊕.
+(19)
+4.2. Asymptotic enumeration. In the following, we derive from the previously obtained
+equations the radii of the different series introduced, the asymptotic behavior of the dif-
+ferent series in R and an equivalent of the number of graphs in GP,P•
+From now on, we assume that P and P • have a positive radius of convergence.
+Let R0
+be the minimum of their radii of convergence. Denote by P(R0) and P •(R0) the limit in
+[0, +∞] of P and P • at R−
+0 .
+In the following, we assume that one of the conditions below is verified:
+• P •(R0) ≥ 1
+• R0 + P(R0) + 2 ln(1 + P •(R0)) − P •(R0) > 2 ln(2) − 1
+
+24
+TH´EO LENOIR
+Note that one of these conditions is verified in the different classes of graphs we study,
+as R0 = +∞.
+Denote by R the only solution in [0, R0) of the equation:
+R + P(R) + 2 ln(1 + P •(R)) − P •(R) = 2 ln(2) − 1
+(20)
+such that P •(R) < 1 (unicity comes from the fact that z �→ 2 ln(1 + z) − z is increasing in
+[0, 1]). Note that by definition, 0 < R < R0.
+Recall that a formal series A is aperiodic if there does not exist two integers r ≥ 0 and
+d ≥ 2 and B a formal series such that A(z) = zrB(zd).
+Lemma 4.8. The functions T, Tnot⊕, T ⊕, T ⊖
+not⊕, T ⊕
+not⊕, T blo, T blo
+not⊕ are aperiodic.
+Proof. One can easily check that for each of the previous series, the coefficients of degree
+3 and 4 are positive, and thus all the series are aperiodic.
+□
+Definition 4.9. A set ∆ is a ∆-domain at 1 if there exist two positive numbers R and
+π
+2 < φ < π such that
+∆ = {z ∈ C||z| ≤ R, z ̸= 1, |arg(1 − z)| < φ}
+For every w ∈ C∗, a set is a ∆-domain at w if it is the image of a ∆-domain by the
+mapping z �→ zw.
+Definition 4.10. A power series U is said to be ∆-analytic if it has a positive radius of
+convergence ρ and there exists a ∆-domain D at ρ such that U has an analytic continuation
+on D.
+Theorem 4.11. Both T and Tnot⊕ have R as radius of convergence and a unique dominant
+singularity at R. They are ∆-analytic. Their asymptotic expansions near R are:
+Tnot⊕(z) = ln
+�
+2
+1 + P •(R)
+�
+− κ
+�
+1 − z
+R + o
+��
+1 − z
+R
+�
+(21)
+T(z) =
+2
+1 + P •(R) − 1 −
+2
+1 + P •(R)κ
+�
+1 − z
+R + o
+��
+1 − z
+R
+�
+(22)
+where κ is the constant given by:
+κ =
+�
+�
+�
+�R
+�
+1 + P ′(R) + (1 − P •(R))(P •)′(R)
+1 + P •(R)
+�
+Proof. We begin with the expansion of Tnot⊕ for which we apply the smooth implicit the-
+orem [8, Theorem VII.3, p.467]. Following [8, Sec VII.4.1] we claim that Tnot⊕ satifies the
+settings of the so-called smooth implicit-function schema: Tnot⊕ is solution of
+T = G(z, T),
+where G(z, w) = z + P(z) + (exp(w) − 1)P •(z) + (exp(w) − 1 − w).
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+25
+The singularity analysis of Tnot⊕ will go through the study of the characteristic system:
+�
+�
+�
+G(r, s) = s,
+Gw(r, s) = 1
+with 0 < r < R, s > 0
+where Fx = ∂F
+∂x .
+Note that (r, s) =
+�
+R, ln
+�
+2
+1+P •(R)
+��
+is a solution of the characteristic system of G since
+• Gw(r, s) = exp(s)(1 + P •(R)) − 1 = 2 − 1 = 1
+• G(r, s) = R+P(R)−P •(R)+∂wG(r, s)−s = 2 ln(2)−1−2 ln(1+P •(R))+1−s =
+2s − s = s
+Moreover
+• Gz(r, s) = 1 + P ′(R) + (exp(s) − 1)(P •)′(R) = 1 + P ′(R) + (1−P •(R))(P •)′(R)
+(1+P •(R))
+• Gw,w(r, s) = exp(s)(1 + P •(r)) = 2
+The expansion of T is then a consequence of Eq. (2) p.19 and of the expansion of
+Tnot⊕.
+□
+Corollary 4.12. The radius of convergence of T ⊕, T ⊖
+not⊕, T ⊕
+not⊕, T blo, and T blo
+not⊕ is R
+and R is the unique dominant singularity of these series. They are ∆-analytic and their
+asymptotic expansions near R are:
+T ⊕ = 1
+2κ
+�
+1 − z
+R
+�− 1
+2 + o
+��
+1 − z
+R
+�− 1
+2
+�
+(23)
+T ⊖
+not⊕ = (1 + P•(R))
+4κ
+�
+1 − z
+R
+�− 1
+2 + o
+��
+1 − z
+R
+�− 1
+2
+�
+(24)
+T ⊕
+not⊕ = (1 + P•(R))
+4κ
+�
+1 − z
+R
+�− 1
+2 + o
+��
+1 − z
+R
+�− 1
+2
+�
+(25)
+T blo =
+1
+(1 + P•(R))κ
+�
+1 − z
+R
+�− 1
+2 + o
+��
+1 − z
+R
+�− 1
+2
+�
+(26)
+T blo
+not⊕ = 1
+2κ
+�
+1 − z
+R
+�− 1
+2 + o
+��
+1 − z
+R
+�− 1
+2
+�
+(27)
+Proof. note that, if |z| ≤ R,
+|(1 + P •(z)) exp(Tnot⊕(z))| ≤ (1 + P •(|z|)) exp(|Tnot⊕(z)|) ≤ (1 + P •(R)) exp(Tnot⊕(R)) = 2
+with equality if and only if z = R by aperiodicity from Daffodil lemma [8, Lemma IV.1]
+and since Tnot⊕(R) ∈ R+.
+Hence, by Theorem 4.11 and by compacity, 2−(1+P •(z)) exp(Tnot⊕(z)) can be extended
+to a ∆-domain D at R with 2 − (1 + P •(z)) exp(Tnot⊕)(z) ̸= 0 for every z ∈ D.
+Eq. (7) shows that T ⊕ can be extended to D and yields the announced expansions when
+z tends to R. These expansions show that all these series have a radius of convergence
+exactly equal to R.
+□
+
+26
+TH´EO LENOIR
+Applying the Transfer Theorem [8, Corollary VI.1 p.392] to the results of Theorem 4.11,
+we obtain an equivalent of the number of trees of size n in TP,P•. Since there is a one-to-one
+correspondence between graphs in GP,P• and trees in TP,P•, we get the following result:
+Corollary 4.13. The number of graphs in GP,P• of size n is asymptotically equivalent to
+C
+n!
+Rnn
+3
+2
+where
+C =
+κ
+√π(1 + P •(R)).
+Here are the numerical approximations of R and C in the different cases:
+class of graph
+R−1
+R
+C
+P4-tidy
+2.90405818
+0.34434572
+0.40883495
+P4-lite
+2.90146936
+0.34465296
+0.40833239
+P4-extendible
+2.88492066
+0.34662998
+0.40351731
+P4-sparse
+2.72743550
+0.36664478
+0.37405701
+P4-reducible
+2.71715531
+0.36803196
+0.37115484
+P4-free
+1
+2 ln(2)−1 ≈ 2.58869945
+2 ln(2) − 1 ≈ 0.38629436
+0.35065840
+5. Enumeration of graphs with a given induced subgraph
+5.1. Induced subtrees and subgraphs. We recall that the size of a graph is its number
+of vertices, and the size of a tree is its number of leaves.
+Definition 5.1 (Induced subgraph). Let G be a graph, k a positive integer and I a partial
+injection from the set of labels of G to N. The labeled subgraph GI of G induced by I is
+defined as:
+• The vertices of GI are the vertices of G whose label ℓ is in the domain of I. For
+every such vertex, we replace the label ℓ of the vertex by I(ℓ);
+• For two vertices v and v′ of GI, (v, v′) is an edge of GI if and only if it is an edge
+of G.
+Definition 5.2 (First common ancestor). Let t be a rooted tree and let ℓ1, ℓ2 be two distinct
+leaves of t. The first common ancestor of ℓ1 and ℓ2 is the internal node of t that is the
+furthest from the root and that belongs to the shortest path from the root to ℓ1, and the
+shortest path from the root to ℓ2.
+Definition 5.3 (Induced subtree). Let (t, I) be a marked tree in T0 (T0 is defined in
+Definition 2.4, and the notion of marked tree in Definition 4.3). The induced subtree tI
+of t induced by I is defined as:
+• The leaves of tI are the leaves of t that are marked. For every such leaf labeled with
+an integer ℓ, the new label of ℓ is I(ℓ);
+• The internal nodes of tI are the internal nodes of t that are first common ancestors
+of two or more leaves of tI;
+• The ancestor-descendent relation in tI is inherited from the one in t;
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+27
+• For every internal node v of t that appears in tI, let H be its decoration in t. Denote
+by J the set of positive integers k such that the k-th tree of tv contains a leaf of tI.
+For every k in J, we define L(k) as the smallest image by I of a marked leaf label
+in the k-th tree of tv. The decoration of v in tI is the reduction of HL.
+For every internal node v (resp. leaf ℓ) of tI, we also define φ(v) to be the only internal
+node (resp. leaf) of t corresponding to v.
+Remark. When (t, I) is a marked tree and t′ is a subtree of t, we will denote t′
+I the tree
+induced by the restriction of I to the set of labels of leaves of t′.
+As a consequence of Definitions 5.1 and 5.3, we obtain:
+Lemma 5.4. Let (t, I) be a marked tree in T0. Then
+Graph(t)I = Graph(tI).
+8
+3
+7
+2
+6
+4
+5
+1
+4
+1
+3
+2
+Graph(t)
+3
+7
+5
+6
+1
+2
+4
+1
+3
+4
+2
+Figure 10. Relations between induced subgraph and induced subtree.
+Definition 5.5. For every pair of graphs (G, H) such that G has no blossom and H has at
+most one blossom, let OccG(H) be the number of partial injection I from the vertex labels
+of G to N such that no blossom is marked and HI is isomorphic to G.
+Definition 5.6. For every pair of graphs (G, H) and a ∈ N such that G has no blossom,
+H has exactly one blossom and a is the label of a vertex of G, let OccG,a(H) be the number
+
+(6) = 1, J(7) = 2, J(3) = 1, J(4) = 3tGraph(ts)Graph(t)>J(6) = 1, (7) = 2, J(1) = 3, J(3) = 428
+TH´EO LENOIR
+of partial injection I from the vertex labels of G to N such that the image of the blossom
+by I is a and HI is isomorphic to G.
+2
+3
+1
+∗
+8
+5
+7
+9
+6
+6
+Figure 11. Two occurences of a P4 in a blossomed graph H. If G is a P4,
+the blue one is counted twice in OccG(H), the red one in counted once in
+OccG,a(H) iff a is the label of an extremity of G.
+Definition 5.7. For every graph G without blossom, and every a ∈ {1, . . . , N(G) = |G|},
+set:
+OccG,P(z) :=
+�
+H∈P
+OccG(H)zN(H)−N(G)
+N(H)!
+;
+OccG,P•(z) :=
+�
+H∈P•
+OccG(H)zN(H)−N(G)
+N(H)!
+OccG,a,P•(z) :=
+�
+H∈P•
+OccG,a(H)zN(H)−N(G)+1
+N(H)!
+Notation. OccG,... will only be used for graphs G with no blossom.
+Proposition 5.8. For every k ≥ 1 and every a ∈ {1, . . . , k}:
+�
+G: N(G)=k
+OccG,P(z) = P (k)(z)
+(28)
+�
+G: N(G)=k
+OccG,P•(z) = (P •)(k)(z)
+(29)
+�
+G: N(G)=k
+OccG,a,P•(z) = (P •)(k−1)(z)
+(30)
+Thus for every graph G with no blossom and every a ∈ {1, . . . , N(G)}, OccG,P, OccG,P•
+and OccG,a,P• have a radius of convergence strictly greater than R, the radius of convergence
+of T.
+Proof. Let H be an element of P. Since there are
+N(H)!
+(N(H)−k)! choices of partial injection
+whose image is {1, . . . , k}, we have:
+�
+G: N(G)=k
+OccG,P(z) =
+�
+H∈P
+�
+G: N(G)=k
+OccG(H)zN(H)−k
+N(H)!
+=
+�
+H∈P
+zN(H)−k
+(N(H) − k)! = P (k)(z)
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+29
+The proofs of Eqs. (29) and (30) are similar. In Eq. (30), since I−1(a) must be ∗, there
+are exactly
+N(H)!
+(N(H)−(k−1))! choices for the partial injection.
+For every graph G, OccG,P has nonnegative coefficients and for every k ≥ 0, as mentioned
+in Section 4.2, P (k) has a radius of convergence at least R0, the minimum of the radii of
+convergence P and P •, which is greater than R. This implies that OccG,P has a radius of
+convergence greater than R. The proof for the other series is similar.
+□
+5.2. Enumerations of trees with a given induced subtree. The key step in the proof
+of our main theorem is to compute the limiting probability (when n → ∞) that a uniform
+induced subtree of a uniform tree in TP,P• with n leaves is a given substitution tree.
+In the following, let τ ∈ T0 be a fixed substitution tree of size at least 2.
+Definition 5.9. We define Tτ to be the set of marked trees (t, I) where t ∈ TP,P• and I
+is such that tI is isomorphic to τ. We also define Tτ to be the corresponding exponential
+generating function (where the size parameter is the total number of leaves, including the
+marked ones).
+The aim now is to decompose a tree admitting τ as a subtree in smaller trees. Let
+(t, I) be in Tτ. A prime node v of τ is such that t[φ(v)] is either in case (D2) or (D4) of
+Definition 2.17: in other word, φ(v) must be a prime node. In constrast, knowing that an
+internal node v′ of τ is decorated with ⊕ or ⊖ does not give any information about the
+decoration of φ(v′).
+In order to state Theorem 5.11 below, we need to partition the internal nodes of τ:
+Definition 5.10. Let (t, I) be in Tτ. We denote by V(t, I) the set of internal nodes v of
+τ such that φ(v) is non-linear. The set V(t, I) can be partitioned in 4 subsets:
+• V0(t, I) the set of internal nodes v such that t[φ(v)] is in case (D2);
+• V1(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and no marked
+leaf is in the i-th tree of tφ(v) (where i is the element such that (D4) holds in
+Definition 2.17);
+• V2(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and exactly
+one marked leaf is in the i-th tree of tφ(v) (where i is the element such that (D4)
+holds in Definition 2.17);
+• V3(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and at least
+two marked leaves are in the i-th tree of tφ(v) (where i is the element such that (D4)
+holds in Definition 2.17).
+Note that the set of non-linear nodes of τ must be included in V(t, I). Since for every
+element v of V(t, I) at most one element of tφ(v) is non trivial, at most one element of τv
+is non trivial. Thus if τ has some non-linear nodes v such that two or more elements of τv
+are not reduced to a single leaf, Tτ = ∅. In the following, we assume that it is not the case
+for τ. If τv has exactly one non trivial subtree, then v ∈ V3(t, I). Otherwise, τv is a union
+of leaves.
+Notation. We denote by U0 (resp. U1) the set of internal nodes v of τ such that no tree
+(resp. exactly one tree) of τv has size greater or equal to 2.
+
+30
+TH´EO LENOIR
+Note that by definition V0(t, I) ∪ V1(t, I) ∪ V2(t, I) ⊂ U0 and V3(t, I) ⊂ U1.
+We also define rkt,I : V2(t, I) �→ N as follows. Let v ∈ V2(t, I), we define rkt,I(v) to
+be the only integer k such that, if ℓ is the label of the k-th leaf of τv then the leaf of label
+I−1(ℓ) in t belongs to the i-th tree of tφ(v) (where i is the element such that (D4) holds in
+Definition 2.17). For every v ∈ V2(t, I), we have 1 ≤ rkt,I(v) ≤ |τv|.
+Theorem 5.11. Let τ be a substitution tree of size at least 2 such that every non-linear
+node of τ is in U0 ∪ U1. Let V0, V1 and V2 be three disjoint subsets of U0 and let V3 be a
+subset of U1 such that every non-linear node of τ is in V := V0 ∪ V1 ∪ V2 ∪ V3. Let rk:
+V2 → N be such that 1 ≤ rk(w) ≤ |τw| for every w ∈ V2.
+Let Tτ,V0,V1,V2,V3,rk be the set of marked trees (t, I) in Tτ such that V0(t, I) = V0, V1(t, I) =
+V1, V2(t, I) = V2, V3(t, I) = V3, rkt,I = rk, and let Tτ,V0,V1,V2,V3,rk be its exponential gener-
+ating function.
+Then
+Tτ,V0,V1,V2,V3,rk = z|t|T root �
+T ⊕
+not⊕
+�d= �
+T ⊖
+not⊕
+�d̸= �
+T blo
+not⊕
+�dV →V �
+T
+′
+not⊕
+�dV →ℓ exp(nLTnot⊕)
+× T |V1|T ′|V2|(T ⊕)n1(T blo)n2F
+where
+F :=
+�
+v∈V0
+Occdec(v),P
+�
+v∈V3
+Occdec(v),br(v),P•
+�
+v∈V1
+Occdec(v),P•
+�
+v∈V2
+Occdec(v),rk(v),P•
+and:
+• d= is the number of edges between two internal nodes not in V with the same
+decoration (⊕ and ⊕, or ⊖ and ⊖);
+• d̸= is the number of edges between two internal nodes not in V decorated with
+different decorations (⊕ and ⊖);
+• dV →V is the number of edges between an internal node not belonging to V and one
+of its children belonging to V ;
+• dV →ℓ is the number of edges between an internal node not in V and a leaf;
+• nL is the number of internal nodes not in V ;
+• dec(v) is the decoration of v;
+• for every v ∈ V3, br(v) is the position of the subtree of τv not reduced to a leaf;
+• n1 (resp. n2) is the number of internal nodes v in V3 such that the root of the
+br(v)-th tree of τv is not in V (resp. is in V );
+• T root = T ⊕ if the root of τ is not in V , T root = T blo otherwise.
+Proof. Let t be a tree in Tτ,V0,V1,V2,V3,rk. We decompose t into several disjoints subtrees.
+The blossoms are nodes where (the root of) an other tree will be glued (and thus they are
+not counted in the generating series, to avoid counting them twice).
+We define t→root to be the tree t blossomed at φ(r0), where r0 is the root of τ.
+We define the tree tv→ in the following way:
+• If v is not in V , tv→ is the subtree of t containing φ(v) and all the subtrees of tφ(v)
+that do not contain a marked leaf of t.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+31
+V0
+V0
+V1
+V2
+V3
+V
+V
+Figure 12. A possible τ and choices of V0, V1, V2, V3
+• If v is in V0 ∪ V1 ∪ V2, tv→ is the tree t[φ(v)].
+• If v is in V3, tv→ is the tree t[φ(v)] obtained after blossoming the root of the non
+trivial tree of tφ(v). The blossom is marked with the smallest mark in the non trivial
+tree of tφ(v).
+For every internal nodes v, v′ in τ such that v is not in V and v′ is a child of v, let tv→v′
+be the unique tree of tφ(v) containing φ(v′), blossomed at φ(v′).
+For every internal node v in τ not in V , and every leaf f which is a child of v in τ, we
+define tv→f to be the subtree of tφ(v) containing φ(f).
+For every internal node v in V3, we define tv→br(v) to be the non trivial tree of tφ(v) blossomed
+at φ(v′), where v′ is the root of the br(v)-th tree of τv.
+Now we need to analyze the properties of the trees that appear in this decomposition
+and compute the corresponding exponential generating function. In the rest of the proof,
+we will say abusively that every blossomed tree belongs to TP,P•, and that two nodes both
+decorated with ⊕ or ⊖ have the same decoration, even if they do not have the same number
+of children.
+(i): analysis of t→root where v ̸∈ V
+The tree t→root is a tree in TP,P•, it has no marked leaf and a unique blossom. If the root
+is not in V and decorated with ⊕ (resp. ⊖), the blossom is ⊕-replaceable (see Definition 4.4)
+(resp. ⊖-replaceable). If the root is in V , the blossom is replaceable.
+
+32
+TH´EO LENOIR
+root
+ii
+iii
+iv
+iv
+iv
+iv
+v
+v
+vii
+vii
+vii
+vi
+ix
+tv,v′
+i
+viii
+viii
+x
+xi
+tv,f
+tv→
+t→root
+tv→br(v)
+Figure 13. The decomposition of a tree admitting the graph τ of Fig. 12
+as an induced tree. The different notations correspond to the different cases
+of the proof of Theorem 5.11.
+The corresponding exponential generating function is equal to T ⊕ if the root is not in
+V and equal to T blo otherwise.
+(ii): analysis of tv→v′ where v ̸∈ V and v′ is a child of v not in V with the same
+decoration
+The tree tv→v′ is a tree in TP,P• whose root is not decorated with the same decoration as
+v and with one blossom ⊕-replaceable if v′ is decorated with ⊕, ⊖-replaceable otherwise
+and no marked leaf.
+The exponential generating function of such trees is either T ⊕
+not⊕ if both nodes are deco-
+rated with ⊕ or T ⊖
+not⊖ if both nodes are decorated with ⊖, which are both equal.
+(iii): analysis of tv→v′ where v ̸∈ V and v′ is a child of v not in V with a different
+decoration
+The tree tv→v′ is a tree in TP,P• whose root is not decorated with the same decoration as
+v and with one blossom ⊕-replaceable if v′ is decorated with ⊕, ⊖-replaceable otherwise
+and no marked leaf.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+33
+The exponential generating function of such trees is either T ⊖
+not⊕ if v is decorated with
+⊕ and v′ with ⊖ or T ⊖
+not⊖ if v is decorated with ⊖ and v′ with ⊕, which are both equal.
+(iv): analysis of tv→v′ where v ̸∈ V and v′ is a child of v in V
+The tree tv→v′ is a tree in TP,P• whose root is not decorated with the decoration of v
+with one blossom and no marked leaf.
+The corresponding exponential generating function is T blo
+not⊕.
+(v): analysis of tv→f where v ̸∈ V and f is a leaf which is a child of v
+The tree tv→f is a tree in TP,P• whose root is not decorated with the decoration of v
+with one marked leaf and no blossom.
+The corresponding exponential generating function is zT ′
+not⊕.
+(vi): analysis of tv→br(v) where v ∈ V3
+The tree tv→br(v) is a tree with a blossom that is replaceable if the root of the br(v)-th
+subtree of t[v] is in V , ⊕-replaceable (resp. ⊖-replaceable) if the root is not in V and
+labeled ⊕ (resp. ⊖), with no marked leaf.
+The corresponding exponential generating function is equal to T ⊕ if the root of the
+br(v)-th tree of τv is not in V and equal to T blo otherwise.
+(vii): analysis of tv→ where v ̸∈ V
+The tree tv→ is a tree whose root denoted is decorated with the same decoration as v, who
+has no marked leaf and no blossom. It verifies all the conditions of being (P, P•)-consistent,
+except that the root can have 0 or 1 child.
+The corresponding exponential generating function is �
+k≥0 T k
+not⊕ = exp(Tnot⊕).
+(viii): analysis of tv→ where v ∈ V0
+The tree tv→ is a tree in TP,P• whose root is decorated with an element of P. The subtree
+induced by the marked leaves of tv→ is τ[v]. Moreover tv→ has only one internal node.
+The corresponding exponential generating function is
+�
+H∈P
+Occdec(v)(H)zN(H)
+N(H)!
+= zN(dec(v))Occdec(v),P.
+Indeed, for a given H ∈ P, the term zN(H)
+N(H)! correspond to the set of leaves and the term
+Occdec(v)(H) to the possible markings.
+(ix): analysis of tv→ where v ∈ V3
+The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.17. The subtree
+induced by the marked leaves of tv→ is τ[v], where the non-trivial tree of τv is replaced by
+a blossom, marked with the smallest mark in the non-trivial tree of τv. Moreover tv→ has
+only one internal node.
+Similarly to case (viii), the corresponding exponential generating function is:
+�
+H∈P•
+Occdec(v),br(v)(H)zN(H)
+N(H)!
+= zN(dec(v))−1Occdec(v),rk(v),P•.
+(x): analysis of tv→ where v ∈ V1
+
+34
+TH´EO LENOIR
+The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.17. The subtree
+induced by the marked leaves of tv→ is τ[v] and no marked leaf belongs to the i-th tree of
+tφ(v) (where i is the element such that (D4) holds in Definition 2.17).
+The corresponding exponential generating function is:
+�
+H∈P•
+Occdec(v)(H)zN(H)
+N(H)!
+× T = zN(dec(v))Occdec(v),P• × T.
+The sum corresponds to the choice of the root (as in the previous cases), and the factor
+T to the potential non trivial tree of tv.
+(
+¯
+xi): analysis of tv→ where v ∈ V2
+The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.17. The subtree
+induced by the marked leaves of tv→ is τ[v] and there is only one marked leaf ℓ in the i-th
+tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.17). Moreover, if
+we denote by j the label of ℓ, the label of the rk(v)-th leaf of τv is I(j).
+Similarly to case (x), the corresponding exponential generating function is:
+�
+H∈P•
+Occdec(v),rk(v)(H)zN(H)
+N(H)!
+× zT ′ = zN(dec(v))Occdec(v),rk(v),P• × T ′.
+All these conditions ensure that we can recover t by gluing all the different trees and that
+the subtree of t induced by I is τ. Thus, Tτ,V0,V1,V2,V3,rk is the product of the generating
+functions corresponding to labeled such trees and concludes the proof of the theorem.
+□
+Corollary 5.12. The series Tτ,V0,V1,V2,V3,rk has radius at least R, is ∆-analytic and its
+asymptotic expansion near R is:
+Tτ,V0,V1,V2,V3,rk = Cτ,V0,V1,V2,V3,rk
+�
+1 − z
+R
+�β
+(1 + o(1))
+where
+Cτ,V0,V1,V2,V3,rk := ακγ(1 + P •(R))θ(1 − P •(R))|V1|2λRµ × F(R)
+with
+β = −1 + d= + d̸= + dV →V + dV →ℓ + |V2| + |V3|
+2
+γ = dV →ℓ + |V2| − d= − d̸= − dV →V − |V3| − 1
+θ = d= + d̸= − |V1| − |V2| − n2 − nL
+λ = −dV →ℓ − n1 − 2d= − 2d̸= + dV →V + nL
+µ = −dV →ℓ − |V2| + l
+and α = 1
+2 if the root is not in V ,
+1
+1+P •(κ) otherwise.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+35
+6. Proof of the main theorems
+6.1. Background on graphons. We now review the necessary material on graphons.
+We refer the reader to [19] for a comprehensive presentation of deterministic graphons,
+while [7] studies specifically the convergence of random graphs in the sens of graphons.
+Here we will only recall the properties needed to prove the convergence of random graphs
+toward the Brownian cographon (see [1]).
+Definition 6.1. A graphon is an equivalence class of symmetric functions f : [0, 1]2 �→
+[0, 1], under the equivalence relation ∼, where f ∼ g if there exists a measurable function
+φ : [0, 1] �→ [0, 1] that is invertible and measure preserving such that, for almost every
+(x, y) ∈ [0, 1]2, f(φ(x), φ(y)) = g(x, y). We denote by ˜
+W the set of graphons.
+Intuitively graphons can be seen as continuous analogous of graph adjacency matrices,
+where graphs are considered up to relabeling (hence the quotient by ∼). There is a natural
+way to embed a finite graph into graphons:
+Definition 6.2. Let G be a (random) graph of size n. We define the (random) graphon
+WG to be the equivalence class of wG : [0, 1]2 �→ [0, 1] defined by:
+∀(x, y) ∈ [0, 1]2
+wG(x, y) := 1⌈nx⌉connected to⌈ny⌉
+There exists a metric δ□ on the set of graphons ˜
+W such that ( ˜
+W, δ□) is compact [19,
+Chapter 8], thus we can define for δ□ the convergence in distribution of a random graphon.
+If (G(n))n≥1 is a sequence of random graphs, there exists a simple criterion [7, Theorem
+3.1] characterizing the convergence in distribution of (WG(n)) with respect to δ□:
+Theorem 6.3 (Rephrasing of [7], Theorem 3.1). For any n, let G(n) be a random graph of
+size n. Denote by WG(n) the random graphon associated to G(n). The following assertions
+are equivalent:
+(a) The sequence of random graphons (WG(n))n≥1 converges in distribution to some
+random graphon W.
+(b) The random infinite vector
+�
+OccG(n)(H)
+n(n−1)...(n−|H|+1)
+�
+H finite graph converges in distribution
+in the product topology to some random infinite vector (ΛH)H finite graph.
+For a finite graph H, the random variable ΛH can be seen as the density of the pattern
+H in the graphon W: the variables (ΛH)H play the roles of margins of W in the space of
+graphons.
+For k ≥ 1 and W a random graphon, we denote by Samplek(W) the unlabeled random
+graph built as follows: Samplek(W) has vertex set {v1, v2, . . . , vk} and, letting (X1, . . . , Xk)
+be i.i.d. uniform random variables in [0, 1], we connect vertices vi and vj with probability
+w(Xi, Xj) (these events being independent, conditionally on (X1, · · · , Xk) and W). The
+construction does not depend on the representation of the graphon.
+With the notations of Theorem 6.3, we have for any finite graph H
+ΛH = P(Sample|H|(W) = H | W).
+
+36
+TH´EO LENOIR
+The article [1] introduces a random graphon W1/2 called the Brownian cographon which
+can be explicitly constructed as a function of a realization of a Brownian excursion. Besides,
+[1, Proposition 5] states that the distribution of the Brownian cographon is characterized2
+by the fact that for every k ≥ 2, Samplek(W1/2) has the same law as the unlabeled
+version of Graph(bk) with bk a uniform labeled binary tree with k leaves and i.i.d. uniform
+decorations in {⊕, ⊖}.
+A consequence of this characterization is a simple criterion for convergence to the Brow-
+nian cographon.
+Lemma 6.4 (Rephrasing of [1] Lemma 4.4). For every positive integer n, let T(n) be a
+uniform random tree in TP,P• with n vertices.
+For every positive integer ℓ, Iℓ
+(n) be a
+uniform partial injection from {1, . . . , n} to N whose image is {1, . . . , ℓ} and independent
+of T(n). Denote by T(n)
+Iℓ(n) the subtree induced by Iℓ
+(n).
+Suppose that for every ℓ and for every binary tree τ with ℓ leaves,
+(31)
+P(T(n)
+I(n) = τ) −−−→
+n→∞
+(ℓ − 1)!
+(2ℓ − 2)!.
+Then WGraph(T(n)) converges as a graphon to the Brownian cographon W1/2 of parameter
+1/2.
+6.2. Conclusion of the proof of Theorem 1.1.
+Proposition 6.5. Let τ be a binary tree with ℓ ≥ 2 leaves. The series Tτ has radius of
+convergence R, is ∆-analytic and its asymptotic expansion near R is:
+Tτ =
+κ
+(1 + P •(R))22ℓ−2
+�
+1 − z
+R
+�− 2ℓ−1
+2
+(1 + o(1)) .
+(32)
+Proof. As
+Tτ =
+�
+τ,V0,V1,V2,V3,rk
+Tτ,V0,V1,V2,V3,rk,
+the asymptotic expansions of the different series Tτ,V0,V1,V2,V3,rk yield the ∆-analyticity of
+Tτ, its asymptotic expansion and its radius of convergence.
+Note that β ≤
+1+e
+2
+where e is the number of edge of τ, with equality if and only if
+V0, V1, V2 and V3 are all empty.
+Therefore, only the series Tτ,∅,∅,∅,∅,rk contributes to the leading term of the asymptotic
+expansion. In this case, dV →ℓ = ℓ, d= + d̸= = ℓ − 2 and nL = ℓ − 1 which gives the
+announced expansion.
+□
+Theorem 6.6. Let τ be a binary tree with ℓ ≥ 2 leaves.
+For n ≥ ℓ and T(n) be a
+uniform random tree in TP,P• with n vertices. Let Iℓ
+(n) be a uniform partial injection from
+{1, . . . , n} to N whose image is {1, . . . , ℓ} and independent of T(n). Denote by T(n)
+Iℓ(n) the
+subtree induced by Iℓ
+(n).
+2This characterization is strongly linked to the remarkable property that k uniform leaves in the CRT
+induce a uniform binary tree with k leaves, see again [1, Section 4.2].
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+37
+Then
+P(T(n)
+Iℓ(n) = τ) −−−→
+n→∞
+(ℓ − 1)!
+(2(ℓ − 1))!.
+Proof. Since Iℓ
+(n) is independent of T(n),
+P(T(n)
+Iℓ(n) = τ) =
+n![zn]Tτ
+n(n − 1) . . . (n − ℓ + 1)n![zn]T =
+[zn]Tτ
+n(n − 1) . . . (n − ℓ + 1)[zn]T
+By applying the Transfer Theorem [8, Corollary VI.1 p.392] to Eq. (32), we get
+[zn]Tτ ∼
+κ
+(1 + P •(R))22ℓ−2
+n
+2ℓ−3
+2
+Γ
+�
+2ℓ−1
+2
+�
+Rn
+and by Corollary 4.13 we obtain
+n × · · · × (n − ℓ + 1)[zn]T ∼ nℓ
+κ
+√π(1 + P•(R))
+1
+Rnn
+3
+2 .
+Thus when n goes to infinity
+P(T(n)
+Iℓ(n) = τ) →
+√π
+22ℓ−2Γ
+�
+2ℓ−1
+2
+� =
+(ℓ − 1)!
+(2(ℓ − 1))!
+□
+Combining Lemma 6.4 and Theorem 6.6 prove Theorem 6.7 of which Theorem 1.1 is a
+particular case.
+Theorem 6.7. Let G(n) be a uniform random graph in GP,P• with n vertices. We have the
+following convergence in distribution in the sense of graphons:
+WG(n)
+n→∞
+−→ W
+1
+2
+where W
+1
+2 is the Brownian cographon of parameter 1
+2.
+6.3. Number of induced prime subgraphs. We now estimate for a prime graph H the
+number OccH(G(n)) of induced occurences of H in G(n) and show that in average it is null,
+linear or of order n
+3
+2.
+We first observe that substitution trees encoding prime graphs have a very simple struc-
+ture.
+Lemma 6.8. Let H be a prime graph. If t is a substitution tree such that H = Graph(t),
+t is reduced to a single internal node decorated with a relabeling of H with |H| leaves.
+Proof. Let t be such a tree and r its root. To every element t′ of tr we can associate a
+module of H by taking the vertices whose labels are the labels of the leaves of t′. Thus tr
+is a union of leaves, and the decoration of the root is a relabeling of H.
+□
+We say that H verifies (A) if there exists a ∈ {1, . . . , ℓ} such that OccG,a,P•(R) > 0.
+
+38
+TH´EO LENOIR
+Theorem 6.9. Let H be a prime graph and let ℓ be its size. For n ≥ ℓ, let G(n) be a
+uniform random graph in GP,P• with n vertices.
+Then if H verifies (A),
+E[OccH(G(n))] ∼ KHn
+3
+2
+with
+KH =
+Rℓ−1√π
+�
+a∈{1,...,ℓ}
+OccH,a,P•(R)
+κ(1 + P •(R))
+otherwise,
+E[OccH(G(n))] ∼ KHn
+with
+KH =
+�1 − P •(R)
+1 + P •(R)OccH,P•(R) + OccH,P(R)
+� Rℓ
+κ2
+Proof. Let T(n) be a uniform random tree in TP,P• with n vertices .
+Let τ be the canonical tree of H and NT(n),τ the number of induced subtrees of Tn
+isomorphic to τ. Since τ is the unique substitution tree of G, E[OccH(G(n))] = E[NT(n),τ].
+By independence
+E[OccH(G(n))] = n![zn]Tτ
+n![zn]T = [zn]Tτ
+[zn]T .
+From Theorem 5.11, since in this case the only node of τ is either in V0, V1 or V2, we
+have that:
+Tτ = zℓT blo
+�
+�T ′
+�
+�
+�
+a∈{1,...,ℓ}
+OccG,a,P•
+�
+� + TOccG,P• + OccG,P
+�
+� .
+Thus
+• in case (A), with Eqs. (22) and (26)
+Tτ ∼
+Rℓ
+R(1 + P•(R))2
+�
+�
+�
+a∈{1,...,ℓ}
+OccH,a,P•(R)
+�
+�
+�
+1 − z
+R
+�−1
+;
+• Otherwise, Tτ ∼
+� 1−P •(R)
+1+P •(R)OccH,P•(R) + OccH,P(R)
+�
+Rℓ
+κ(1+P •(R))
+�
+1 − z
+R
+�− 1
+2 .
+By applying the Transfer Theorem [8, Corollary VI.1 p. 392],
+• In case (A),
+[zn]Tτ ∼
+Rℓ
+Rn+1(1 + P•(R))2
+�
+a∈{1,...,ℓ}
+OccH,a,P•(R)
+• Otherwise,
+[zn]Tτ ∼
+�1 − P •(R)
+1 + P •(R)OccH,P•(R) + OccH,P(R)
+�
+Rℓ
+√πκ(1 + P•(R))
+1
+Rnn
+1
+2
+.
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+39
+By Corollary 4.13,
+[zn]T ∼
+κ
+√π(1 + P•(R))
+1
+Rnn
+3
+2 .
+Thus:
+• In case (A),
+E[OccG(G(n))] ∼
+Rℓ−1√π
+�
+a∈{1,...,ℓ}
+OccG,a,P•(R)
+κ(1 + P •(R))
+n
+3
+2,
+• Otherwise,
+E[OccG(G(n))] ∼
+�1 − P •(R)
+1 + P •(R)OccG,P•(R) + OccG,P(R)
+� Rℓ
+κ2 n,
+concluding the proof.
+□
+An interesting application of this theorem is the computation of the asymptotic number
+of ˜P4’s in a random uniform graph of each of the graph classes of Section 3, where ˜P4 is
+the only labeling of P4 with endpoints 1 and 4 and 2 connected to 1.
+Lemma 6.10. A prime spider has exactly |K|(|K| − 1) induced ˜P4. A pseudo-spider of
+size k has exactly (|K| + 2)(|K| − 1) induced ˜P4.
+Proof. One can check that for a prime spider, the P ′
+4s are induced by the partial injections
+I whose domain is {k, k′, f(k), f(k′)} for every (k, k′) ∈ K2 with k ̸= k′. In the 24 such
+partial injections, only 2 are such that the graph induced is ˜P4. Since every induced ˜P4 is
+counted twice, we have |K|(|K| − 1) induced ˜P4.
+For a pseudo-spider, let d be the duplicate and d0 the original node. The P ′
+4s are induced
+by the partial injections I whose domain is {k, k′, f(k), f(k′)} for every (k, k′) ∈ K2 with
+k ̸= k′, or by the partial injections I whose domain is {d, k′, f(d0), f(k′)} (resp. {f −1(d0), k′, d, f(k′)})
+for every k′ ∈ K with k′ ̸= d0 (resp. k′ ̸= f −1(d0)) if d0 is in K (resp. in S). In the 24 such
+partial injections, only 2 are such that the graph induced is ˜P4. Since every induced ˜P4
+not containing d is counted twice, we have |K|(|K| − 1) + 2(|K| − 1) = (|K| + 2)(|K| − 1)
+induced ˜P4.
+□
+Remark. Note that this lemma implies that Occ ˜
+P4,a,P• = 0 for all the graph classes men-
+tionned in Section 3.
+Theorem 6.11. For each graph class introduced in Section 3, we have the following ex-
+pressions for Occ ˜
+P4,P and Occ ˜
+P4,P•:
+
+40
+TH´EO LENOIR
+P4-tidy
+Occ ˜
+P4,P•
+tidy(z) = (2 + 16z + 4z3) exp(z2) − 1 − 8z
+Occ ˜
+P4,Ptidy(z) = Occ ˜
+P4,P•
+tidy(z) + 5z
+P4-lite
+Occ ˜
+P4,P•
+lite(z) = (2 + 16z + 4z3) exp(z2) − 1 − 8z
+Occ ˜
+P4,Plite(z) = Occ ˜
+P4,P•
+lite(z) + 4z
+P4-extendible
+Occ ˜
+P4,P•
+ext(z) = 1 + 8z
+Occ ˜
+P4,Pext = Occ ˜
+P4,P•
+ext(z) + 5z
+P4-sparse
+Occ ˜
+P4,P•spa(z) = Occ ˜
+P4,Pspa(z) = 2 exp(z2) − 1
+P4-reducible
+Occ ˜
+P4,P•
+red(z) = Occ ˜
+P4,Pred = 1
+P4-free
+Occ ˜
+P4,P•cog(z) = Occ ˜
+P4,Pcog(z) = 0
+Proof. We only detail the computation of Occ ˜
+P4,P•
+tidy and Occ ˜
+P4,Ptidy for P4-tidy graphs as
+this is the most involved case. Note that, with the notations of Section 4.1,
+Occ ˜
+P4,P(z) =
+�
+n∈N
+�
+H∈RPn
+�
+H′∼H
+Occ ˜
+P4(H)zN(H)−4
+N(H)!
+=
+�
+n∈N
+�
+H∈RPn
+Occ ˜
+P4(H)zN(H)−4
+|Aut(H)|
+and similarly
+Occ ˜
+P4,P•(z) =
+�
+n∈N
+�
+H∈RP•n
+�
+H′∼H
+Occ ˜
+P4(H)zN(H)−4
+N(H)!
+=
+�
+n∈N
+�
+H∈RPn
+Occ ˜
+P4(H)zN(H)−4
+|Aut(H)|
+According to Theorem 3.7, Ptidy is composed of one C5 that has 10 automorphisms and
+10 induced ˜P4 and all its relabelings, one P5, and one P5 that both have 2 automorphisms
+and 4 induced ˜P4’s and all their relabelings.
+For k ≥ 3 (resp. k = 2), there are thin and fat spiders corresponding to the 2 (resp. 1)
+different orbits of the action Φ2k over prime spiders of size 2k, each having k! automorphisms
+and k(k − 1) ˜P4’s.
+For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can
+come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4)
+cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over pseudo-spiders
+of size 2k + 1, each having 2(k − 1)! automorphisms and (k + 2)(k − 1) ˜P4’s.
+Thus we have
+Occ ˜
+P4,P(z) = z + 4z
+2 + 4z
+2 + 2
+2 + 2
+�
+k≥3
+k(k − 1)z2k−4
+k!
++ 44z
+2 + 8
+�
+k≥3
+(k + 2)(k − 1)z2k−3
+2(k − 1)!
+= 5z + 1 + 2
+�
+k≥1
+z2k
+k! + 8z + 4
+�
+k≥1
+(k + 4)z2k+1
+k!
+= 5z + 1 + 2 exp(z2) − 2 + 8z + 4
+�
+k≥0
+z2k+3
+k!
++ 16
+�
+k≥1
+z2k+1
+k!
+
+GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS
+41
+= 5z + 2 exp(z2) − 1 + 4z3 exp(z2) + 16z exp(z2) − 8z
+= 5z + (2 + 16z + 4z3) exp(z2) − 1 − 8z
+Now let’s compute Occ ˜
+P4,P•(z). For k ≥ 3 (resp. k = 2), there are thin and fat spiders
+with blossom corresponding to the 2 (resp. 1) different orbits of the action Φ2k over blos-
+somed prime spiders G with 2k non blossomed vertices, each having k! automorphisms and
+k(k − 1) ˜P4’s.
+For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can
+come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4)
+cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over blossomed
+pseudo-spiders with 2k + 1 non blossomed vertices, each having 2(k − 1)! automorphisms
+and (k + 2)(k − 1) ˜P4’s.
+Hence
+Occ ˜
+P4,P•(z) = 2
+2 + 2
+�
+k≥3
+k(k − 1)z2k−4
+k!
++ 44z
+2 + 8
+�
+k≥3
+(k + 2)(k − 1)z2k−3
+2(k − 1)!
+Thus Occ ˜
+P4,P•(z) + 5z = Occ ˜
+P4,P(z) which gives the announced result.
+□
+Combining Theorem 6.9, Theorem 6.11 and the remark above, we get that ˜P4 does not
+verify (A), thus ˜P4 belongs to the linear case of Theorem 6.9:
+Corollary 6.12. Let G(n) be a uniform graph of size n taken uniformly at random in one
+of the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible, P4-reducible or P4-free.
+Then E[Occ ˜
+P4(G(n))] ∼ K ˜
+P4n where K ˜
+P4 is defined in Theorem 6.9.
+Here are the numerical approximations of K ˜
+P4 in the different cases:
+class of graph
+K ˜
+P4
+P4-tidy
+0.29200322
+P4-lite
+0.28507010
+P4-extendible
+0.24959979
+P4-sparse
+0.10280703
+P4-reducible
+0.08249263
+P4-free
+0
+Acknowledgements. I would like to thank Lucas Gerin and Fr´ed´erique Bassino for
+useful discussions and for carefully reading many earlier versions of this manuscript.
+References
+[1] F. Bassino, M. Bouvel, V. F´eray, L. Gerin, M. Maazoun, and A. Pierrot. Random cographs: Brownian
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+graphs I: Subgraph frequencies, metric properties and testing. Adv. Math., 219(6):1801–1851, 2008.
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+[15] B. Jamison and S. Olariu. P4-reducible graphs—class of uniquely tree-representable graphs. Stud.
+Appl. Math., 81(1):79–87, 1989.
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+Math., 34(1-3):151–164, 1991.
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+Th´eo Lenoir theo.lenoir@polytechnique.fr
+Cmap, Cnrs, ´Ecole polytechnique,
+Institut Polytechnique de Paris,
+91120 Palaiseau, France
+

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+Model selection in atomistic simulation
+Jonathan E. Moussa
+Molecular Sciences Software Institute, Virginia Tech, Blacksburg, Virginia 24060, USA
+(*Electronic mail: godotalgorithm@gmail.com)
+There are many atomistic simulation methods with very different costs, accuracies, transferabilities, and numbers of empirical parameters.
+I show how statistical model selection can compare these methods fairly, even when they are very different. These comparisons are also
+useful for developing new methods that balance cost and accuracy. As an example, I build a semiempirical model for hydrogen clusters.
+I.
+INTRODUCTION
+Scientists have been building quantitative atomistic models for
+over a century1. In that time, many atomistic models have evolved
+into sophisticated computer simulations2.
+While there are now
+models based on a wide variety of atomistic simulation methods,
+most development has focused on two contradictory goals. Classi-
+cal molecular mechanics (MM) methods focus on minimizing cost
+to access phenomena at large length scales and long time scales3.
+However, the use of MM methods is limited by the availability and
+accuracy of system-specific interatomic potentials4. In contrast,
+first-principles quantum mechanics (QM) methods focus on mini-
+mizing error for general-purpose simulations5, which can get very
+expensive. MM methods can achieve simulation costs of less than
+10−5 CPU-seconds per atom6, while high-accuracy QM methods
+have asymptotic costs greater than 104 CPU-seconds per atom7.
+Because of the large gaps in cost and utility, there are many
+atomistic simulation tasks for which QM methods are too expensive
+and MM methods have no suitable interatomic potential. In this
+situation, a scientist needs an affordable model and must either
+develop their own or use an existing one such as a semiempirical
+QM (SQM) model8,9. In either case, they need to collect evidence
+to support their model. They must either gather enough reference
+data to fit a new model, or find enough examples of scientists
+using an existing model for similar tasks to be confident that it will
+work for them. This type of model selection process is a common
+occurrence in atomistic science, and yet it remains rather informal
+and subjective much of the time.
+In this paper, I advocate for using statistical model selection10
+to develop and compare models for atomistic simulation. All else
+being equal, a scientist should fit or choose a model to maximize
+the probability that they will succeed at their simulation task. Since
+the exact probability will be more expensive to compute than the
+simulation task itself, they must rely on a proxy probability based
+on related but simpler simulation tasks. Assumptions about the
+transferability of a method’s accuracy between related simulation
+tasks are unavoidable in atomistic science. Also, when considering
+methods with different numbers of fitting parameters or costs, extra
+penalties are needed to avoid overfitting or exceeding computa-
+tional budgets. These same principles apply to the development
+of general-purpose models that are intended to be used by many
+scientists over a broad distribution of simulation tasks.
+As an example, I apply statistical model selection to the task
+of simulating random hydrogen clusters. First, I generate high-
+accuracy QM reference data. Second, I compare the accuracy of
+some popular SQM models and density functionals from density
+functional theory (DFT)11. Third, I build new SQM models by
+correcting this SQM and QM data with atomic pair potentials. Here,
+model selection determines the optimal number of parameters in
+the pair potentials and the computational budget thresholds for
+switching between models.
+II.
+STATISTICAL MODEL SELECTION
+The standard practice in fitting atomistic models with param-
+eters is to minimize a distance between model predictions and
+reference data. I consider vectors of 𝑚 reference data points x and
+model predictions y(λ), which are determined by 𝑛 real parame-
+ters λ. The value of λ is usually chosen by minimizing the mean
+absolute error (MAE),
+∥x − y(λ)∥1 =
+𝑚
+∑︁
+𝑖=1
+|𝑥𝑖 − 𝑦𝑖(λ)|,
+(1)
+or the root-mean-square deviation (RMSD),
+∥x − y(λ)∥2 =
+�
+� 𝑚
+∑︁
+𝑖=1
+[𝑥𝑖 − 𝑦𝑖(λ)]2.
+(2)
+The general expectation is that smaller distances correspond to bet-
+ter accuracy and thus a higher chance of success when these models
+are used for other simulation tasks. However, this relationship is
+indirect because these distances are not operational measures of
+success. An operational measure would describe the application of
+a model by scientists in a more explicit and direct way, including
+how successful they are. Directly optimizing an operational mea-
+sure should produce a more successful model if the operational
+measure itself is sufficiently accurate.
+To use statistical model
+selection as an operational measure in this context, I must first
+introduce two distinct sources of randomness.
+The first source of randomness is in the model predictions.
+I consider a generalization of the reference information from data
+points x to simulation tasks X. Each reference simulation task 𝑋𝑖
+defines one or more physical systems and calculations to perform,
+together with reference output data and success criteria. The con-
+ditional probability of success, 𝑝(λ|𝑋𝑖), after choosing a task 𝑋𝑖
+and using a model with parameters λ replaces a distance between
+𝑥𝑖 and 𝑦𝑖(λ). The only constraint on the success criteria is that the
+success probability for the method used to generate the reference
+data must be one.
+Viable models must always have a nonzero
+success probability, which requires the model output or success
+criteria to have a random component.
+The second source of randomness is in the choice of reference
+simulation tasks. I relate a set of reference simulation tasks to
+the actual simulation task that a scientist wants to succeed at by
+arXiv:2301.05287v1  [physics.chem-ph]  12 Jan 2023
+
+2
+considering them to be randomly drawn from a common distribution
+of simulation tasks. The probability of choosing a simulation task
+𝑋 is 𝑝(𝑋), and the probability of choosing this task and then
+succeeding with the model is
+𝑝(λ, 𝑋) = 𝑝(λ|𝑋)𝑝(𝑋).
+(3)
+It is not strictly necessary for the simulation tasks to have been
+randomly drawn from this distribution. Such a distribution is still
+formally useful even when it is an artificial context and not even
+precisely defined. It is simply the mathematical representation of
+a computational scientist as a distribution over simulation tasks.
+A.
+Maximum likelihood estimation
+I now apply the framework of maximum likelihood estimation
+(MLE)10 to determine the best model in this randomized setting.
+The operational measure of modeling success is the probability of
+succeeding at all 𝑚 reference simulation tasks,
+𝑃(λ) =
+𝑚
+�
+𝑖=1
+𝑝(λ|𝑋𝑖).
+(4)
+It is related to a statistical likelihood function,
+𝐿(λ) =
+𝑚
+�
+𝑖=1
+𝑝(λ, 𝑋𝑖) = 𝑃(λ)
+𝑚
+�
+𝑖=1
+𝑝(𝑋𝑖),
+(5)
+over the joint distribution of simulation tasks and modeling success
+or failure events. I follow the common convention of considering
+the negative logarithm of the probability or likelihood,
+− log 𝑃(λ) = −
+𝑚
+∑︁
+𝑖=1
+log 𝑝(λ|𝑋𝑖)
+= − log 𝐿(λ) +
+𝑚
+∑︁
+𝑖=1
+log 𝑝(𝑋𝑖),
+(6)
+which replaces the product over reference simulation tasks with a
+more convenient sum. The negative logarithm is a strictly mono-
+tonically decreasing function, and maximizing it corresponds to
+maximizing the likelihood. Since 𝑝(𝑋𝑖) has no dependence on λ,
+𝑃(λ) and 𝐿(λ) are maximized by the same value of λ.
+The familiar case of minimizing RMSD follows from a simple
+success criterion and error model. I assume that each simulation
+task 𝑋𝑖 produces a single model output 𝑦𝑖(λ) that must be within 𝜖
+of a reference value 𝑥𝑖 for success. I further adjust each model output
+by a Gaussian error model with mean 𝜇 and standard deviation 𝜎
+to guarantee a finite success probability. Each success probability
+reduces to a quadratic penalty for small 𝜖 values,
+− log 𝑝(λ|𝑋𝑖) = − log
+∫
+𝑥𝑖+𝜖
+𝑥𝑖−𝜖
+𝑒−0.5[𝑧−𝜇−𝑦𝑖 (λ)]2/𝜎2
+𝜎
+√
+2𝜋
+𝑑𝑧
+≈ [𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2
+2𝜎2
++ 1
+2 log 𝜋𝜎2
+2𝜖2 + 𝑂(𝜖).
+(7)
+While not clear from this notation, error model parameters such as
+𝜇 and 𝜎 are also considered to be part of the parameter vector λ.
+In the small-𝜖 limit, the operational measure of success reduces to
+an RMSD-like formula,
+− log 𝑃(λ) ≈ 𝑚
+2 log 𝜋𝜎2
+2𝜖2 +
+𝑚
+∑︁
+𝑖=1
+[𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2
+2𝜎2
+.
+(8)
+When minimizing this formula over 𝜇 and 𝜎, the minimizers are
+the mean and standard deviation of the model error distribution,
+𝜇 =
+𝑚
+∑︁
+𝑖=1
+𝑥𝑖 − 𝑦𝑖(λ)
+𝑚
+,
+𝜎 =
+�
+� 𝑚
+∑︁
+𝑖=1
+[𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2
+𝑚
+.
+(9)
+The remaining minimization over λ is equivalent to minimizing
+the RMSD with a model bias correction of 𝜇. The minimum value
+of the small-𝜖 success measure is
+− log 𝑃(λ) ≈ 𝑚
+2 + 𝑚
+2 log 𝜋𝜎2
+2𝜖2
+(10)
+for 𝜎 in Eq. (9), which is a monotonically increasing function of the
+bias-corrected RMSD. In the absence of model bias, the RMSD
+and success measure thus produce the same minimizing models
+and rank them in the same order.
+Using a Gaussian distribution to approximate model errors is
+justified when they come from an accumulation of many small,
+independent errors. A non-zero mean suggests that these small
+errors are biased on average. The same small-𝜖 analysis can relate
+a similar success measure to the MAE if the underlying error model
+is a Laplace distribution,
+𝜌(𝑥) = 𝑒−
+√
+2|𝑥−𝜇|/𝜎
+𝜎
+√
+2
+.
+(11)
+However, non-Gaussian error distributions suggest a small number
+of dominant, independent error sources that avoid the inevitable
+consequences of the central limit theorem.
+Also, the Laplace
+distribution has a fatter tail than a Gaussian distribution, which
+implies an increased tolerance of large error outliers. Ultimately,
+the choice of distributions in an error model should be informed
+by the observed distribution of errors between model and data.
+A more sophisticated MLE example is a multi-Gaussian error
+model. Here, we partition the reference simulation tasks into 𝑟
+groups of similar tasks, each with their model errors described
+by a different Gaussian distribution. Such grouping is appropriate
+when different groups of tasks are observed to have different error
+statistics for models under consideration12. The small-𝜖 limit of
+the success measure generalizes from Eq. (8) to
+− log 𝑃(λ) ≈
+𝑟∑︁
+𝑖=1
+𝑚𝑖
+2 log 𝜋𝜎2
+𝑖
+2𝜖2
++
+𝑟∑︁
+𝑖=1
+𝑚𝑖
+∑︁
+𝑗=1
+[𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) − 𝜇𝑖]2
+2𝜎2
+𝑖
+,
+(12)
+where the extra index is for the groups. The minimizing 𝜇𝑖 and 𝜎𝑖
+values generalize from Eq. (9) to
+𝜇𝑖 =
+𝑚𝑖
+∑︁
+𝑗=1
+𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ)
+𝑚𝑖
+,
+𝜎𝑖 =
+�
+� 𝑚𝑖
+∑︁
+𝑖=1
+[𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) − 𝜇𝑖]2
+𝑚𝑖
+.
+(13)
+
+3
+The minimization over λ is now equivalent to a weighted, bias-
+corrected RMSD with weights proportional to the inverse error
+variance. However, the minimum value of the success measure,
+− log 𝑃(λ) ≈ 𝑚
+2 +
+𝑟∑︁
+𝑖=1
+𝑚𝑖
+2 log 𝜋𝜎2
+𝑖
+2𝜖2 ,
+(14)
+no longer ranks minimizing models in the same order as the cor-
+responding weighted RMSD. Thus MLE rapidly deviates from
+minimizing simple distances between model and reference data as
+success criteria and error models get more complicated.
+Beyond these simple examples, MLE can provide a lot of
+flexibility to the model-fitting process. It is possible to fit low-
+cost models that are designed to have only qualitative accuracy
+by choosing success criteria that tolerate large but well-shaped
+errors. For example, conformer searches only need to preserve the
+order of conformer energies, which can be tested by the Spearman
+rank correlation coefficient13. When fitting very accurate models,
+many reference simulation tasks may have success probabilities
+very close to one and effectively vanish from log 𝑃(λ). In this
+highly successful regime, error outliers in a model will have a
+greatly enhanced influence on the success measure and MLE may
+become functionally equivalent to minimax optimization.
+B.
+Information criteria
+Simple MLE is capable of selecting the best model from one
+family of models parameterized by λ, but it cannot reliably compare
+models from different families. Adding more free parameters to an
+existing model and optimizing them can only improve the success
+measure, and nested models with more parameters will always be
+preferred. This can eventually cause the modeling phenomenon
+of fitting noise rather than data, and there needs to be additional
+modeling criteria for eliminating parameters that are not useful.
+The most common approach is to introduce a penalty for adding
+model parameters that is overcome by useful parameters. Such
+measures of model accuracy with penalties for parameters are
+called information criteria (IC), the oldest and most famous of
+which is the Akaike information criterion (AIC)14. The Takeuchi
+information criterion (TIC)15 is a more complicated generalization
+of the AIC that does not assume model accuracy. Here, I provide
+a minimal motivation and derivation of the TIC and AIC to justify
+their use in fitting models for atomistic simulation.
+An implicit assumption about both the IC derivations and
+MLE itself is that 𝑃(λ) can be optimized over λ effectively in
+practice. The mathematical structure of 𝑃(λ) depends on both
+the model family and the success criteria of simulation tasks. I
+specifically assume that 𝑃(λ) is twice differentiable with respect to
+λ and that derivative information is used to find local minimizers.
+I also assume that it is possible to choose initial values for λ in
+the basin of convergence for the global minimizer. While there is
+not enough structure here to guarantee or verify global minima,
+there are often physical considerations to guide reasonable choices
+of initial λ values.
+Both the AIC and TIC come from attempting to change the
+modeling success measure from Eq. (6) to
+𝐷(λ) = −𝑚
+∑︁
+𝑋
+𝑝(𝑋) log 𝑝(λ|𝑋),
+(15)
+which is 𝑚 times the Kullback-Leibler divergence16 of the always
+successful reference distribution from the model distribution that
+can fail at simulation tasks. Minimizing this divergence maximizes
+the asymptotic success probability for any large number of simula-
+tion tasks drawn from the model distribution16. While this is more
+reliable than only maximizing the success probability for a specific
+set of 𝑚 simulation tasks, 𝐷(λ) and its minimizer ˆλ cannot be
+calculated efficiently in general. The practical alternative is to use
+− log 𝑃(λ) and its minimizer ˆλX to approximate these inaccessi-
+ble quantities. To clarify their relationship, I use two convenient
+intermediates,
+𝐷X(λ) = −
+𝑛
+∑︁
+𝑖=1
+log 𝑝(λ|𝑋𝑖),
+∑︁
+X
+=
+∑︁
+𝑋1
+𝑝(𝑋1) · · ·
+∑︁
+𝑋𝑛
+𝑝(𝑋𝑛),
+(16)
+to simplify the notation during the IC derivations.
+For a constant value of λ, 𝐷X(λ) is an unbiased estimator of
+𝐷(λ) when averaged over sets of 𝑚 simulation tasks X,
+𝐷(λ) =
+∑︁
+X
+𝐷X(λ).
+(17)
+Since I cannot efficiently calculate ˆλ, I would like to evaluate 𝐷(λ)
+at one λ = ˆλX value that I can calculate. If this was repeated and
+averaged over sets of 𝑚 simulation tasks, it would be an unbiased
+estimator of
+𝐷min−ave =
+∑︁
+X
+𝐷(ˆλX).
+(18)
+However, with a single X, I can only evaluate 𝐷X(λ) at its own
+minimum, λ = ˆλX, which is an unbiased estimator of the average
+minimum,
+𝐷ave−min =
+∑︁
+X
+𝐷X(ˆλX).
+(19)
+This has a negative bias relative to 𝐷(ˆλX) because each 𝐷(λ) is
+evaluated at its own minimum instead of a common λ. A single
+𝐷X(ˆλX) can be unbiased as an estimator of 𝐷(ˆλX) by adding a
+bias correction,
+Δ = 𝐷min−ave − 𝐷ave−min =
+∑︁
+X
+[𝐷(ˆλX) − 𝐷X(ˆλX)].
+(20)
+I approximate Δ with several simplifying assumptions.
+The first IC assumption is that 𝐷(λ) and 𝐷X(λ) are both
+slowly changing in a region containing ˆλ and ˆλX. Both functions
+can be extrapolated from their minimum to the other function’s
+minimum with a second-order Taylor expansion,
+𝐷(ˆλX) ≈ 𝐷(ˆλ) + 1
+2 (ˆλX − ˆλ)𝑇 F(ˆλX − ˆλ),
+𝐷X(ˆλ) ≈ 𝐷X(ˆλX) + 1
+2 (ˆλ − ˆλX)𝑇 FX(ˆλ − ˆλX),
+[F]𝑖, 𝑗 =
+𝜕2𝐷
+𝜕𝜆𝑖𝜕𝜆 𝑗
+(ˆλ),
+[FX]𝑖, 𝑗 = 𝜕2𝐷X
+𝜕𝜆𝑖𝜕𝜆 𝑗
+(ˆλX).
+(21)
+
+4
+These extrapolations can be combined using Eq. (17) to simplify
+the bias correction in Eq. (20) to
+Δ ≈ 1
+2
+∑︁
+X
+(ˆλ − ˆλX)𝑇 (F + FX)(ˆλ − ˆλX).
+(22)
+Similarly, I can extrapolate 𝐷X(λ) from λ = ˆλ to λ = ˆλX,
+𝐷X(ˆλX) ≈ 𝐷X(ˆλ) + (ˆλX − ˆλ)𝑇 𝜕𝐷X
+𝜕λ (ˆλ)
++ 1
+2 (ˆλX − ˆλ)𝑇 F′
+X(ˆλX − ˆλ),
+[F′
+X]𝑖, 𝑗 = 𝜕2𝐷X
+𝜕𝜆𝑖𝜕𝜆 𝑗
+(ˆλ),
+(23)
+and minimize the quadratic form for the parameter variations,
+ˆλ − ˆλX ≈ (F′
+X)−1 𝜕𝐷X
+𝜕λ (ˆλ).
+(24)
+The second IC assumption is that F ≈ FX ≈ F′
+X, which allows for
+the removal of FX and F′
+X from Eq. (22) after substituting Eq. (24),
+Δ ≈ tr[ ˜FF−1],
+[ ˜F]𝑖, 𝑗 =
+∑︁
+X
+𝜕𝐷X
+𝜕𝜆𝑖
+(ˆλ) 𝜕𝐷X
+𝜕𝜆 𝑗
+(ˆλ).
+(25)
+The validity of these two assumptions can be increased by adding
+more reference data to reduce finite-sample effects until 𝐷X(λ)
+and 𝐷(λ) have small differences in their gradients and negligible
+differences in their Hessians at λ = ˆλX.
+The TIC follows from a related assumption about small finite-
+sampling effects. As a useful reference, I rearrange F and ˜F into a
+similar form by rewriting ˜F as a sum over simulation tasks rather
+than over groups of 𝑚 simulation tasks,
+[ ˜F]𝑖, 𝑗 = 𝑚
+∑︁
+𝑋
+𝑝(𝑋)
+� 𝜕 log 𝑝(λ|𝑋)
+𝜕𝜆𝑖
+𝜕 log 𝑝(λ|𝑋)
+𝜕𝜆 𝑗
+�
+λ=ˆλ
+,
+[F]𝑖, 𝑗 = −𝑚
+∑︁
+𝑋
+𝑝(𝑋)
+� 𝜕2 log 𝑝(λ|𝑋)
+𝜕𝜆𝑖𝜕𝜆 𝑗
+�
+λ=ˆλ
+.
+(26)
+The TIC bias correction is a direct approximation of Eq. (25) by
+Δ ≈ ΔTIC = tr[ ˜FXF−1
+X ],
+[ ˜FX]𝑖, 𝑗 =
+𝑚
+∑︁
+𝑘=1
+� 𝜕 log 𝑝(λ|𝑋𝑘)
+𝜕𝜆𝑖
+𝜕 log 𝑝(λ|𝑋𝑘)
+𝜕𝜆 𝑗
+�
+λ=ˆλX
+,
+(27)
+which again assumes that the 𝑚 samples in X are sufficient to
+converge expectation values so that ˜F ≈ ˜FX and F ≈ FX.
+The AIC follows from additional assumptions about model
+accuracy. I can simplify the difference between F and ˜F in Eq. (26)
+by rearranging and combining the logarithmic derivatives into
+[ ˜F − F]𝑖, 𝑗 = 𝑚
+∑︁
+𝑋
+𝑝(𝑋)
+𝑝(ˆλ|𝑋)
+� 𝜕2𝑝(λ|𝑋)
+𝜕𝜆𝑖𝜕𝜆 𝑗
+�
+λ=ˆλ
+.
+(28)
+Next, I consider a modified form of 𝐷(λ) from Eq. (15) in which
+the reference simulation tasks are assigned a failure rate 𝛿,
+𝐷(λ) = −𝑚
+∑︁
+𝑋
+(1 − 𝛿)𝑝(𝑋) log 𝑝(λ|𝑋)
+− 𝑚
+∑︁
+𝑋
+𝛿𝑝(𝑋) log(1 − 𝑝(λ|𝑋)).
+(29)
+The original form is recovered in the 𝛿 → 0 limit.
+If the IC
+derivation is repeated for the modified form, Eq. (28) becomes
+[ ˜F − F]𝑖, 𝑗 = 𝑚
+∑︁
+𝑋
+(1 − 𝛿)𝑝(𝑋)
+𝑝(ˆλ|𝑋)
+� 𝜕2𝑝(λ|𝑋)
+𝜕𝜆𝑖𝜕𝜆 𝑗
+�
+λ=ˆλ
++ 𝑚
+∑︁
+𝑋
+𝛿���(𝑋)
+1 − 𝑝(ˆλ|𝑋)
+� 𝜕2[1 − 𝑝(λ|𝑋)]
+𝜕𝜆𝑖𝜕𝜆 𝑗
+�
+λ=ˆλ
+.
+(30)
+The final AIC assumption is that the optimized model can recover
+the reference distribution, resulting in 𝑝(ˆλ|𝑋) ≈ 1 − 𝛿 here. I can
+then cancel the 𝛿 factors and combine the two terms in Eq. (30),
+[ ˜F − F]𝑖, 𝑗 ≈ 𝑚
+�
+𝜕2
+𝜕𝜆𝑖𝜕𝜆 𝑗
+∑︁
+𝑋
+𝑝(𝑋)
+�
+λ=ˆλ
+= 0.
+(31)
+The difference between ˜F and F disappears for any value of 𝛿. In
+this scenario, ˜F and F are 𝑚 times the Fisher information matrix16
+of 𝑝(λ, 𝑋). The AIC bias correction corresponds to ignoring this
+difference and keeping only the trace of the identity matrix over
+the 𝑛-dimensional parameter space,
+Δ = 𝑛 + tr[( ˜F − F)F−1] ≈ ΔAIC = 𝑛.
+(32)
+The validity of the good model assumption can be increased by
+improving the model family and relaxing the success criteria to
+increase all optimized success probabilities towards one.
+C.
+Transferability
+A statistical framework for model selection can also sup-
+port more precise statistical statements about model transferability.
+Here, I briefly contrast a notion of statistical transferability from
+that of physical transferability, which is frequently discussed when
+building models for atomistic simulation17. I argue that while sta-
+tistical transferability is the more desirable goal of model building,
+it is often impractical to avoid physical transferability assumptions
+given the present state of atomistic simulation methods.
+Statistical transferability can directly predict the average future
+success of a model when simulation tasks can be interpreted as
+being drawn from the same distribution that was used to fit the
+model. This is a form of model transferability to future simulation
+tasks that were not part of the reference data. For a model fit with
+𝑚 reference simulation tasks to a minimum divergence 𝐷(ˆλ) in
+Eq. (15), the asymptotic fraction of successful simulations will be
+exp(−𝐷(ˆλ)/𝑚).
+(33)
+If the task distribution is designed to predict or approximate typical
+workloads of typical users of a model, then the model fitting process
+provides a direct operational statement about how effective the
+model should be for its users.
+Statistical transferability can also be used to recycle refer-
+ence data by transferring it between task distributions. Reference
+data sampled from a second distribution 𝑝′(𝑋) over a superset of
+simulation tasks can be reused to estimate 𝐷(λ) for 𝑝(𝑋),
+𝐷(λ) ≈ −
+�
+min
+𝑖
+𝑝′(𝑋𝑖)
+𝑝(𝑋𝑖)
+�
+𝑚
+∑︁
+𝑖=1
+𝑝(𝑋𝑖)
+𝑝′(𝑋𝑖) log 𝑝(λ|𝑋𝑖).
+(34)
+
+5
+This is an implicit form of rejection sampling, and it requires the
+ability to calculate probability ratios between two task distribu-
+tions.
+It can also be used heuristically to reduce the influence
+of data that is necessary to fit a model but not representative of
+its typical applications. Operationally, this can be interpreted as
+rare instances when users validate the model for themselves on the
+original reference data. The effective sample size associated with
+this resampling procedure is
+𝑚′ =
+�
+min
+𝑖
+𝑝′(𝑋𝑖)
+𝑝(𝑋𝑖)
+�
+𝑚
+∑︁
+𝑖=1
+𝑝(𝑋𝑖)
+𝑝′(𝑋𝑖) ,
+(35)
+which can be small if 𝑝(𝑋) and 𝑝′(𝑋) are very different.
+Physical transferability is a set of observations and assumptions
+about the spatial locality of physics at an atomistic length scale. It
+assumes that some model details and parameters describing short-
+range interatomic effects will be insensitive to distant changes in
+a large system with many atoms and then observes the varying
+degrees to which this is true. The underlying first-principles QM
+equations are completely local and transferable when long-range
+interactions are mediated by local fields. Unfortunately, locality
+and transferability are both degraded when encapsulating many-
+body effects and non-essential degrees of freedom to build simpler
+models. Physical transferability assumptions are essential for justi-
+fying the use of methods that decompose large systems into a set of
+small fragments and simulate them individually, often embedded
+in simpler model environments. Such methods include implicit
+solvation models18, QM/MM embedding19, and the use of periodic
+supercells20. However, the effectiveness of these methods can be
+highly system dependent, an important example being the reduced
+locality of electronic effects in metallic systems that complicate
+efforts to develop low-cost methods21.
+In the context of statistical model selection, physical transfer-
+ability assumptions are unavoidable when generating reference data
+for task distributions containing large systems. Reliable methods
+for reference data generation generally have large cost prefactors or
+poor cost scaling with system size that prevent their direct use on
+the task distribution. Physical transferability can be used to justify
+the use of more accessible reference data corresponding to a proxy
+task distribution over small embedded fragments. Tasks from the
+original distribution can be decomposed into sets of proxy tasks
+on fragments to generate the proxy distribution. While these small
+proxy tasks may all be contained within the original task distri-
+bution, the proxy distribution is over a strict subset of simulation
+tasks. It is statistically impossible to sample from a distribution
+by weighting samples from a second distribution over a subset of
+events, but this is avoided by the physical fragmentation process.
+While rigorous error analysis of this process is difficult, the general
+expectation is that the use of larger system fragments increases the
+validity of physical transferability assumptions.
+D.
+Cost penalties
+The primary purpose of fitting models in statistics is to explain
+data in the absence of a prior explanation. In contrast, the purpose
+of fitting models for atomistic simulation is to avoid the large cost of
+evaluating a known first-principles model. Statistics is concerned
+with efficiency, but its main consideration is in getting the most
+value out of limited data to avoid the potentially high cost of
+collecting or generating data. Without some penalty for the cost
+of models, the inevitable conclusion of statistical model selection
+in atomistic simulation is to choose the expensive model that was
+used to generate the reference data. The IC already add penalties
+to the success measure that limits the number of model parameters,
+and the simplest approach is to introduce a cost penalty with a
+similar form. The linear parameter penalty in Eq. (32) looks like a
+Lagrange multiplier, except that the coefficient is not adjustable and
+the number of parameters is a trivial function of parameter values.
+The average model evaluation cost can be a non-trivial function of
+model parameter values, and it can be controlled using a Lagrange
+multiplier that penalizes excessive cost.
+With both cost and parameter penalties added, the operational
+measure of modeling success is
+˜𝐷(λ) = 𝛾(𝑡 − 𝑡0) + Δ −
+𝑚
+∑︁
+𝑖=1
+log 𝑝(λ|𝑋𝑖).
+(36)
+Here, 𝛾 is a Lagrange multiplier, 𝑡 is the total cost of applying
+the model to the 𝑚 simulation tasks, 𝑡0 is the target computational
+budget, and Δ is an IC penalty approximating Eq. (20). Between
+multiple model families with different costs and parameters, the
+family that produces the minimum value of ˜𝐷(λ) for a common 𝛾
+value should be selected. The stationary condition of the Lagrange
+multiplier,
+𝜕
+𝜕𝛾
+˜𝐷(λ) = 𝑡 − 𝑡0 = 0,
+(37)
+should be applied to the model family with the smallest minimum
+˜𝐷(λ) value. If this best model family has a parameter-invariant 𝑡
+value, then 𝛾 should be adjusted until the minimum cost-penalized
+˜𝐷(λ) is equal for two different models.
+In this scenario, the
+cost of the two best models, 𝑡1 and 𝑡2, should bracket 𝑡0 as 𝑡1 ≤
+𝑡0 ≤ 𝑡2. A new, hybrid model can then achieve the target cost by
+randomly switching tasks between the two bracketing models with
+probabilities (𝑡2 − 𝑡0)/(𝑡2 − 𝑡1) and (𝑡0 − 𝑡1)/(𝑡2 − 𝑡1). It is often
+more practical to minimize Eq. (36) over λ without any penalties
+and then add in the penalties with no further optimization of λ.
+The use of cost penalties may be more complicated if applied
+to proxy distributions of fragmented simulation tasks as described
+in the previous subsection. If sets of fragmented simulation tasks
+are meant to represent a larger simulation task, then the model eval-
+uation cost for the larger simulation task may not be approximated
+well by the sum of costs for the proxy tasks. In this situation, a
+proxy cost penalty could be constructed from resource estimates
+that approximate the unknown cost of the larger simulation task
+from the known costs of the proxy tasks and other task-specific
+data. Models usually have a well-understood scaling with system
+size and cost prefactors can be estimated from the proxy calcu-
+lations. More detailed, model-specific resource estimation is also
+possible22. The estimated total simulation cost of the model on the
+large simulation tasks could then be used as 𝑡 in Eq. (36) instead
+of the total proxy simulation cost that is directly observed.
+
+6
+III.
+HYDROGEN CLUSTER EXAMPLE
+To demonstrate the principles of statistical model selection, I
+consider a simple set of simulation tasks on randomly generated
+hydrogen clusters. By only considering hydrogen atoms, I keep
+the elemental diversity at a minimum to simplify the process of
+fitting SQM models with element-specific parameters. I keep the
+phenomenological diversity high by considering two distributions
+of clusters. A “dense” distribution of clusters forces the minimum
+interatomic distance between hydrogen atoms to be less than the
+Coulson-Fischer point23 near 1 Å, while a “sparse” distribution
+allows larger minimum separations.
+Molecular orbitals tend to
+remain grouped into pairs with opposite spin and similar spatial
+character in the dense distribution, while the sparse distribution
+generates many clusters that favor spin-polarized, atom-localized
+orbitals.
+Because of limitations in methods and software that
+generate accurate and reliable reference data, the only observable
+that I consider is the total energy of clusters.
+I consider three
+simulation tasks to calculate energies for three cluster modifications:
+removal of an atom, removal of an electron, and addition of an
+electron. A success is defined as the calculation of one such energy
+with an error of 1 kcal/mol or less. While these distributions and
+tasks are artificial and not directly motivated by any application,
+there is some experimental interest in positively24 and negatively25
+charged hydrogen clusters.
+I generate the dense and sparse distribution of hydrogen clus-
+ters by sequential rejection sampling. Atoms are assigned uniformly
+random positions in a box containing the valid domain, and the
+atom is rejected and repositioned if it violates a distance constraint.
+The minimum allowed interatomic distance for both distributions
+is 0.3 Å, near the classical turning point of the H2 potential energy
+surface. The maximum allowed value for the minimum interatomic
+distance is 1 Å for the dense distribution and 4 Å for the sparse
+distribution. The sparse distribution is a strict superset of the dense
+distribution, and some sparse clusters could be recycled as dense
+clusters. However, the recycling rate of two-atom clusters is only
+0.016, and it decreases rapidly with increasing cluster size. This is
+an example of low recycling efficiency between distributions that
+are very different. For the reference data set, I generate 10,000
+nested sequences of clusters between two and seven atoms for each
+distribution, resulting in 120,000 distinct structures. The three sim-
+ulation tasks require calculations of three different charge states –
+0, 1, and -1 – corresponding to 360,003 total energy calculations
+including an isolated hydrogen atom.
+I briefly compare this reference data set with the MGCDB84
+data set that is popular for testing DFT functionals26. Both data
+sets are restricted to total energies of small, isolated groups of
+atoms.
+MGCDB84 corresponds to 5,931 total energy calcula-
+tions of structures that are 52.6% hydrogen, 29.2% carbon, 8.8%
+oxygen, 5.5% nitrogen, and less than 1% each of main-group ele-
+ments from the first four rows of the periodic table. Thus, while
+it is not restricted to only hydrogen atoms, hydrogen is the most
+well-represented element in MGCDB84. MGCDB84 is organized
+into 84 subsets of data corresponding to different simulation tasks,
+including non-covalent binding energies, isomerization energies,
+formation energies, and barrier heights.
+However, this data set
+lacks diversity by some measures, such as 95.2% of the structures
+being closed-shell singlets and 93.0% being charge neutral. Also,
+0
+1
+2
+3
+4
+5
+101
+103
+105
+MGCDB84
+0
+1
+2
+3
+4
+5
+101
+103
+105
+dense
+0
+1
+2
+3
+4
+5
+H-H distance
+101
+103
+105
+sparse
+FIG. 1. Histograms of interatomic distances between hydrogen atoms in
+the structures from three reference data sets.
+MGCDB84 mostly contains structures and properties of interest to
+organic chemistry with structures at equilibrium or saddle points.
+MGCDB84 is thus a reasonable proxy for the interests of organic
+chemists, while the hydrogen cluster data sets broadly sample from
+the potential energy surface of many hydrogen atoms. Of partic-
+ular interest when fitting distance-dependent parameters such as
+pair potentials and one-body matrix elements is the distribution
+of interatomic distances between hydrogen atoms. These distance
+distributions are shown in Fig. 1 for MGCDB84 and the two distri-
+butions of hydrogen clusters considered here. MGCDB84 has poor
+coverage at distances less than 1.4 Å and is not a good reference
+to fit distance-dependent parameters for hydrogen interactions.
+A.
+Reference data
+I gather high-level reference data for the hydrogen clusters at
+the CCSD(T) level of theory27 with the def2-QZVPP basis set28. I
+also record data at the Hartree-Fock (HF), MP2, and CCSD levels
+of theory during the CCSD(T) calculations. In addition to the high-
+level reference data, I also gather data using several popular SQM
+models and DFT functionals to test their transferability. There are
+too many SQM models and DFT functionals to test all of them, and
+this study is limited to a few important representative examples.
+AM129 was the most popular SQM thermochemistry model of the
+last century, and PM730 is the most recent model from that family
+of MNDO-like models31.
+GFN132 and GFN233 are two recent
+SQM models from the density functional tight-binding (DFTB)
+framework34. PBE35 is the most popular DFT functional in solid-
+state physics and materials science. B3LYP36 is the most popular
+
+7
+DFT functional in chemistry. 𝜔B97M-V37 is claimed to be the most
+accurate DFT functional without including terms from many-body
+perturbation theory. While a smaller basis set might be sufficient,
+I perform all DFT calculations using the def2-QZVPP basis set
+for consistency. In total, I gather data from eleven QM and SQM
+models, which corresponds to 3,960,033 total energy calculations.
+All QM calculations use a post-2.1.1 development version of
+PySCF38–40. All calculations use spin-unrestricted orbitals. For
+HF theory and every DFT functional, the large-basis calculations
+are initialized by projecting a converged density matrix from a
+calculation in the smaller def2-SVP basis set28.
+The def2-SVP
+density matrix is taken from the calculation with the lowest total
+energy from a systematic ground-state search for each structure
+and charge state. First, a def2-SVP calculation is performed for
+every spin state from the standard spin-averaged independent-atom
+density matrix guess. Second, a custom density matrix guess is
+constructed from spin-polarized independent-atom density matrices
+with every combination of atomic charges and spin orientations.
+Third, after performing all of these small-basis calculations with the
+default DIIS algorithm41, they are all repeated with an alternative
+ADIIS algorithm42.
+The large-basis calculation uses the same
+algorithm, either DIIS or ADIIS, as the small-basis calculation that
+is used to initialize it. Even with all of this redundancy, it is not
+possible to converge a self-consistent field (SCF) cycle for every
+charge and spin state of every structure.
+While the variational
+nature of SCF calculations guarantees the existence of stable local
+energy minima, DIIS-based algorithms provide no guarantees of
+convergence. All large-basis DFT calculations use a (99,590) local
+grid and a SG-1 nonlocal grid, following the recommendations for
+the 𝜔B97M-V functional37.
+For SQM calculations, MOPAC 22.0.543 is used for AM1 and
+PM7 calculations, and xTB 6.5.19 is used for GFN1 and GFN2
+calculations. MOPAC calculations follow the same ground-state
+search procedure as the PySCF calculations except with only DIIS
+and without any projection into a larger basis. There are fewer
+points of failure in minimal-basis calculations, and MOPAC is able
+to converge an SCF calculation for every structure and charge state.
+xTB calculations do not contain Fock exchange and depend on an
+initial electronic density guess rather than a density matrix guess.
+I only use the default spin-averaged density guess and restrict the
+ground-state search to total spin values. There is a high failure rate
+for SCF convergence in xTB with the default options for this data
+set. However, it is possible to converge every structure and charge
+state in xTB with calculations at elevated electronic temperatures
+of 3000 K and then 1500 K followed by linear extrapolation of the
+total energies to zero temperature.
+At this level of automation and scale of data generation, it is not
+possible to converge every iterative solve for HF, DFT, and CCSD
+calculations in PySCF. The choice of solver options is important
+as it changes success statistics and average run times. I did not
+try to optimize these choices in a systematic way, but they were
+adjusted during the implementation of the workflow to improve
+success rates44. In addition to convergence failures, a DFT or HF
+calculation is considered to fail if the def2-QZVPP total energy
+is more than 10 kcal/mol larger than the smallest def2-SVP total
+energy. Total energies tend to be lower for larger basis sets because
+they have more variational degrees of freedom. I attribute these
+energy increases to the DIIS phenomenon of escaping from the
+0.00
+0.01
+0.02
+0.03
+0.04
+101
+103
+105
+dense
+0.00
+0.01
+0.02
+0.03
+0.04
+101
+103
+105
+sparse
+FIG. 2. Histograms of the maximum deviation from zero and one of the
+unrelaxed MP2 1RDM eigenvalues for all structures and charge states from
+the reference data sets.
+basin of convergence of a ground state and converging to a very
+different stationary state with a larger energy. The failure rate of
+DFT calculations is 3.4%, the failure rate of CCSD(T) calculations
+is 1.0%, and the overall failure rate of the simulation tasks is 4.3%.
+If at least one model fails to produce an output for a simulation
+task, then that task is omitted from the final data set and statistical
+analysis. Such failures distort the distribution of simulation tasks
+because they act as a form of rejection sampling.
+I also validate the CCSD(T)/def2-QZVPP level of theory for
+this data set while gathering data. The main validity concern is
+strong electron correlation effects, which are known to occur in
+hydrogen clusters45. These effects are caused by multi-reference
+ground states that come from a superposition of many electronic
+spin configurations with nearly degenerate energies in the atomic
+limit. Randomly generated hydrogen clusters are unlikely to have
+many degenerate spin configurations, and they are expected to be
+more weakly correlated on average. The most direct validity test
+would be the overlap between the normalized HF and CCSD many-
+body wave-functions, but this quantity is not efficiently computable.
+Instead, I use the eigenvalues of the one-particle density matrix
+(1RDM) at the unrelaxed MP2 level of theory as an accessible
+proxy for this overlap. The maximum deviation of the eigenvalues
+from zero and one is strictly zero when the overlap is one, and
+the deviation increases as the overlap is reduced. This deviation
+is plotted for every structure in every charge state in Fig. 2. The
+sparse distribution that is expected to be more susceptible to multi-
+reference effects because of spin symmetry breaking does not have
+larger deviations than the dense distribution.
+The other major validity concern is the basis-set convergence
+of CCSD(T)/def2-QZVPP. A quadruple-zeta basis such as def2-
+QZVPP does not typically converge absolute post-HF energies to
+chemical accuracy of 1 kcal/mol or less without basis-set extrapo-
+lation or explicit correlation corrections. However, the simulation
+tasks considered here only require energy differences between struc-
+
+8
+−2.5
+0.0
+2.5
+101
+103
+105
+CCSD
+–0.5
+±0.5
+remove atom
+dense
+−5
+0
+5
+–0.0
+±0.1
+sparse
+−2.5
+0.0
+2.5
+–0.5
+±0.5
+remove electron
+dense
+0
+25
+0.1
+±0.7
+sparse
+0.0
+2.5
+0.1
+±0.5
+add electron
+dense
+0
+20
+0.7
+±1.0
+sparse
+−20
+0
+101
+103
+105
+MP2
+–2.7
+±3.1
+−20
+0
+–2.3
+±4.7
+−20
+0
+20
+–2.0
+±3.4
+−25
+0
+25
+–3.0
+±5.7
+−10
+0
+10
+0.0
+±0.9
+−10
+0
+10
+6.6
+±4.2
+−50
+0
+101
+103
+105
+HF
+–18.8
+±8.9
+−25
+0
+–4.1
+±8.1
+−25
+0
+–17.1
+±9.7
+−50
+0
+–5.0
+±8.3
+0
+25
+1.2
+±4.2
+0
+25
+17.8
+±6.3
+−25
+0
+25
+101
+103
+105
+ωB97M-V
+0.7
+±2.3
+−10
+0
+10
+–1.3
+±2.4
+−25
+0
+25
+0.5
+±2.6
+−50
+0
+50
+–29.1
+±8.8
+0
+25
+–0.2
+±1.1
+−20
+0
+–7.6
+±3.4
+−25
+0
+25
+101
+103
+105
+B3LYP
+2.2
+±2.8
+0
+10
+–0.1
+±1.6
+0
+25
+4.2
+±3.3
+−50
+0
+–36.9
+±13.4
+0
+20
+–0.4
+±1.4
+−50
+0
+–21.5
+±6.1
+−25
+0
+25
+101
+103
+105
+PBE
+0.9
+±4.5
+−10
+0
+10
+–0.8
+±2.0
+−25
+0
+25
+0.7
+±4.6
+0
+200
+–47.6
+±19.0
+0
+25
+–0.2
+±1.1
+−25
+0
+25
+–15.6
+±5.7
+−1000
+0
+1000
+101
+103
+105
+GFN2
+–32.7
+±96.5
+−100
+0
+6.8
+±9.3
+−200
+0
+21.0
+±54.0
+0
+100
+21.4
+±17.8
+−100
+0
+–19.1
+±33.2
+−200
+0
+–157.1
+±19.7
+−250
+0
+101
+103
+105
+GFN1
+–7.7
+±54.3
+−200
+0
+9.4
+±13.2
+0
+100
+68.5
+±26.5
+0
+100
+36.1
+±21.5
+−100
+0
+–10.7
+±21.6
+−200
+0
+–153.3
+±23.4
+−2000
+0
+101
+103
+105
+PM7
+–23.5
+±150.1
+−1000
+0
+–1.3
+±20.6
+−100
+0
+–42.1
+±15.9
+−100
+0
+–50.4
+±7.7
+0
+50
+1.3
+±4.6
+0
+50
+17.9
+±6.4
+0
+250
+101
+103
+105
+AM1
+15.6
+±42.9
+0
+200
+–2.0
+±7.5
+−100
+0
+–21.8
+±22.2
+−100
+0
+–41.6
+±8.5
+0
+50
+1.3
+±4.6
+0
+50
+17.9
+±6.4
+FIG. 3. Error histograms in kcal/mol for all models and tasks along with their means, standard deviations, and moment-matching Gaussian model fits.
+
+9
+−25
+0
+25
+101
+103
+105
+ωB97M-V
+0.7
+±1.6
+remove atom
+dense
+−10
+0
+10
+–1.3
+±2.4
+sparse
+−25
+0
+25
+0.5
+±2.0
+remove electron
+dense
+−50
+0
+50
+–27.7
+±8.9
+sparse
+0
+25
+–0.2
+±0.7
+add electron
+dense
+−20
+0
+–6.9
+±2.9
+sparse
+−25
+0
+25
+101
+103
+105
+B3LYP
+2.2
+±2.3
+0
+10
+–0.1
+±1.6
+0
+25
+4.1
+±3.0
+−50
+0
+–33.3
+±14.0
+0
+20
+–0.5
+±1.2
+−50
+0
+–19.5
+±6.2
+−25
+0
+25
+101
+103
+105
+PBE
+0.8
+±4.2
+−10
+0
+10
+–0.8
+±2.1
+−25
+0
+25
+0.6
+±4.4
+0
+200
+–42.8
+±14.9
+0
+25
+–0.2
+±0.9
+−25
+0
+25
+–14.0
+±5.8
+FIG. 4. Error histograms in kcal/mol for DFT models and all tasks along with the means, standard deviations, and moment-matching Gaussian model
+fits of the marked data with consistent total spin values between HF and DFT.
+tures that differ by at most one atom, which should be less sensitive
+to finite-basis errors. The most basis-set sensitive structures are
+correlation-bound anions, which account for 5.3% of the anions
+in the dense distribution and 45.2% in the sparse distribution.
+Correlation-bound anions do not have a proper complete basis set
+(CBS) limit with HF orbitals because the overlap between the HF
+and CCSD wave-functions tends to zero as the unbound HF orbital
+delocalizes. A formally correct treatment of correlation-bound an-
+ions in the CBS limit requires Brueckner orbitals46. However, I
+do not expect the def2-QZVPP basis set to be large enough for the
+pathological CBS limit to have a substantial effect on this data set.
+B.
+Anomaly detection
+Anomaly detection is a natural part of error analysis when
+gathering large amounts of data within a statistical framework.
+The basic expectation of a good model is that its errors are an
+accumulation of a large number of small, independent errors, which
+tend to induce Gaussian distributions of model errors. Errors in the
+hydrogen cluster data organized by model and task are shown in
+Fig. 3 with moment-matching Gaussian fits. While many errors are
+effectively described by the Gaussian model, there are also several
+fat error tails, many of which are rare enough to be unlikely to
+appear in data generation at smaller scales. What is not shown are
+some even larger error tails that were present in earlier versions of
+the data set as the workflow was being refined to detect and avoid
+more failure events and silent errors.
+This statistical overview
+of error distributions along with metadata collected during the
+primary data generation are essential for detecting and correcting
+rare failures. Unfortunately, sufficiently rare failures are unlikely
+to occur in small-scale preliminary testing of a workflow precisely
+because of how rare they are.
+There is not necessarily a clean partition between model,
+algorithm, and software errors in large-scale data generation. For
+example, the lack of reliability in DIIS-based SCF solvers causes
+enough gaps in the ground-state searches that the wrong total
+spin is assigned in some DFT calculations.
+As a result, some
+DFT calculations produce total energies that are too high, which
+are likely a source of some rare error outliers.
+However, there
+is no guarantee that the DFT and HF ground states for a given
+structure and charge state will have the same total spin. There is
+not enough information to distinguish model from algorithm errors
+here without more reliable SCF solver algorithms to fill gaps in
+data. Similarly, software bugs may cause failures in one algorithm
+implementation that are not reproduced by other implementations,
+and custom improvements to algorithms may cause successes that
+are also not reproducible in other software. To see the impact of
+spin inconsistency, the DFT data is shown in Fig. 4 with spin-
+consistent calculations marked and fit to Gaussian error models.
+The spin-inconsistent data contains most of the error outliers but
+does not substantially change the overall error statistics since the
+spin-consistent data has similar means and standard deviations.
+The broadest error distributions in Fig. 3 are in the SQM atom
+removal data from the dense distribution. It is likely that errors in
+short-range pair potentials and matrix elements account for much
+of this error since these SQM models are mostly fit to data from
+near-equilibrium structures. I test this hypothesis by separating data
+in Fig. 5 based on the minimum interatomic distance in a structure
+being greater than or less than 0.74 Å, the equilibrium bond length
+of H2. There is a clear narrowing of the error distributions for the
+structures without short interatomic distances, which supports the
+error hypothesis.
+It may not be possible to detect or explain all error outliers.
+The CCSD error tails from the sparse distribution in Fig. 3 imply
+rare instances of large perturbative triples corrections to the total
+energy. In these cases, the exact ground-state wave-function may
+have strong multi-reference character. However, the multi-reference
+test in Fig. 2 has no corresponding outliers, and a variety of multi-
+reference tests may be needed to increase detection reliability47.
+
+10
+−1000
+0
+1000
+101
+103
+105
+GFN2
+–1.6
+±37.2
+remove atom
+dense
+−100
+0
+6.9
+±9.3
+sparse
+−200
+0
+51.3
+±21.6
+remove electron
+dense
+0
+100
+21.3
+±17.7
+sparse
+−100
+0
+–25.4
+±38.8
+add electron
+dense
+−200
+0
+–157.3
+±19.0
+sparse
+−250
+0
+101
+103
+105
+GFN1
+15.1
+±35.6
+−200
+0
+9.5
+±12.6
+0
+100
+88.0
+±19.0
+0
+100
+35.9
+±21.4
+−100
+0
+–16.7
+±29.4
+−200
+0
+–153.6
+±22.8
+−2000
+0
+101
+103
+105
+PM7
+26.5
+±14.5
+−1000
+0
+–1.0
+±6.2
+−100
+0
+–37.5
+±9.7
+−100
+0
+–50.4
+±7.6
+0
+50
+0.7
+±3.4
+0
+50
+17.9
+±6.4
+0
+250
+101
+103
+105
+AM1
+5.3
+±20.3
+0
+200
+–2.4
+±5.7
+−100
+0
+–28.8
+±17.3
+−100
+0
+–41.6
+±8.4
+0
+50
+0.7
+±3.4
+0
+50
+17.9
+±6.4
+FIG. 5. Error histograms in kcal/mol for SQM models and all tasks along with the means, standard deviations, and moment-matching Gaussian model
+fits of the marked data from structures with minimum interatomic distances greater than 0.74 Å.
+The failures that statistical model selection in Sec. II seeks to
+avoid are silent failures. Anomaly detection implies an ability to
+detect and herald some types of failures. For the example data set in
+this paper, I havechosen to removesome heralded failuresassociated
+with algorithm-specific SCF convergence problems to increase the
+emphasis on errors in the physical models. This formally changes
+the underlying task distributions by a small amount. To be faithful to
+the original task distributions, a more complete model would always
+produce a viable output by branching to less accurate but more
+reliable calculations and eventually resorting to a random guess.
+When trying to increase a model’s overall success probability,
+improving the ability to detect and respond to rare failures and
+error outliers can be just as important as improving the average
+model accuracy for typical inputs.
+C.
+Model fitting
+I now consider a minimal representative example of using
+model selection to fit SQM models. First, I highlight the benefits of
+using more complicated error models to improve success measures.
+Second, I fit an atomic pair potential to all QM and SQM data,
+primarily to correct the large error outliers in the SQM data. Pair
+potentials are one of the most common and basic elements of both
+interatomic potentials and SQM models. While pair potentials are
+often restricted to a simple form before fitting them, I consider a
+general form and rely on model selection to limit the number of
+parameters that define the pair potential.
+Because some models being considered are near chemical
+accuracy, the small-𝜖 approximation used in Eq. (7) is not always
+accurate. Instead, I use the exact success probability,
+𝑝(λ|𝑋𝑖) =
+∫
+𝑥𝑖+𝜖
+𝑥𝑖−𝜖
+𝑒−0.5[𝑧−𝜇−𝑦𝑖 (λ)]2/𝜎2
+𝜎
+√
+2𝜋
+𝑑𝑧
+= erf
+�
+𝑥𝑖−𝑦𝑖 (λ)−𝜇+𝜖
+√
+2𝜋𝜎
+�
+− erf
+�
+𝑥𝑖−𝑦𝑖 (λ)−𝜇−𝜖
+√
+2𝜋𝜎
+�
+,
+(38)
+for a success interval [𝑥𝑖 − 𝜖, 𝑥𝑖 + 𝜖] around a reference data value
+𝑥𝑖. For chemical accuracy, 𝜖 = 1 kcal/mol. This interval needs
+to be adjusted for electron addition and removal energies that are
+near their vacuum-limited values. The energy to add an electron
+cannot be greater than zero, and the energy to remove an electron
+cannot be less than zero. If the success interval crosses into this
+physically forbidden region, then I ignore the unphysical end point
+and consider a semi-infinite success interval in Eq. (38). The form
+of the pair potential is a polynomial at short range that goes to
+zero at an adjustable cutoff 𝑅 and strictly zero beyond that. The
+success measure in Eq. (36) and its analytical first and second
+derivatives with respect to λ are tedious but straightforward to
+evaluate. I minimize the success measure with a sequence of line
+searches that use this derivative information to achieve asymptotic
+quadratic convergence.
+As I increase the polynomial degree, I
+use the minimizing model with one fewer degree as the initial
+guess for minimization. For degree one, I use the moment-based
+approximations of the error model in Eq. (13) and a zero pair
+potential with 𝑅 = 4 Å as the initial guess. The TIC bias correction
+in Eq. (27) is calculated at the penalty-free minimum of the success
+measure instead of being included in the minimization process.
+The models that minimize the success measure are summarized
+in Table I. There is a clear benefit to using a richer error model
+with a separate Gaussian error model for each type of simulation
+
+11
+model
+˜𝐷1g
+˜𝐷6g
+𝜇
+𝜎
+𝜇rad
+𝜎rad
+𝜇ras
+𝜎ras
+𝜇red 𝜎red
+𝜇res
+𝜎res
+𝜇aed 𝜎aed
+𝜇aes 𝜎aes 𝑡
+CCSD+PP 1.57 × 105 9.20 × 104
+0.2
+0.7
+0.1
+0.4 -0.1
+0.3
+-0.6
+0.3
+0.1
+0.8
+1.3
+0.4
+0.8
+0.9 3.64 × 108
+CCSD
+1.72 × 105 9.57 × 104
+0.0
+0.7
+-0.6
+0.4 -0.2
+0.3
+-0.6
+0.3
+0.1
+0.8
+1.3
+0.4
+0.8
+0.9 3.64 × 108
+MP2
+7.41 × 105 6.17 × 105
+0.0
+5.5
+-2.7
+3.0 -2.3
+4.7
+-2.0
+3.4
+-3.0
+5.6
+2.0
+1.8
+6.9
+3.8 2.10 × 108
+HF
+1.06 × 106 8.17 × 105
+-2.9
+16.0
+-18.9
+8.8 -4.1
+8.1
+-1.7
+9.7
+-5.0
+8.3
+12.5
+6.0
+18.3
+5.5 5.42 × 107
+𝜔B97M-V 9.82 × 105 5.63 × 105
+-4.5
+12.3
+0.7
+2.2 -1.3
+2.3
+0.5
+2.5
+-29.1
+8.8
+2.8
+2.8
+-7.5
+3.2 2.57 × 108
+B3LYP
+1.09 × 106 6.28 × 105
+-6.2
+17.8
+2.2
+2.7 -0.1
+1.5
+4.2
+3.3
+-36.9 13.4
+4.3
+4.3
+-21.4
+6.0 8.51 × 107
+PBE
+1.14 × 106 7.00 × 105
+-7.5
+21.1
+0.9
+4.5 -0.8
+2.0
+0.7
+4.6
+-47.6 19.0
+2.9
+2.8
+-15.4
+5.6 1.14 × 108
+GFN2+PP
+1.53 × 106 1.17 × 106
+-9.0
+82.3
+-52.5
+92.0
+2.2
+5.2
+21.0 54.0
+21.4 17.8 111.2 98.9 -156.8 20.0 9.31 × 104
+GFN2
+1.54 × 106 1.21 × 106
+-15.1
+85.1
+-32.7
+96.5
+6.8
+9.3
+21.0 54.0
+21.4 17.8 111.2 98.9 -156.8 20.0 9.31 × 104
+GFN1+PP
+1.52 × 106 1.13 × 106
+5.2
+79.4
+-23.2
+42.7
+4.7
+8.4
+68.5 26.5
+36.1 21.5
+69.4 60.7 -153.0 23.9 9.08 × 104
+GFN1
+1.52 × 106 1.17 × 106
+2.2
+81.2
+-7.7
+54.3
+9.4 13.2
+68.5 26.5
+36.1 21.5
+69.4 60.7 -153.0 23.9 9.08 × 104
+PM7+PP
+1.27 × 106 9.11 × 105
+-6.5
+33.2
+13.8
+29.7 -0.7
+7.3
+-42.1 15.9
+-50.4
+7.7
+13.7
+6.8
+18.4
+5.7 1.22 × 106
+PM7
+1.50 × 106 1.07 × 106
+-7.3
+75.0
+-23.5 150.1 -1.3 20.6
+-42.1 15.9
+-50.4
+7.7
+13.7
+6.8
+18.4
+5.7 1.22 × 106
+AM1+PP
+1.21 × 106 9.00 × 105
+-5.0
+26.8
+15.1
+25.1 -0.9
+4.6
+-21.8 22.2
+-41.6
+8.5
+13.7
+6.8
+18.4
+5.7 1.18 × 106
+AM1
+1.26 × 106 9.60 × 105
+-0.8
+32.1
+15.6
+42.9 -2.0
+7.5
+-21.8 22.2
+-41.6
+8.5
+13.7
+6.8
+18.4
+5.7 1.18 × 106
+PP
+1.69 × 106 1.15 × 106 -100.8 135.5 143.8 121.8 -2.4
+9.7 -228.1 76.2 -293.1 17.7
+13.7
+6.8
+18.4
+5.7 5.25 × 10−2
+none
+1.72 × 106 1.18 × 106
+-62.7 154.5
+20.8 100.3 -7.5 19.4 -228.1 76.2 -293.1 17.7
+13.7
+6.8
+18.4
+5.7 3.38 × 10−2
+TABLE I. Comparison of minimized success measures over 𝑚 = 344, 513 simulation tasks for various models, including a pair potential (PP) correction
+when the improvement is greater than one percent. This comparison includes one-Gaussian (1g) error models (𝜇, 𝜎) and six-Gaussian (6g) error models
+fit to atom removal (ra), electron removal (re), and electron addition (ae) on both dense (d) and sparse (s) distributions. The success measures do not
+include parameter or cost penalties. The error model parameters are in kcal/mol and the total model evaluation times 𝑡 are in CPU-seconds. The cost of
+generating the reference data is 𝑡 = 4.13 × 108.
+task. Many of the large standard deviations in the overall error
+are better explained as biases in a specific task type with a smaller
+standard deviation per type. Some of these biases are obvious and
+expected, but it is still useful to quantify them. The GFN1 and
+GFN2 models predict a very large electron binding energy for most
+hydrogen clusters, while AM1 and PM7 do not predict any binding
+of excess electrons to any hydrogen cluster. The HF model has
+biases associated with the absence of electron correlation energy,
+which is always negative and usually proportional to the number of
+electrons. The DFT models are known to have large delocalization
+errors48 that are likely to be biasing the electron removal energies of
+the sparse distribution. If an error model is used to improve success
+probabilities by adding random numbers to a model’s outputs, then
+an improvement to the error model is an improvement to the model
+as a whole.
+The effects of the IC penalties on the selection of the pair
+potential are shown for two representative model families in Fig. 6.
+CCSD is a more accurate model than PM7, and the AIC is likewise
+a better approximation of the TIC for CCSD. For PM7, the AIC
+is unable to compensate for the parameter bias enough to create
+a local minimum in the success measure. For CCSD, the AIC is
+able to create a local minimum, but its location is different than for
+the TIC. In this example, the TIC correction introduces significant
+numerical noise, which appear as values above the smoother trend
+line.
+The TIC is a response property that depends sensitively
+on the numerical quality of the success measure minimum. The
+derivative discontinuity that I allow at the large-distance cutoff
+point 𝑅 of the pair potential introduces derivative discontinuities
+in the 𝑅 dependence of the success measure that complicates the
+minimization. Even under such non-ideal conditions, the TIC is
+0
+20
+40
+60
+80
+100
+−3700
+−3690
+−3680
+−3670
+−3660
+−3650
+−3640
+˜D6g change for CCSD
+AIC
+TIC
+0
+20
+40
+60
+80
+100
+polynomial degree
+−159500
+−159000
+−158500
+−158000
+−157500
+−157000
+−156500
+˜D6g change for PM7
+AIC
+TIC
+FIG. 6. Reduction of the success measure ˜𝐷6g with a six-Gaussian error
+model as the polynomial degree of the pair potential is increased. The
+TIC is regularized by replacing small and negative eigenvalues of the ˜𝐷6g
+Hessian with 10−9 times the largest eigenvalue when that value is greater.
+
+12
+0.5
+1.0
+1.5
+2.0
+2.5
+−0.4
+−0.2
+0.0
+0.2
+PP for CCSD (kcal/mol)
+no TIC
+TIC
+0.3
+0.4
+0.5
+0.6
+0.7
+distance (Å)
+−200
+−150
+−100
+−50
+0
+PP for PM7 (kcal/mol)
+no TIC
+TIC
+FIG. 7. Short-range polynomial pair potential corrections for the CCSD
+and PM7 models. With the TIC penalty, the minimizing polynomial has
+degree 22 for CCSD and degree 14 for PM7. Without an IC penalty, there
+is no local minimum in polynomial degree and best degree 100 polynomial
+is shown as an example of overfitting. Outside of the plotted range, the
+PM7 pair potential decreases to -1790 kcal/mol at 0.3 Å.
+still functional for model selection with appropriate regularization
+of the success measure Hessian.
+The TIC is more challenging
+to calculate for parameterized QM calculations that involve QM
+response properties in the parameter derivatives of the success
+measure.
+The benefits of a pair potential correction are not uniform
+over models or tasks. Since the pair potential only depends on
+the atomic structure and not electronic structure, it cannot correct
+the electron addition and removal tasks. For many models, the
+overall reduction of the success measure is one percent or less, and
+these minor improvements are omitted from Table I. The largest
+improvement comes from the PM7 pair potential, shown in Fig.
+7. Apparently, the short-range hydrogen-hydrogen pair potential
+in PM7 is much too repulsive at distances just below the bond
+length of H2. In contrast, the CCSD pair potential is much longer
+in range and much smaller in magnitude. It is not surprising that
+the largest correction occurs near the Coulson-Fischer point around
+1 Å. However, it is surprising that something as complicated as
+the CCSD(T) triples correction can be partially approximated by a
+pair potential. The IC penalties succeed in suppressing the high-
+frequency oscillations typically attributed to overfitting noise, but
+there are still some artifacts near the edges of the polynomial’s
+domain. There are other ways to reduce unphysical oscillations
+in pair potentials, such as considering reference tasks that depend
+directly on derivatives of a pair potential or explicit functional
+regularization49. As shown in Fig. 8, the pair potential corrections
+eliminate most of the large error outliers in SQM models except for
+GFN2 on the dense distribution. I expect that the persistent error
+in GFN2 is from the short-range part of either a 3-body potential
+term or a Hamiltonian matrix element, neither of which can be
+repaired by a pair potential.
+This example demonstrates the benefits of having an excessive
+amount of data available when fitting models. As the amount of
+data increases, the utility and reliability of statistical tools and con-
+cepts increases. The abundance of data creates a comfortable safety
+buffer between the number of parameters needed to fit a model ac-
+curately and the maximum number of parameters that can be fit with
+statistical significance. The model selection process then enables
+an accurate model to be carved from an accessible set of redundant,
+overfit models. Such large amounts of data are accessible because
+of the massive scale of modern high-performance computing, an
+ability to generate data sets procedurally, and careful use of phys-
+ical transferability assumptions. This strongly contrasts with how
+SQM models such as PM730 and GFN233 have been developed.
+They prescribe simple model forms with a few tens of parameters
+per element and collect enough reference data to fit those forms
+specifically. They do not gather enough data to consider or rule out
+more complicated models with more parameters, and many SQM
+model design choices have remained frozen for decades.
+PM7
+still uses the MNDO model form31 proposed in 1977, just with
+the addition of more complicated classical correction terms. De-
+spite being from a much newer model family, GFN2 also contains
+old model forms such as the Wolfsberg–Helmholz approximation50
+from 1952 relating Hamiltonian and overlap off-diagonal matrix
+elements. With an increasing amount of data, model forms can
+−2.5
+0.0
+2.5
+101
+103
+105
+CCSD+PP
+0.1
+±0.4
+remove atom
+dense
+−5
+0
+5
+0.0
+±0.1
+sparse
+−1000
+0
+1000
+101
+103
+105
+GFN2+PP
+–52.3
+±92.0
+−50
+0
+2.2
+±5.2
+−100
+0
+100
+101
+103
+105
+GFN1+PP
+–22.9
+±42.7
+−50
+0
+4.7
+±8.5
+−200
+−100
+0
+100
+101
+103
+105
+PM7+PP
+13.7
+±29.7
+−50
+0
+50
+–0.7
+±7.4
+−100
+0
+100
+101
+103
+105
+AM1+PP
+15.0
+±25.2
+−50
+0
+50
+–0.9
+±4.7
+FIG. 8. Revisions of error histograms from Fig. 3 in kcal/mol for the
+models and tasks that benefit from a pair potential correction.
+
+13
+shift more towards what is objectively supported by the data and
+farther from the subjective technical opinions of specific model
+builders.
+D.
+Cost budgeting
+Considerations of model cost are always more complicated
+than model accuracy because they are much more sensitive to
+software, hardware, and fine details of a workflow. All calculations
+reported in Table I are performed on the same computing cluster,
+with AMD EPYC 7702 CPU cores and two gigabytes of memory
+per core. Except for MP2, CCSD, and CCSD(T) calculations, all
+calculations are performed on a single CPU core for maximum
+throughput. Some of the MP2, CCSD, and CCSD(T) calculations
+exceed the memory budget of a single CPU core, and they are run
+with four cores per calculation for a safety buffer of memory usage.
+Parts of the calculation are threaded and make use of multiple
+cores, but the thread scaling is limited. This complicates some
+cost comparisons. For example, the HF and MP2 calculations have
+very similar run times under similar conditions, and the large cost
+difference reported in Table I is caused by the different number
+of cores required.
+Also, the AM1 and PM7 calculations have
+similar run times as GFN1 and GFN2 calculations for an individual
+calculation, but their workflow requires a combinatorial search over
+atomic spin configurations. The limited sensitivity of GFN1 and
+GFN2 calculations to spin order makes this search unnecessary and
+reduces their overall run time per simulation task, but may also be
+related to their relatively poor accuracy here.
+A visual way to compare success measures with varying cost
+penalties is to plot them versus cost as in Fig. 9 and draw the
+convex hull connecting minimum-cost models with various rates
+of success. Models on the convex hull are optimal for a range
+of computational budgets, and models above the convex hull are
+not worth using for these simulation tasks according to this cost
+analysis. In this example, the convex hull connects PP, AM1+PP,
+B3LYP, and CCSD+PP, with GFN1+PP also just on the convex
+hull. As noted in Sec. II D, any model cost versus accuracy along
+the convex hull can be achieved by randomly switching between the
+models on the end points with a probability varying linearly between
+zero and one along the facet. Thus, there is a natural continuum of
+hybrid models between the cheapest and most expensive models.
+The practice of randomly mixing models with different accu-
+racies as suggested here is usually avoided in atomistic simulation.
+Models often rely on some form of error cancellation or a study
+of qualitative trends rather than precise quantitative predictions
+that can be disrupted by comparing data between different models.
+These behaviors can be described within a framework of statisti-
+cally independent simulation tasks by carefully defining tasks as
+groups of simulations with success based on comparisons rather
+than the absolute value of outputs. In the simple example of a
+ranking, random pairs of simulations might be performed with the
+output being the decision of which system had the larger value for
+a specific output. Such a grouping forces comparisons to remain
+within a specific model while still allowing for the use of a different
+model for each independent ranking decision. A more common
+practice of mixing models is to filter a larger number of systems
+with a cheap, inaccurate model and then filter the remains systems
+10−2
+100
+102
+104
+106
+108
+1010
+1012
+t (CPU-seconds)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+˜D6g
+×106
+CCSD+PP
+MP2
+HF
+ωB97M-V
+B3LYP
+PBE
+GFN2+PP
+GFN1+PP
+PM7+PP
+AM1+PP
+PP
+FIG. 9. Success measure versus total cost of models from Table I with the
+convex hull denoting the most cost-effective models in the example.
+that pass the first filter with a more expensive and accurate model.
+The intention of this practice is to approximate the effect of ap-
+plying the more expensive filter to all of the systems with a lower
+overall cost. However, this requires the cheaper model to have a
+sufficiently low false positive rate that the overall cost is actually
+reduced while maintaining a very low false negative rate to avoid
+distorting the outcome.
+Simultaneous considerations of model cost and accuracy at
+a large enough scale that reliability also matters as in Fig. 9 is a
+very challenging test for models. It is much easier to show cost
+benchmarks of a model or software under ideal conditions, accuracy
+benchmarks under a different set of ideal conditions, and ignore
+problematic cases altogether. Even the hydrogen cluster example
+considered here is artificially generous because a small fraction
+of structures that caused SCF convergence failures were omitted
+from the set of simulation tasks. While the models are depicted
+as points on the plot, they are more generally going to be regions
+corresponding to the set of possible changes in a workflow that alter
+both cost and accuracy. For example, the combinatorial search over
+atomic spin configurations for the hydrogen cluster example could
+have been avoided, which would have substantially reduced the
+cost of many models. However, many of the calculations would
+have failed to find the lowest energy ground state, and the overall
+accuracy would have been reduced as a result. Cost and accuracy
+could have been balanced more carefully by randomly sampling a
+limited set of spin configurations rather than using an exhaustive
+combinatorial search. Adjusting details of a model workflow to
+improve the convex hull of optimal models requires a careful balance
+of these cost, accuracy, and reliability considerations.
+
+14
+IV.
+CONCLUSION
+As scientists continue to develop more diverse and sophisti-
+cated models for atomistic simulation, how models are compared
+and how their successes are judged become increasingly important.
+Progress in method development can slow down or stop if scientists
+have different, incompatible definitions for what success is51. This
+paper has presented an operational success measure for judging
+atomistic models that is based on statistical model selection. Using
+simple simulation tasks on hydrogen clusters as an example, I have
+shown how this measure can be used to compare the cost and accu-
+racy of a diverse set of QM and SQM models. I have also used it to
+fit a minimal SQM model that applies a pair potential correction to
+this QM and SQM data and select the potential form that best fits
+the data. The TIC provides a reliable parameter penalty to avoid
+selecting over-complicated models, while the AIC is not a reliable
+penalty because some atomistic models are too inaccurate for its
+assumptions to hold. For a computational budget that is too small
+for a high-accuracy model but excessive for a low-accuracy model,
+the success measure predicts the efficacy of splitting a workload
+between models to match the budget. By adjusting the operational
+definition of success for simulation tasks, this success measure
+can be equally good for designing expensive models to succeed at
+difficult tasks and cheap models to succeed at easy tasks.
+An essential aspect of model building in atomistic simulation
+is the availability of high-quality reference data for fitting and test-
+ing. While models have historically relied on reference data from
+experiments, it is now possible to generate accurate data using ex-
+pensive QM models. As shown in the hydrogen cluster example,
+CCSD(T) data is affordable for small molecular fragments, and
+less accurate DFT data remains affordable for larger molecules and
+periodic systems. For data generation at scales larger than what
+has been presented in this paper, reliability issues will become
+increasingly important alongside cost and accuracy considerations.
+SCF convergence problems can cause heralded failures, while SCF
+convergence to excited states can cause silent failures. Without
+more fundamentally reliable algorithms to reduce failure rates, a
+fixed rate of failure means an increasing number of failure events
+as data sets grow larger in size. There are increasingly sophisti-
+cated tools52 for remote, automated computing of large workloads
+and organizing large data sets with modern database standards.
+However, limitations in the reliability of the underlying tasks being
+automated may have a strongly negative influence on the cost and
+accuracy of generating large data sets as failures persist against
+increasing computational redundancy.
+The hydrogen cluster example considered here is sufficiently
+different from typical reference data sets that it serves as a challeng-
+ing test of physical transferability. There is a significant difference
+in the apparent progress that DFT and SQM models have made in
+developing transferable models. The improvement in transferability
+from PBE to B3LYP to 𝜔B97M-V is consistent with the develop-
+ment roadmap of DFT functionals with increasing complexity53.
+While likely a coincidence, the SQM models considered here have
+systematically degrading performance in chronological order of
+their development. A simple explanation of this difference might
+be that DFT functionals are fundamentally more transferable than
+minimal-basis SQM models. However, it is also important to con-
+sider the vastly differing amounts of technical effort that have been
+invested in these two approaches.
+The development path from
+B3LYP to 𝜔B97M-V includes the development of hundreds of
+DFT functionals from numerous research groups over more than
+three decades26. In contrast, the development path from AM1 to
+PM7 consists of only a few other models developed by a single
+scientist – Dr. James J. P. Stewart – working mostly in isolation
+outside of academia for more than three decades. GFN1 and GFN2
+were developed much more recently by a single academic group
+– the research group of Prof. Stefan Grimme at the University of
+Bonn. While there are other SQM models outside of the GFN and
+MNDO-like model families, these are the two most widely used
+families and the only non-commerical models54 to be fit for com-
+binations of elements over most of the periodic table. The GFN
+models incorporate ideas from both MNDO-like models (multipole
+expansions of electrostatics, avoidance of diatomic parameters) and
+DFTB models (expansion around an atomic limit, DFT-like cor-
+relation models). All of the SQM models considered here have
+similar superficial complexity, similar numbers of parameters per
+element, and are fit to similar amounts of reference data. Except
+for a belief in the superiority of DFT-like models, there is no com-
+pelling theoretical reason why any SQM model from this set should
+perform any better than any other on systems that are very different
+from their training data.
+The concepts presented in this paper are meant to inform the
+process of designing, fitting, and selecting models for atomistic
+simulation tasks. If a simulation task is not going to be repeated a
+very large number of times, then the formal process of gathering
+reference data and calculating a success measure might not be
+worth the amount of human effort required. However, the statistical
+model selection process can still be useful as a conceptual guide
+even when it is not worthwhile to perform it carefully or explicitly.
+For tasks that are performed frequently by many scientists, it may
+be worthwhile to capture that activity as a distribution of tasks
+and a representative sampling from that distribution.
+Quantum
+chemistry has a tradition of curating reference data sets to guide
+method development26. Expanding that tradition to accommodate
+larger data sets, statistical interpretations, and success measures that
+capture the real needs of applied scientists could create an even
+better guide for method development. It is difficult for a scientist
+to characterize real application needs while also developing novel
+simulation methods to satisfy those needs, and it would be helpful
+to decouple those important research activities from each other.
+ACKNOWLEDGMENTS
+J. E. M. thanks Jimmy Stewart for helpful discussions. The
+Molecular Sciences Software Institute is supported by NSF Grant
+No. ACI-1547580. The computational resources used in this work
+were provided by Advanced Research Computing at Virginia Tech.
+AUTHOR DECLARATIONS
+Conflict of Interest
+The author has no conflicts to disclose.
+
+15
+Author Contributions
+Jonathan E. Moussa: Conceptualization (equal); Data curation
+(equal); Formal analysis (equal); Investigation (equal); Method-
+ology (equal); Resources (equal); Software (equal); Validation
+(equal); Visualization (equal); Writing – original draft (equal);
+Writing – review & editing (equal).
+DATA AVAILABILITY
+The data and software that support the findings of this study
+are available on Zenodo at the DOI 10.5281/zenodo.7530231.
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+arXiv:2301.04852v1  [hep-ph]  12 Jan 2023
+Evaluation of one type scalar one loop three-point amplitude
+inspired by H → gg decay in the standard model
+Jin Zhang∗
+School of Physics and Engineering, Yuxi Normal University,
+Yuxi, Yunnan, 653100, P. R. China
+Abstract
+Motivated by the Higgs boson decaying to gg at one loop approximation, the amplitude of scalar one
+loop three-point diagram with two different internal masses are evaluated and fully analytic results are
+obtained. The main ingredient of the evaluation is a integral in which the integrand is product of the
+reciprocal of the integral variable and a logarithm, where the argument of the logarithm is a quadratic
+function of the general form. The results depend on the choice of the masses of the propagators and
+the massive external line. In the first case the amplitude contains an infinite series in which each
+term is a hypergeometric function, in the second case the result is expressed through dilogarithms. In
+particular, if the three internal lines are taking the same mass, the results will reduce to the known
+functions in one loop evaluation of Higgs decaying to gg or γγ.
+PACS numbers:
+∗ jinzhang@yxnu.edu.cn
+1
+
+I.
+INTRODUCTION
+In framework of the Standard Model(SM) and its minimal supersymmetric extension, the
+evaluation of scalar one loop three-point amplitudes play a fundamental role in deciphering the
+property of the Higgs boson[1, 2] though its decaying to gg(or γγ) and the inverse process, i.e.,
+production of the Higgs boson by gluon fusion[3–7]. Owing to the coupling of the Higgs boson
+to the fermions gHf ¯f and the coupling of fermions to gluon, the three propagators take the
+same mass at the leading order of perturbative theory. The evaluated amplitude, if the high
+energy approximation[9, 10] is not exploited, will be expressed as function of mf/mH, where
+mf and mH are the masses of the internal fermion and the Higgs boson, respectively. At the
+final stage of the evaluation, a integral of the following form must be handled carefully
+I1 =
+� 1
+0
+dx ln(ax2 − ax + 1 − iε)
+x
+,
+a > 0
+(1)
+where ε is positive infinitesimal. The integral in Eq.(1) necessarily arises both in the evaluation
+of the amplitude of Higgs boson decaying to gg and its production via gluon fusion at one loop.
+In addition to the top quark, there is considerable mass hierarchy between the Higgs boson
+and other quarks, an economic way to compute Eq.(1) is taking the limit that the masses of
+the propagators in the triangle are negligible compared with mass of the Higgs boson, then
+the result of Eq.(1) tends to constant. However, in this manner we can not tell the different
+contributions from various competing processes to the amplitude. Thus, fully analytic results
+is essential to analyze H → gg, then results will be distinguished the cases 0 < a < 4 from the
+case a > 4 as detailed in the later works[5, 11–17].
+As a natural generalization of Eq.(1), let us consider the following integral
+I2 =
+� 1
+0
+dx ln(ax2 − bx + 1 − iε)
+x
+,
+(2)
+the parameters a and b satisfy
+a > 0,
+b > 0,
+a ̸= b
+(3)
+It is obvious that if a = b, Eq.(2) reduce to Eq.(1). This integral can be derived from the
+evaluation of the scalar one loop three-point diagram depicted in fig.1, in this case a and b will
+be functions of the masses of the propagators ω1 and ω1 as well as the mass of external line m1.
+Unfortunately, a close inspection to fig.1 indicates that it does not connect with real decaying
+2
+
+p1
+p2
+p3
+k
+ω2
+ω2
+ω1
+FIG. 1: Massive triangle with two massless external lines. The solid lines and dashed lines denote
+massive and massless particles, respectively.
+processes of the Higgs boson even though the contribution from Higgs-Kibble ghosts associated
+with the W ± and Z bosons are taken into account. Maybe it is the reason that rarely can
+we look up the evaluation of integral displayed in Eq.(2) in the one loop evaluation of Higgs
+boson decaying to gg1. An thorough investigation of the integral in Eq.(2) on the footing of
+perturbative theory is necessary. Therefore, in this paper we will present a systematic study on
+Eq.(2) based on the evaluation of scalar one loop three-point amplitude, the complete analytic
+results are derived. We hope that the results can be applied to some decaying process under
+reasonable approximation addition to H → gg, but also enrich the results of scalar one loop
+three-point diagram from the viewpoint of analytic evaluation.
+The paper is organized as follows.
+In section II we introduce the integral in Eq.(2) by
+evaluation the amplitude depicted in fig.1 in a scalar field theory, some general results are
+derived. In section III the analytic results of fig.1 are obtained and the special case a = b
+are discussed. A short summary are presented in IV. Some useful formulas are listed in the
+appendix.
+1 Integral of this type has been computed in Eq.(23) of Ref.[8], but only the case b2 − 4a > 0 is considered.
+3
+
+II.
+THE FORMULAS
+A.
+The massive triangle with two massless external lines
+To start with, we write down the amplitude corresponding to fig.1
+I =
+�
+d4k
+(2π)4
+1
+A1A2A3
+,
+(4)
+where the three denominators are defined by
+A1 = k2 − ω2
+1 + iε
+A2 = (p1 − k)2 − ω2
+2 + iε
+A3 = (p1 − p2 − k)2 − ω2
+2 + iε,
+(5)
+and ε is real positive infinitesimal, the three external momentum satisfy
+p2
+1 = m2
+1,
+p2
+2 = p2
+3 = 0.
+(6)
+Using the Feynman’s trick, Eq.(4) can be written as
+I =
+�
+d4k
+(2π)4
+�
+dxdydz2! δ(1 − x − y − z)
+[D(x, y, z)]3
+,
+(7)
+where
+D(x, y, z) = x(k2 − ω2
+1 + iε) + y[(p1 − k)2 − ω2
+2 + iε] + z[(p1 − p2 − k)2 − ω2
+2 + iε],
+(8)
+Since the amplitude given by Eq.(4) is both ultraviolet and infrared finite, thus regularization
+is unnecessary, the evaluation can be carried out in the four-dimensional space-time. We first
+perform the integral over k and z, obtaining
+I = −
+i
+16π2
+� 1
+0
+dx
+� 1−x
+0
+dy
+1
+−yxm2
+1 + x(ω2
+1 − ω2
+2) + ω2
+2 − iε
+(9)
+The integral over y in Eq.(9) is trivial, combining with Eq.(A1), we arrive at the following
+intermediate result
+I =
+i
+16π2m2
+1
+� 1
+0
+dx1
+x
+�
+ln
+�m2
+1
+ω2
+2
+x2 − m2
+1 − ω2
+1 + ω2
+2
+ω2
+2
+x + 1 − iε
+�
+− ln
+�ω2
+1 − ω2
+2
+ω2
+2
+x + 1 − iε
+��
+=
+i
+16π2m2
+1
+�
+Li2
+�
+1 − ω2
+1
+ω2
+2
+�
++
+� 1
+0
+dx1
+x ln
+�m2
+1
+ω2
+2
+x2 − m2
+1 − ω2
+1 + ω2
+2
+ω2
+2
+x + 1 − iε
+��
+.
+(10)
+The remaining work is the evaluation of the last integral in Eq.(10). For brevity, in the forth-
+coming sections the pre-factor i/(16π2) will be suppressed while 1/m2
+1 will be preserved so as
+to maintain the correct dimension of the primitive amplitude displayed in Eq.(4).
+4
+
+B.
+evaluation of integral with logarithms
+The evaluation of the last term in Eq.(10) motivates a general investigation on the integral
+of the following type
+F =
+� 1
+0
+dx ln(ax2 − bx + 1 − iε)
+x
+,
+a > 0,
+b > 0
+(11)
+Since the argument of the logarithm is quadratic in x, we first consider the case b2 − 4a < 0,
+in this case the argument of the logarithm is positive definite thus the iε term can be safely
+dropped. A feasible way to calculate the integral of Eq.(11) turns out to be expressing it as
+F =
+� 1
+0
+dx
+� 1
+0
+dz
+ax − b
+1 + zx(ax − b)
+= a
+� 1
+0
+dz
+� 1
+0
+dx
+x
+1 + zx(ax − b) − b
+� 1
+0
+dz
+� 1
+0
+dx
+1
+1 + zx(ax − b)
+= 1
+2
+� 1
+0
+dzln[1 + z(a − b)]
+z
+− b
+2
+� 1
+0
+dz
+� 1
+0
+dx
+1
+1 + zx(ax − b))
+= −1
+2Li2(b − a) − b
+2
+� 1
+0
+dz
+� 1
+0
+dx
+1
+1 + zx(ax − b).
+(12)
+Now we concentrate on the last integral in Eq.(12), for later convenience we label it as A, the
+integral over x can be calculated[18]
+A =
+� 1
+0
+dz
+� 1
+0
+dx
+1
+1 + zx(ax − b)
+=
+� 1
+0
+dz
+2
+√
+4az − b2z2
+�
+arctan
+bz
+√
+4az − b2z2 + arctan
+2az − bz
+√
+4az − b2z2
+�
+,
+(13)
+In deriving Eq.(13) we employ the property that arctan(x) is odd
+arctan(−x) = − arctan(x),
+(14)
+To proceed we separate the integral in Eq.(13) into two parts
+A = 2(A1 + A2),
+(15)
+where
+A1 =
+� 1
+0
+dz
+1
+√
+4az − b2z2 arctan
+bz
+√
+4az − b2z2,
+5
+
+A2 =
+� 1
+0
+dz
+1
+√
+4az − b2z2 arctan
+2az − bz
+√
+4az − b2z2.
+(16)
+It is not difficult to demonstrate that
+�
+arcsin
+�
+b
+� z
+4a
+��′ =
+�
+arctan
+bz
+√
+4az − b2z2
+�′ = b
+2
+1
+√
+4az − b2z2,
+(17)
+Hence, the integral in A1 is easy to calculate
+A1 = 1
+b
+�
+arctan
+b
+√
+4a − b2
+�2,
+(18)
+Using the identity[21]
+arctan x = arcsin
+x
+√
+1 + x2,
+(19)
+leads to the final result for A1
+A1 = 1
+b
+�
+arcsin
+b
+2√a
+�2.
+(20)
+Next, we evaluate A2, by making use integration by parts, getting
+A2 = 2
+b arcsin(
+b
+2√a) arctan
+2a − b
+√
+4a − b2 − 2a − b
+b
+� 1
+0
+arcsin
+�
+b� z
+4a
+�
+[1 + (a − b)z]
+√
+4az − b2z2 dz,
+(21)
+We label the second term in Eq.(21) as B and define
+u = b
+� z
+4a,
+(22)
+such that this term casts into the following form
+B = 2a − b
+b2
+� b/(2√a)
+0
+arcsin u
+(1 + αu2)
+√
+1 − u2 du,
+α = 4a(a − b)
+b2
+(23)
+Since 0 < u < 1, we employ the following expansion[22]
+arcsin u
+√
+1 − u2 =
++∞
+�
+n=0
+2nn!
+(2n + 1)!!u2n+1,
+(24)
+Plugging Eq.(24) into Eq.(23), interchanging the order of integration and summation, yields
+B = 2a − b
+b2
++∞
+�
+n=0
+2nn!
+(2n + 1)!!
+� b/(2√a)
+0
+u2n+1
+1 + αu2 du,
+(25)
+It is convenient to make variable transformation for the second time
+ξ = u2,
+(26)
+6
+
+Then we obtain
+B = 2a − b
+b2
++∞
+�
+n=0
+2nn!
+(2n + 1)!!
+� b2/(4a)
+0
+ξn
+1 + αξdξ,
+(27)
+Changing the variable further
+ξ = b2
+4aρ,
+(28)
+Eq.(27) can be written as
+B = 2a − b
+b2
++∞
+�
+n=0
+2nn!
+(2n + 1)!!
+� b2
+4a
+�n+1 � 1
+0
+ρn�
+1 + αb2
+4a ρ
+�−1
+dρ,
+(29)
+Comparing with Eq.(A4), we identify
+α = 1,
+β = n + 1,
+γ = n + 2,
+z = b − a,
+(30)
+which generates the following result for B
+B = 2a − b
+2a
++∞
+�
+n=0
+2nn!
+(n + 1)(2n + 1)!!
+� b2
+4a
+�n
+2F1(1, n + 1; n + 2; b − a),
+(31)
+Combining Eq.(20), Eq.(21) and Eq.(31), we arrive at the final result of F in the case b2−4a < 0
+F = −1
+2Li2(b − a) −
+�
+arcsin
+b
+2√a
+�2 − 2 arcsin(
+b
+2√a) arctan
+2a − b
+√
+4a − b2
++ (2a − b)b
+2a
++∞
+�
+n=0
+2nn!
+(n + 1)(2n + 1)!!
+� b2
+4a
+�n
+2F1(1, n + 1; n + 2; b − a).
+(32)
+Next, we consider the case b2 − 4a > 0. In this case there are two zeros of the logarithm in
+the range [0, 1], the iε prescription must be retained appropriately. By exploiting integration
+by parts, it is easy to get
+F = −
+� 1
+0
+(2ax − b) ln x
+a(x − x1)(x − x2) dx,
+(33)
+where x1 and x2 are the two roots of the argument of the logarithm
+x1 = x+ + iε,
+x+ = b +
+√
+b2 − 4a
+2a
+,
+x2 = x− − iε,
+x− = b −
+√
+b2 − 4a
+2a
+,
+(34)
+Making use partial fraction expansion and Eq.(A3), we obtain the following result
+F =
+1
+x1 − x2
+( b
+a − 2x1)
+� 1
+0
+ln x
+x − x1
+dx +
+1
+x1 − x2
+(2x2 − b
+a)
+� 1
+0
+ln x
+x − x2
+dx
+=
+1
+x1 − x2
+�
+( b
+a − 2x1)Li2[ 1
+x+
+− iε sgn(x+)] − ( b
+a − 2x2)Li2[ 1
+x−
++ iε sgn(x−)]
+�
+.
+(35)
+with the function sgn(x) defined in Eq.(A3).
+7
+
+III.
+RESULTS AND DISCUSSIONS
+We shall now apply the results obtained in Section 2 to calculate the integral left in Eq.(10).
+Comparing Eq.(10) and Eq.(11), we identify that
+a = m2
+1
+ω2
+2
+,
+b = m2
+1 − ω2
+1 + ω2
+2
+ω2
+2
+,
+(36)
+In the case b2 − 4a < 0 which implies that λ(m2
+1, ω2
+1, ω2
+2) < 0, where λ(x, y, z) is the well-known
+K¨allen function
+λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz,
+(37)
+By employing Eq.(32), yields the following explicit result
+I =
+1
+m2
+1
+�1
+2 Li2
+�
+1 − ω2
+1
+ω2
+2
+�
+−
+�
+arcsin m2
+1 − ω2
+1 + ω2
+2
+2m1ω2
+�2
+− 2 arcsin m2
+1 − ω2
+1 + ω2
+2
+2m1ω2
+arctan m2
+1 + ω2
+1 − ω2
+2
+�
+λ(m2
+1, ω2
+1, ω2
+2)
++ m4
+1 − (ω2
+1 − ω2
+2)2
+8m2
+1ω2
+2
+� +∞
+�
+n=0
+2nn!
+(n + 1)(2n + 1)!!
+�(m2
+1 − ω2
+1 + ω2
+2)2
+4m2
+1ω2
+1
+�n
+× 2F1
+�
+1, n + 1; n + 2; 1 − ω2
+1
+ω2
+2
+���
+.
+(38)
+In order to apply the result in Eq.(38) correctly, the following comments are necessary. First,
+from the conditions of a and b declared in Eq.(11), to guarantee Eq.(38) holds true for the
+evaluation, in addition to λ(m2
+1, ω2
+1, ω2
+2) < 0, the masses must obey
+m2
+1 > ω2
+1 − ω2
+2,
+(39)
+Second, since Eq.(38) is summed over hypergeometric functions, a crucial issue is that if the
+summation of the infinite series is convergent. Due to λ(m2
+1, ω2
+1, ω2
+2) < 0, it is obvious that
+0 < (m2
+1 − ω2
+1 + ω2
+2)2
+4m2
+1ω2
+1
+< 1.
+(40)
+and the hypergeometric functions are always taking finite value, thus the summation is conver-
+gent. Finally, in considering the analytic property of the dilogarithm in Eq.(A2), a question is
+that if Eq.(38) can develop imaginary part. In other words, if the argument of the dilogarithm
+can be greater than 1. But it is impossible since both ω1 and ω2 are assumed to be real, thus
+0 < 1 − (ω2
+1/ω2
+2) < 1 is always satisfied, therefore there is no imaginary part can be developed.
+8
+
+In the case b2 − 4a > 0, this implies that λ(m2
+1, ω2
+1, ω2
+2) > 0, exploiting Eq.(35) we get
+I = 1
+m2
+1
+�1
+2 Li2(1 − ω2
+1
+ω2
+2
+) − Li2[ 1
+x+
+− iε sgn(x+)] − Li2[ 1
+x−
++ iε sgn(x−)]
+�
+,
+(41)
+where
+x+ = (m2
+1 − ω2
+1 + ω2
+2) + λ1/2(m2
+1, ω2
+1, ω2
+2)
+2m2
+1
+,
+x− = (m2
+1 − ω2
+1 + ω2
+2) − λ1/2(m2
+1, ω2
+1, ω2
+2)
+2m2
+1
+,
+(42)
+Since m2
+1 > ω2
+1 − ω2
+2 as presented in Eq.(39), both x+ and x− are positive definite, thus Eq.(41)
+simplified to
+I = 1
+m2
+1
+�1
+2 Li2(1 − ω2
+1
+ω2
+2
+) − Li2( 1
+x+
+− iε) − Li2( 1
+x−
++ iε)
+�
+.
+(43)
+In order to explore phenomenological implications of the results presented in Eq.(38) and
+Eq.(43), it is instructive to consider the special case that the coefficients in Eq.(11) are con-
+strained by the following conditions
+a = b > 0,
+(44)
+This is the integral indispensable in the evaluation of H → gg decay. studied in past. Supposing
+the mass of each propagator of the triangle is ω1, setting
+a = b = m2
+1
+ω2
+1
+,
+(45)
+Hence, Eq.(10) reduces to
+I = 1
+m2
+1
+� 1
+0
+dxln(ax2 − ax + 1 − iε)
+x
+,
+a = m2
+1
+ω2
+1
+(46)
+First consider the case 0 < a < 4, i.e., 0 < m1 < 2ω1. To get the correct results we must trace
+back to Eq.(12) and Eq.(13), other than taking advantage of Eq.(32) and simply setting a = b,
+otherwise it will make some mistakes. By exploiting Eq.(17), we immediately obtain
+I = − a
+2m2
+1
+� 1
+0
+dz
+� 1
+0
+dx
+1
+1 + zax(x − 1)
+= − a
+2m2
+1
+� 1
+0
+dz
+4
+√
+4az − a2z2 arctan
+az
+√
+4az − a2z2
+= − 2
+m2
+1
+�
+arctan
+a
+√
+4a − a2
+�2
+,
+(47)
+9
+
+By using Eq.(19), we get the well-known function appeared in one loop evaluation of H → gg
+I = − 2
+m2
+1
+�
+arcsin m1
+2ω1
+�2
+,
+0 < m1 < 2ω1.
+(48)
+Next, we consider the case of a > 4, i.e., m1 > 2ω1. From Eq.(35) we get
+I = 1
+m2
+1
+�
+− Li2( 1
+x−
++ iε) − Li2( 1
+x+
+− iε)
+�
+,
+(49)
+x+ and x− are given by
+x+ = 1
+2
+�
+1 +
+�
+1 − 4
+a
+�
+,
+x− = 1
+2
+�
+1 −
+�
+1 − 4
+a
+�
+,
+(50)
+Since both x+ and x− are less than 1, in order to simplify the Eq.(49), defining
+α = 1
+x−
+> 1,
+1
+x+
+=
+1
+1 − x−
+=
+α
+α − 1 > 1,
+(51)
+Then Eq.(49) can be written as
+I = 1
+m2
+1
+�
+− Li2(α + iε) − Li2(
+α
+α − 1 − iε)
+�
+,
+(52)
+Combining Eq.(A2) we obtain
+I =
+1
+m2
+1
+�
+−
+�
+Re Li2(α) + iπ ln α
+�
+−
+�
+Re Li2(
+α
+α − 1) − iπ ln
+α
+α − 1
+�
+=
+1
+m2
+1
+�
+− Re
+�
+Li2(α) + Li2(
+α
+α − 1)
+�
+− iπ ln(α − 1)
+�
+,
+(53)
+In considering the property od dilogarithm
+Re
+�
+Li2(x) + Li2(
+x
+x − 1)
+�
+= π2
+2 − ln2(x − 1)
+2
+,
+x > 1
+(54)
+which leads to
+I = 1
+m2
+1
+�
+− π2
+2 + ln2(α − 1)
+2
+− iπ ln(α − 1)
+�
+,
+(55)
+Substituting Eq.(51) into Eq.(55) and noticing that
+α − 1 = x+
+x−
+,
+(56)
+we get
+I = 1
+m2
+1
+�
+− π2
+2 + 1
+2 ln2 x+
+x−
+− iπ ln x+
+x−
+�
+,
+(57)
+Plugging Eq.(50) into Eq.(57) we arrive at the well-known function in the one loop evaluation
+of H → gg decay
+I = − 1
+2m2
+1
+�
+π + i ln
+1 +
+�
+1 − 4ω2
+1
+m2
+1
+1 −
+�
+1 − 4ω2
+1
+m2
+1
+�2
+.
+m1 > 2ω1
+(58)
+10
+
+IV.
+SUMMARY
+In this paper, the amplitude of scalar one loop three-point diagram in which the masses of
+the three internal propagators are assigned two different masses is evaluated. A general type
+integral is extracted and analytic results are obtained. Conditions of the validity of the results
+are discussed in details. As a check to the results, we consider the case that each propagator of
+the triangle taking the same mass, we find that the general results will reduce to the functions
+obtained in the lowest order evaluation of H → gg decay. We also notice that the results are
+mathematically preferred in that the diagram does not corresponds to real process in H → gg
+in the SM. However, we still hope that the results and techniques may be found its applications
+in triangle mediated decays addition to H → gg.
+Appendix A: The dilogarithm function and integral representation of Gauss hyperge-
+ometric function
+In this section we list some necessary formula in our evaluation. The dilogarithm is defined
+as[19]
+Li2(x) =
++∞
+�
+k=1
+xk
+k2 = −
+� 1
+0
+ln(1 − xt)
+t
+dt,
+|x| < 1
+(A1)
+There is a branch cut from 1 to ∞, for ε → 0
+Li2(x + iε) = Re Li2(x) + iπ sgn(ε)Θ(x − 1) ln x,
+(A2)
+where Θ is the step function, the sgn(x) is
+sgn(x) =
+
+
+
+
+
+1
+x > 0
+−1
+x < 0
+Another two formulas we need are the following equation of dilogarithm[20]
+� 1
+0
+ln x
+a + bx dx = 1
+bLi2
+�
+− b
+a
+�
+,
+(A3)
+and the integral representation of the Gauss hypergeometric function[23, 24]
+2F1(α, β; γ; z) =
+Γ(γ)
+Γ(β)Γ(γ − β)
+� 1
+0
+tβ−1(1 − t)γ−β−1(1 − zt)−αdt,
+(A4)
+11
+
+where
+Re(γ) > Re(β) > 0,
+|arg(1 − z)| < π.
+(A5)
+[1] G. Aad et al.(ATLAS), PhysLett. B 716, 1(2012).
+[2] S. Chatrchyan et al.(CMS), B 716, 30(2012).
+[3] J. F. Gunion, H. E. Haber, G. Kane, S. Dawson, The Higgs Hunter’s Guide (Perseus Publishing,
+Cambridge, Massachusetts, 1990).
+[4] B. A. Kniehl, Phys. Rept. 240, 211(1994).
+[5] A. Djouadi, Phys. Rept. 457, 1(2008), Phys. Rept. 459, 1(2008).
+[6] M. Spira, Prog. Part. Nucl. Phys. 95, 98(2017).
+[7] M. Carena, C. Grojean, M. Kado et al, “Status of Higgs Boson Physics”, R. L. Workman et
+al.(Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01(2022).
+[8] M. Roth, A. Denner, Nucl. Phys. B 479, 495(1996).
+[9] J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B 106, 292(1976).
+[10] T. G. Rizzo, Phys. Rev. D 22, 178(1980), Phys. Rev. D 22, 1824(1980).
+[11] A. I. Vainshtein, M. B. Voloshin, V. I. Zakharov et al, Sov. J. Nucl. Phys. 30, 711(1979).
+[12] L. B. Okun, Leptons and Quarks(North-Holland, Amsterdam, 1982).
+[13] J. F. Gunion and H. E. Haber, Nucl. Phys. B 278, 449(1986), Erratum: Nucl. Phys. B 402,
+569(1993).
+[14] R. K. Ellis, I. Hinchliffe, M. Soldate et al, Nucl. Phys. B 297, 221(1988).
+[15] D. Huang, Y. Tang and Y-L. Wu, Commun. Theor. Phys. 57, 427(2012).
+[16] M. Shifman, A. Vainshtein, M. B. Voloshin, et al, Phys. Rev. D 85, 013015(2012).
+[17] W. J. Marciano, C. Zhang and S. Willenbrock, Phys. Rev. D 85, 013002(2012).
+[18] H. B. Dwight, Tables of Integrals and Other Mathematical Data(The Macmillan Company, New
+York, 1957), Third edition.
+[19] L. Lewin, Polylogarithms and Associated Functions(North Holland, New York, 1981), Second
+Edition.
+[20] A. Devoto and D. W. Duke, Riv. Nuovo Cim. 7N6, 1(1984).
+12
+
+[21] I. S. Gradshteyn I. M. Ryzhik, Table of Integrals, Series, and Products, Eighth Edition(Academic
+Press, London, 2014).
+[22] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with formulas, Graphs and
+Mathematical Tables(Dover Publications, New York,1972).
+[23] A. Erdelyi, Higher Transcendental Functions, Vol.I (McGrill-Hall Book Company, New York,
+1953).
+[24] Z. X. Wang, D. R. Guo, Special Functions(World Scientific, Singapore, 1989).
+13
+

5tE4T4oBgHgl3EQfBgtg/content/tmp_files/load_file.txt

@@ -0,0 +1,333 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf,len=332
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='04852v1  [hep-ph]  12 Jan 2023  Evaluation of one type scalar one loop three-point amplitude  inspired by H → gg decay in the standard model  Jin Zhang∗  School of Physics and Engineering, Yuxi Normal University,  Yuxi, Yunnan, 653100, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' China  Abstract  Motivated by the Higgs boson decaying to gg at one loop approximation, the amplitude of scalar one  loop three-point diagram with two different internal masses are evaluated and fully analytic results are  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' The main ingredient of the evaluation is a integral in which the integrand is product of the  reciprocal of the integral variable and a logarithm, where the argument of the logarithm is a quadratic  function of the general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' The results depend on the choice of the masses of the propagators and  the massive external line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' In the first case the amplitude contains an infinite series in which each  term is a hypergeometric function, in the second case the result is expressed through dilogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' In  particular, if the three internal lines are taking the same mass, the results will reduce to the known  functions in one loop evaluation of Higgs decaying to gg or γγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  PACS numbers:  ∗ jinzhang@yxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='cn  1  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  INTRODUCTION  In framework of the Standard Model(SM) and its minimal supersymmetric extension, the  evaluation of scalar one loop three-point amplitudes play a fundamental role in deciphering the  property of the Higgs boson[1, 2] though its decaying to gg(or γγ) and the inverse process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=',  production of the Higgs boson by gluon fusion[3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Owing to the coupling of the Higgs boson  to the fermions gHf ¯f and the coupling of fermions to gluon, the three propagators take the  same mass at the leading order of perturbative theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' The evaluated amplitude, if the high  energy approximation[9, 10] is not exploited, will be expressed as function of mf/mH, where  mf and mH are the masses of the internal fermion and the Higgs boson, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' At the  final stage of the evaluation, a integral of the following form must be handled carefully  I1 =  � 1  0  dx ln(ax2 − ax + 1 − iε)  x  ,  a > 0  (1)  where ε is positive infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' The integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (1) necessarily arises both in the evaluation  of the amplitude of Higgs boson decaying to gg and its production via gluon fusion at one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  In addition to the top quark, there is considerable mass hierarchy between the Higgs boson  and other quarks, an economic way to compute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (1) is taking the limit that the masses of  the propagators in the triangle are negligible compared with mass of the Higgs boson, then  the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (1) tends to constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' However, in this manner we can not tell the different  contributions from various competing processes to the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Thus, fully analytic results  is essential to analyze H → gg, then results will be distinguished the cases 0 < a < 4 from the  case a > 4 as detailed in the later works[5, 11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  As a natural generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (1), let us consider the following integral  I2 =  � 1  0  dx ln(ax2 − bx + 1 − iε)  x  ,  (2)  the parameters a and b satisfy  a > 0,  b > 0,  a ̸= b  (3)  It is obvious that if a = b, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (2) reduce to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' This integral can be derived from the  evaluation of the scalar one loop three-point diagram depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='1, in this case a and b will  be functions of the masses of the propagators ω1 and ω1 as well as the mass of external line m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  Unfortunately, a close inspection to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='1 indicates that it does not connect with real decaying  2  p1  p2  p3  k  ω2  ω2  ω1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' 1: Massive triangle with two massless external lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' The solid lines and dashed lines denote  massive and massless particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  processes of the Higgs boson even though the contribution from Higgs-Kibble ghosts associated  with the W ± and Z bosons are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Maybe it is the reason that rarely can  we look up the evaluation of integral displayed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (2) in the one loop evaluation of Higgs  boson decaying to gg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' An thorough investigation of the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (2) on the footing of  perturbative theory is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Therefore, in this paper we will present a systematic study on  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (2) based on the evaluation of scalar one loop three-point amplitude, the complete analytic  results are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' We hope that the results can be applied to some decaying process under  reasonable approximation addition to H → gg, but also enrich the results of scalar one loop  three-point diagram from the viewpoint of analytic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  In section II we introduce the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (2) by  evaluation the amplitude depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='1 in a scalar field theory, some general results are  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' In section III the analytic results of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='1 are obtained and the special case a = b  are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' A short summary are presented in IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Some useful formulas are listed in the  appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  1 Integral of this type has been computed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (23) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' [8], but only the case b2 − 4a > 0 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  THE FORMULAS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  The massive triangle with two massless external lines  To start with, we write down the amplitude corresponding to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='1  I =  �  d4k  (2π)4  1  A1A2A3  ,  (4)  where the three denominators are defined by  A1 = k2 − ω2  1 + iε  A2 = (p1 − k)2 − ω2  2 + iε  A3 = (p1 − p2 − k)2 − ω2  2 + iε,  (5)  and ε is real positive infinitesimal, the three external momentum satisfy  p2  1 = m2  1,  p2  2 = p2  3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (6)  Using the Feynman’s trick, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (4) can be written as  I =  �  d4k  (2π)4  �  dxdydz2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' δ(1 − x − y − z)  [D(x, y, z)]3  ,  (7)  where  D(x, y, z) = x(k2 − ω2  1 + iε) + y[(p1 − k)2 − ω2  2 + iε] + z[(p1 − p2 − k)2 − ω2  2 + iε],  (8)  Since the amplitude given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (4) is both ultraviolet and infrared finite, thus regularization  is unnecessary, the evaluation can be carried out in the four-dimensional space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' We first  perform the integral over k and z, obtaining  I = −  i  16π2  � 1  0  dx  � 1−x  0  dy  1  −yxm2  1 + x(ω2  1 − ω2  2) + ω2  2 − iε  (9)  The integral over y in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (9) is trivial, combining with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (A1), we arrive at the following  intermediate result  I =  i  16π2m2  1  � 1  0  dx1  x  �  ln  �m2  1  ω2  2  x2 − m2  1 − ω2  1 + ω2  2  ω2  2  x + 1 − iε  �  − ln  �ω2  1 − ω2  2  ω2  2  x + 1 − iε  ��  =  i  16π2m2  1  �  Li2  �  1 − ω2  1  ω2  2  �  +  � 1  0  dx1  x ln  �m2  1  ω2  2  x2 − m2  1 − ω2  1 + ω2  2  ω2  2  x + 1 − iε  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (10)  The remaining work is the evaluation of the last integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' For brevity, in the forth-  coming sections the pre-factor i/(16π2) will be suppressed while 1/m2  1 will be preserved so as  to maintain the correct dimension of the primitive amplitude displayed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  evaluation of integral with logarithms  The evaluation of the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (10) motivates a general investigation on the integral  of the following type  F =  � 1  0  dx ln(ax2 − bx + 1 − iε)  x  ,  a > 0,  b > 0  (11)  Since the argument of the logarithm is quadratic in x, we first consider the case b2 − 4a < 0,  in this case the argument of the logarithm is positive definite thus the iε term can be safely  dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' A feasible way to calculate the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (11) turns out to be expressing it as  F =  � 1  0  dx  � 1  0  dz  ax − b  1 + zx(ax − b)  = a  � 1  0  dz  � 1  0  dx  x  1 + zx(ax − b) − b  � 1  0  dz  � 1  0  dx  1  1 + zx(ax − b)  = 1  2  � 1  0  dzln[1 + z(a − b)]  z  − b  2  � 1  0  dz  � 1  0  dx  1  1 + zx(ax − b))  = −1  2Li2(b − a) − b  2  � 1  0  dz  � 1  0  dx  1  1 + zx(ax − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (12)  Now we concentrate on the last integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (12), for later convenience we label it as A, the  integral over x can be calculated[18]  A =  � 1  0  dz  � 1  0  dx  1  1 + zx(ax − b)  =  � 1  0  dz  2  √  4az − b2z2  �  arctan  bz  √  4az − b2z2 + arctan  2az − bz  √  4az − b2z2  �  ,  (13)  In deriving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (13) we employ the property that arctan(x) is odd  arctan(−x) = − arctan(x),  (14)  To proceed we separate the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (13) into two parts  A = 2(A1 + A2),  (15)  where  A1 =  � 1  0  dz  1  √  4az − b2z2 arctan  bz  √  4az − b2z2,  5  A2 =  � 1  0  dz  1  √  4az − b2z2 arctan  2az − bz  √  4az − b2z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (16)  It is not difficult to demonstrate that  �  arcsin  �  b  � z  4a  ��′ =  �  arctan  bz  √  4az − b2z2  �′ = b  2  1  √  4az − b2z2,  (17)  Hence, the integral in A1 is easy to calculate  A1 = 1  b  �  arctan  b  √  4a − b2  �2,  (18)  Using the identity[21]  arctan x = arcsin  x  √  1 + x2,  (19)  leads to the final result for A1  A1 = 1  b  �  arcsin  b  2√a  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (20)  Next, we evaluate A2, by making use integration by parts, getting  A2 = 2  b arcsin(  b  2√a) arctan  2a − b  √  4a − b2 − 2a − b  b  � 1  0  arcsin  �  b� z  4a  �  [1 + (a − b)z]  √  4az − b2z2 dz,  (21)  We label the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (21) as B and define  u = b  � z  4a,  (22)  such that this term casts into the following form  B = 2a − b  b2  � b/(2√a)  0  arcsin u  (1 + αu2)  √  1 − u2 du,  α = 4a(a − b)  b2  (23)  Since 0 < u < 1, we employ the following expansion[22]  arcsin u  √  1 − u2 =  +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='u2n+1,  (24)  Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (24) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (23), interchanging the order of integration and summation, yields  B = 2a − b  b2  +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  � b/(2√a)  0  u2n+1  1 + αu2 du,  (25)  It is convenient to make variable transformation for the second time  ξ = u2,  (26)  6  Then we obtain  B = 2a − b  b2  +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  � b2/(4a)  0  ξn  1 + αξdξ,  (27)  Changing the variable further  ξ = b2  4aρ,  (28)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (27) can be written as  B = 2a − b  b2  +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  � b2  4a  �n+1 � 1  0  ρn�  1 + αb2  4a ρ  �−1  dρ,  (29)  Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (A4), we identify  α = 1,  β = n + 1,  γ = n + 2,  z = b − a,  (30)  which generates the following result for B  B = 2a − b  2a  +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (n + 1)(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  � b2  4a  �n  2F1(1, n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' b − a),  (31)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (20), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (31), we arrive at the final result of F in the case b2−4a < 0  F = −1  2Li2(b − a) −  �  arcsin  b  2√a  �2 − 2 arcsin(  b  2√a) arctan  2a − b  √  4a − b2  + (2a − b)b  2a  +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (n + 1)(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  � b2  4a  �n  2F1(1, n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' b − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (32)  Next, we consider the case b2 − 4a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' In this case there are two zeros of the logarithm in  the range [0, 1], the iε prescription must be retained appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' By exploiting integration  by parts, it is easy to get  F = −  � 1  0  (2ax − b) ln x  a(x − x1)(x − x2) dx,  (33)  where x1 and x2 are the two roots of the argument of the logarithm  x1 = x+ + iε,  x+ = b +  √  b2 − 4a  2a  ,  x2 = x− − iε,  x− = b −  √  b2 − 4a  2a  ,  (34)  Making use partial fraction expansion and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (A3), we obtain the following result  F =  1  x1 − x2  ( b  a − 2x1)  � 1  0  ln x  x − x1  dx +  1  x1 − x2  (2x2 − b  a)  � 1  0  ln x  x − x2  dx  =  1  x1 − x2  �  ( b  a − 2x1)Li2[ 1  x+  − iε sgn(x+)] − ( b  a − 2x2)Li2[ 1  x−  + iε sgn(x−)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (35)  with the function sgn(x) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  7  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  RESULTS AND DISCUSSIONS  We shall now apply the results obtained in Section 2 to calculate the integral left in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (11), we identify that  a = m2  1  ω2  2  ,  b = m2  1 − ω2  1 + ω2  2  ω2  2  ,  (36)  In the case b2 − 4a < 0 which implies that λ(m2  1, ω2  1, ω2  2) < 0, where λ(x, y, z) is the well-known  K¨allen function  λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz,  (37)  By employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (32), yields the following explicit result  I =  1  m2  1  �1  2 Li2  �  1 − ω2  1  ω2  2  �  −  �  arcsin m2  1 − ω2  1 + ω2  2  2m1ω2  �2  − 2 arcsin m2  1 − ω2  1 + ω2  2  2m1ω2  arctan m2  1 + ω2  1 − ω2  2  �  λ(m2  1, ω2  1, ω2  2)  + m4  1 − (ω2  1 − ω2  2)2  8m2  1ω2  2  � +∞  �  n=0  2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (n + 1)(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  �(m2  1 − ω2  1 + ω2  2)2  4m2  1ω2  1  �n  × 2F1  �  1, n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' 1 − ω2  1  ω2  2  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (38)  In order to apply the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (38) correctly, the following comments are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' First,  from the conditions of a and b declared in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (11), to guarantee Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (38) holds true for the  evaluation, in addition to λ(m2  1, ω2  1, ω2  2) < 0, the masses must obey  m2  1 > ω2  1 − ω2  2,  (39)  Second, since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (38) is summed over hypergeometric functions, a crucial issue is that if the  summation of the infinite series is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Due to λ(m2  1, ω2  1, ω2  2) < 0, it is obvious that  0 < (m2  1 − ω2  1 + ω2  2)2  4m2  1ω2  1  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (40)  and the hypergeometric functions are always taking finite value, thus the summation is conver-  gent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Finally, in considering the analytic property of the dilogarithm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (A2), a question is  that if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (38) can develop imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' In other words, if the argument of the dilogarithm  can be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' But it is impossible since both ω1 and ω2 are assumed to be real, thus  0 < 1 − (ω2  1/ω2  2) < 1 is always satisfied, therefore there is no imaginary part can be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  8  In the case b2 − 4a > 0, this implies that λ(m2  1, ω2  1, ω2  2) > 0, exploiting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (35) we get  I = 1  m2  1  �1  2 Li2(1 − ω2  1  ω2  2  ) − Li2[ 1  x+  − iε sgn(x+)] − Li2[ 1  x−  + iε sgn(x−)]  �  ,  (41)  where  x+ = (m2  1 − ω2  1 + ω2  2) + λ1/2(m2  1, ω2  1, ω2  2)  2m2  1  ,  x− = (m2  1 − ω2  1 + ω2  2) − λ1/2(m2  1, ω2  1, ω2  2)  2m2  1  ,  (42)  Since m2  1 > ω2  1 − ω2  2 as presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (39), both x+ and x− are positive definite, thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (41)  simplified to  I = 1  m2  1  �1  2 Li2(1 − ω2  1  ω2  2  ) − Li2( 1  x+  − iε) − Li2( 1  x−  + iε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (43)  In order to explore phenomenological implications of the results presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (38) and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (43), it is instructive to consider the special case that the coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (11) are con-  strained by the following conditions  a = b > 0,  (44)  This is the integral indispensable in the evaluation of H → gg decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' studied in past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Supposing  the mass of each propagator of the triangle is ω1, setting  a = b = m2  1  ω2  1  ,  (45)  Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (10) reduces to  I = 1  m2  1  � 1  0  dxln(ax2 − ax + 1 − iε)  x  ,  a = m2  1  ω2  1  (46)  First consider the case 0 < a < 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=', 0 < m1 < 2ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' To get the correct results we must trace  back to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (12) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (13), other than taking advantage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (32) and simply setting a = b,  otherwise it will make some mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' By exploiting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (17), we immediately obtain  I = − a  2m2  1  � 1  0  dz  � 1  0  dx  1  1 + zax(x − 1)  = − a  2m2  1  � 1  0  dz  4  √  4az − a2z2 arctan  az  √  4az − a2z2  = − 2  m2  1  �  arctan  a  √  4a − a2  �2  ,  (47)  9  By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (19), we get the well-known function appeared in one loop evaluation of H → gg  I = − 2  m2  1  �  arcsin m1  2ω1  �2  ,  0 < m1 < 2ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (48)  Next, we consider the case of a > 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=', m1 > 2ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (35) we get  I = 1  m2  1  �  − Li2( 1  x−  + iε) − Li2( 1  x+  − iε)  �  ,  (49)  x+ and x− are given by  x+ = 1  2  �  1 +  �  1 − 4  a  �  ,  x− = 1  2  �  1 −  �  1 − 4  a  �  ,  (50)  Since both x+ and x− are less than 1, in order to simplify the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (49), defining  α = 1  x−  > 1,  1  x+  =  1  1 − x−  =  α  α − 1 > 1,  (51)  Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (49) can be written as  I = 1  m2  1  �  − Li2(α + iε) − Li2(  α  α − 1 − iε)  �  ,  (52)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (A2) we obtain  I =  1  m2  1  �  −  �  Re Li2(α) + iπ ln α  �  −  �  Re Li2(  α  α − 1) − iπ ln  α  α − 1  �  =  1  m2  1  �  − Re  �  Li2(α) + Li2(  α  α − 1)  �  − iπ ln(α − 1)  �  ,  (53)  In considering the property od dilogarithm  Re  �  Li2(x) + Li2(  x  x − 1)  �  = π2  2 − ln2(x − 1)  2  ,  x > 1  (54)  which leads to  I = 1  m2  1  �  − π2  2 + ln2(α − 1)  2  − iπ ln(α − 1)  �  ,  (55)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (51) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (55) and noticing that  α − 1 = x+  x−  ,  (56)  we get  I = 1  m2  1  �  − π2  2 + 1  2 ln2 x+  x−  − iπ ln x+  x−  �  ,  (57)  Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (50) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' (57) we arrive at the well-known function in the one loop evaluation  of H → gg decay  I = − 1  2m2  1  �  π + i ln  1 +  �  1 − 4ω2  1  m2  1  1 −  �  1 − 4ω2  1  m2  1  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  m1 > 2ω1  (58)  10  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  SUMMARY  In this paper, the amplitude of scalar one loop three-point diagram in which the masses of  the three internal propagators are assigned two different masses is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' A general type  integral is extracted and analytic results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' Conditions of the validity of the results  are discussed in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' As a check to the results, we consider the case that each propagator of  the triangle taking the same mass, we find that the general results will reduce to the functions  obtained in the lowest order evaluation of H → gg decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' We also notice that the results are  mathematically preferred in that the diagram does not corresponds to real process in H → gg  in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' However, we still hope that the results and techniques may be found its applications  in triangle mediated decays addition to H → gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  Appendix A: The dilogarithm function and integral representation of Gauss hyperge-  ometric function  In this section we list some necessary formula in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' The dilogarithm is defined  as[19]  Li2(x) =  +∞  �  k=1  xk  k2 = −  � 1  0  ln(1 − xt)  t  dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  |x| < 1  (A1)  There is a branch cut from 1 to ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' for ε → 0  Li2(x + iε) = Re Li2(x) + iπ sgn(ε)Θ(x − 1) ln x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (A2)  where Θ is the step function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' the sgn(x) is  sgn(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  x > 0  −1  x < 0  Another two formulas we need are the following equation of dilogarithm[20]  � 1  0  ln x  a + bx dx = 1  bLi2  �  − b  a  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (A3)  and the integral representation of the Gauss hypergeometric function[23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' 24]  2F1(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content=' z) =  Γ(γ)  Γ(β)Γ(γ − β)  � 1  0  tβ−1(1 − t)γ−β−1(1 − zt)−αdt,  (A4)  11  where  Re(γ) > Re(β) > 0,  |arg(1 − z)| < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
+page_content='  (A5)  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}
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+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'}

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+First Realization of Quantum Energy Teleportation on Quantum Hardware
+Kazuki Ikeda1, ∗
+1Co-design Center for Quantum Advantage & Center for Nuclear Theory,
+Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA
+Teleporting physical quantities to remote locations is a remaining key challenge for quantum
+information science and technology. Quantum teleportation has enabled the transfer of quantum
+information, but teleportation of quantum physical quantities has not yet been realized.
+Here
+we report the first realization and observation of quantum energy teleportation on real quantum
+hardware.
+We achieve this by using several IBM’s superconducting quantum computers.
+The
+results are consistent with the exact solution of the theory and are improved by the mitigation
+of measurement error. Quantum energy teleportation requires only local operations and classical
+communication. Therefore our results provide a realistic benchmark that is fully achievable with
+current quantum computing and communication technologies.
+I.
+QUANTUM ENERGY TELEPORTATION
+While it is fairly widely known that information about
+quantum states can be transported to remote loca-
+tions [1–4], it is less well known that quantum state
+energy can be similarly transmitted, despite its impact
+and potential for future applications. Quantum informa-
+tion transferred by quantum teleportation is not a phys-
+ical quantity, but energy is a distinct physical quantity.
+Transferring physical quantities to remote locations is an
+unexplored area of technology. Quantum Energy Tele-
+portation (QET) was proposed by Hotta about 15 years
+ago and has been studied theoretically for spin chains [5–
+7], an ion trap system [8], a quantum Hall system [9], and
+other various theoretical systems [10, 11]. It is surprising
+that (to the best of knowledge of the author) QET has
+never been confirmed by any experiment on any system
+before, even though it can be achieved with a very simple
+quantum system. The purpose of this paper is to make
+the first experimental realization of QET in actual quan-
+tum hardware and to establish the quantum circuits that
+make it possible. We achieved the realization of QET us-
+ing some IBM quantum computers by applying quantum
+error mitigation [12–14]. The methods we have estab-
+lished can be applied to any system capable of QET.
+In what follows, we explain that QET is a universal
+means of quantum energy transfer, just as quantum tele-
+portation is a universal means of quantum information
+transfer. Any non-trivial local operations, including mea-
+surements on the ground state of a quantum many-body
+system give rise to excited states, which in turn increase
+the energy expectation value. Note that the increase in
+energy is supplied by the experimental devices. An im-
+portant property of the ground state of a quantum many-
+body system is that it has entanglement, which brings
+about quantum fluctuations in the global ground state
+energy. In other words, quantum fluctuations in the en-
+ergy of the local systems are entangled. Local measure-
+∗kazuki7131@gmail.com
+ment of the quantum state at a subsystem A partially de-
+stroys this ground state entanglement. At the same time,
+energy EA from the device making the measurement is
+injected into the entire system. The injected energy EA
+stays around the subsystem A in the very early stages
+of time evolution, but operations around A alone cannot
+extract EA from the system. This is because informa-
+tion about EA is also stored in remote locations other
+than A due to the entanglement that exists prior to the
+measurement. In other words, the locally injected energy
+EA can be partially extracted at any location other than
+A [15]. QET is the protocol that makes this possible. Up
+to this point, no special assumptions about the system
+have been used.
+The crucial property of QET is that
+it can be realized entirely by the general nature of the
+ground state of the quantum many-body system and the
+universal fact of measurement.
+We work on the minimal QET model given in [16].
+One of the purposes of this paper is to give a quantum
+circuit that utilizes QETs with real quantum computers
+and quantum networks. The complete form of quantum
+circuits we used for QET is displayed in Fig. 1. The max-
+imum circuit depth is 10 and the number of qubits used
+is 2.
+Hence, current quantum computers are powerful
+enough to implement QET.
+Let k, h be positive real numbers. The Hamiltonian of
+the minimal model is
+Htot = H0 + H1 + V,
+(1)
+Hn = hZn +
+h2
+√
+h2 + k2 , (n = 0, 1)
+(2)
+V = 2kX0X1 +
+2k2
+√
+h2 + k2 .
+(3)
+The ground state of Htot is
+|g⟩ =
+1
+√
+2
+�
+1 −
+h
+√
+h2 + k2 |00⟩− 1
+√
+2
+�
+1 +
+h
+√
+h2 + k2 |11⟩ ,
+(4)
+The constant terms in the Hamiltonians are added so
+that the ground state |g⟩ of Htot returns the zero mean
+arXiv:2301.02666v1  [quant-ph]  7 Jan 2023
+
+2
+FIG. 1: Quantum gate operations used for quantum energy teleportation. (A) preparation of ground state and Alice’s X0
+measurement to deposit her energy. She tells Bob via classical communication whether µ = −1 or µ = +1 was observed. (B)
+Bob’s conditional operations to receive energy. He selects an operation U1(+1) or U1(−1) based on µ = +1 or −1, corresponding
+to the Maxwell demon operation. (C) Equivalent implementation of Bob’s operations on a quantum computer.
+energy for all local and global Hamiltonians:
+⟨g| Htot |g⟩ = ⟨g| H0 |g⟩ = ⟨g| H1 |g⟩ = ⟨g| V |g⟩ = 0. (5)
+However it should be noted that |g⟩ is neither a ground
+state nor an eigenstate of Hn, V, Hn + V (n = 0, 1). The
+essence of QET is to extract negative ground state energy
+of those local and semi-local Hamiltonians.
+The QET protocol is as follows. First, Alice makes a
+measurement on her Pauli operator X0 by P0(µ) = 1
+2(1+
+(−1)µX0) and then she obtains either µ = −1 or +1. At
+this point, Alice’s expectation energy is E0 =
+h2
+√
+h2+k2 .
+Via a classical channel, Alice then sends her measure-
+ment result µ to Bob, who applies an operation U1(µ) to
+his qubit and measures H1 and V . The density matrix
+ρQET after Bob operates U1(µ) to P0(µ) |g⟩ is
+ρQET =
+�
+µ∈{−1,1}
+U1(µ)P0(µ) |g⟩ ⟨g| P0(µ)U †
+1(µ).
+(6)
+Using ρQET, the expected local energy at Bob’s subsys-
+tem is evaluated as ⟨E1⟩ = Tr[ρQET(H1 + V )], which
+is negative in general.
+Due to the conservation of en-
+ergy, EB = −⟨E1⟩(> 0) is extracted from the system by
+the device that operates U1(µ) [17]. In this way, Alice
+and Bob can transfer the energy of the quantum system
+only by operations on their own local system and classical
+communication (LOCC).
+II.
+QUANTUM CIRCUIT IMPLEMENTATION
+OF QUANTUM ENERGY TELEPORTATION
+A.
+Preparation of Ground State
+Here we explain how to construct a quantum circuit
+(Fig. 1 (A)) that generates the exact ground state |g⟩.
+Let us begin with a Bell state |Φ−⟩ = |00⟩−|11⟩
+√
+2
+since |g⟩
+is resemble to it. |Φ−⟩ can be prepared by
+|00⟩ − |11⟩
+√
+2
+= (Z ⊗ I)CNOT(H ⊗ I |00⟩
+(7)
+where CNOT= |0⟩ ⟨0|⊗I +|1⟩ ⟨1|⊗X. Using Y = SXS†,
+we can perform a gate operation that maps |Φ−⟩ to the
+ground state |g⟩ (eq. (4)) by a combination of one- and
+two-qubit operators
+|g⟩ = exp(−iαX ⊗ Y ) |Φ−⟩
+=
+1
+√
+2(cos α + sin α) |00⟩ − 1
+√
+2(cos α − sin α) |11⟩ .
+(8)
+where
+α
+is
+designed
+to
+satisfy
+cos α + sin α
+=
+�
+1 −
+h
+√
+h2+k2 and cos α − sin α =
+�
+1 +
+h
+√
+h2+k2 .
+Those quantum operations are implemented by the
+quantum circuit in Fig. 1 (A).
+B.
+Step 1: Deposit Energy
+We use the following projective measurement operator
+P0(µ) = 1
+2(1 + (−1)µX0).
+(9)
+
+H
+H
+st
+Rx(2α)
+s
+H
+Ui(+1)
+Ui(-1)
+Ui(+1)
+Ui(-1)
+H
+Ui(+1)
+Ui(-1)
+Ui(+1)
+Ui(-1)3
+We measure Alice’s X operator, by which we obtain a
+state |+⟩ or |−⟩. This operation does not affect Bob’s
+energy since [X0, V ] = [X0, H1] = 0. Using [P0(µ), V ] =
+0 and ⟨+| Z |+⟩ = ⟨−| Z |−⟩ = 0, we find that Alice’s
+mean energy to deposit is
+⟨E0⟩ =
+�
+µ∈{−1,1}
+⟨g| P0(µ)HtotP0(µ) |g⟩ =
+h2
+√
+h2 + k2 .
+(10)
+Alice’s operation can be implemented on a quantum
+circuit in Fig 1 (A). ⟨E0⟩ can be calculated with the out-
+put bit-strings 00, 01, 10, 11. Analytical values ⟨E0⟩ and
+results with quantum computers for different pairs of k
+and h are summarized in Table I.
+C.
+Step 2: Receive Energy
+As soon as Alice observes µ ∈ {0, 1}, she tells her result
+to Bob who operates UB(µ) to his qubit and measures his
+energy. Here UB(µ) is
+U1(µ) = cos θI − iµ sin θY1 = RY (2µθ),
+(11)
+where θ obeys
+cos(2θ) =
+h2 + k2
+�
+(h2 + 2k2)2 + h2k2
+(12)
+sin(2θ) =
+hk
+�
+(h2 + 2k2)2 + h2k2 .
+(13)
+The average quantum state ρQET eq.(6) is obtained af-
+ter Bob operates U1(µ) to P0(µ) |g⟩. Then the average
+energy Bob measures is
+⟨E1⟩ = Tr[ρQET(H1 + V )] = Tr[ρQETHtot] − ⟨E0⟩, (14)
+where we used [U1(µ), H1] = 0. It is important that the
+map �
+µ∈{−1,1} P0(µ) |g⟩ ⟨g| P0(µ) → ρQET is not a uni-
+tary transformation. Therefore eq. (14) can be negative.
+This is in contrast to eq. (A7).
+Now let us explain quantum circuits for the QET pro-
+tocol. Since V and H1 do not commute, measurement on
+those terms should be done separately. In other words,
+Bob measures V and H1 independently and obtains ⟨V ⟩
+and ⟨H1⟩ statistically. As the figures show, ⟨V ⟩ is always
+negative and ⟨H1⟩ is always positive. Therefore is suffi-
+cient for Bob to measure only ⟨V ⟩ to receive energy by
+QET.
+We consider V (µ) = ⟨g| P0(µ)U †
+1(µ)V U1(µ)P0(µ) |g⟩.
+The quantum circuit to measure V (µ) is shown in the
+right panel of Fig. 1 (B). It is important to note that,
+since Bob knows µ which contains Alice’s information, he
+can obtain VQET(µ) by local measurement only, although
+V is not a local operator. Similarly we can measure H1
+in Z-basis as in the left panel of Fig. 1 (B). The corre-
+sponding quantum circuit is obtained by removing the
+second Hadamard gate from the previous circuit Fig. 1
+(C). On average the circuit generates
+⟨E1⟩ =
+�
+µ∈{−1,1}
+⟨g| P0(µ)U †
+1(µ)(H1 + V )U1(µ)P0(µ) |g⟩
+= −
+1
+√
+h2 + k2 [hk sin(2θ) − (h2 + 2k2)(1 − cos(2θ))].
+(15)
+If θ is small, ⟨E1⟩ is negative.
+Bob receives energy
+⟨EB⟩ = −⟨E1⟩ on average.
+In Appendix B, we per-
+formed measurement of V (µ) and H1 based on the quan-
+tum circuit Fig. 1 (B) and summarized data in Table II,
+where numerical values are compared with analytical val-
+ues given in eq. (15).
+D.
+QET on Real Quantum Hardware
+Here we describe how to implement the conditional
+operations that may not be natively supported by many
+quantum computers and quantum devices. In the QET
+protocol, Bob’s operation must be selected according to
+the results of Alice’s measurements, as shown in Fig. 1
+(B). Even in environments where conditional statements
+are not supported, QET can be implemented without
+problems through the technique of deferred measure-
+ment.
+We can postpone Alice’s measurement until the end
+of the circuit, and obtain the same results. The condi-
+tional operations can be created by a controlled U gate
+Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U and an anti-controlled U
+gate (X ⊗ I)Λ(U)(X ⊗ I). One would find the equiva-
+lence between the following two circuits. We use the right
+circuit enclosed by the orange dashed frame in Fig. 1 (C).
+We performed quantum computation using 6 dif-
+ferent types of IBM quantum hardware ibmq lima,
+ibmq jakarta, ibmq hanoi, ibm cairo, ibm auckland
+and ibmq montreal.
+The properties of each quantum
+computers can be seen from Fig. 2.
+ibmq lima con-
+sists of 5 qubits (Fig. 2 [Left]) and ibmq jakarta has
+7 qubits (Fig. 2 [Middle]).
+ibm cairo is a 27-qubit
+hardware, and ibmq hanoi, ibm cairo, ibm auckland
+and ibmq montreal have the same graph structure as
+ibm cairo (Fig. 2 [Right]). A direct CNOT gate can be
+applied to two qubits connected at the edge.
+We can
+choose two qubits placed on the graph of the hardware
+to perform a quantum computation. We conducted the
+experiment by choosing two qubits connected at the edge
+with relatively small errors.
+We also performed a simulation using a simulator
+qasm simulator, which can classically emulate gate op-
+erations on the same quantum circuits we used for
+quantum computation.
+We summarize results with
+ibmq lima, ibmq jakarta and ibm cairo in Table I. The
+results using the simulator agreed with the analytical so-
+lution with high accuracy, confirming that the quantum
+circuit was implemented correctly.
+More experimental
+
+4
+FIG. 2: (A) properties of quantum computers we used. Each graph of qubits corresponds to the layout of the hardware. A
+direct CNOT gate can be applied to two qubits connected at the edge. (B) Distribution of states compared with a simulator
+qasm simulator and a quantum computer ibm cairo (raw results and mitigated results)
+results are summarized in Table IV in Appendix D. We
+describe details of machine properties and experimental
+conditions in Table III in Appendix C.
+The most significant achievement in this study is the
+observation of negative energy ⟨E1⟩ < 0. The value of
+⟨V ⟩ that was closest to the exact analysis value was -
+0.1079 (h = 1.5, k = 1 with ibmq jakarta), which is
+about 76% accurate.
+As emphasised in Hotta’s origi-
+nal works [5–11, 16], after Alice observes her X0, no
+unitary operation can make ⟨E1⟩ negative (eq. (A7)).
+In order for Bob to obtain the correct ⟨E1⟩, Alice and
+Bob must repeat the experiment an enormous number of
+times, and the correct value of ⟨V ⟩ and ⟨H1⟩ can be ob-
+tained only when Alice and Bob communicate correctly
+in the quantum circuit in Fig. 1 (C). Distributions of
+states obtained by a quantum computer ibm cairo are
+shown in Fig. 2 (B), where distributions of raw results
+and error mitigated results are compared with a simu-
+lator qasm simulator. We used a simple measurement
+error mitigation to determine the effects of measurement
+errors.
+We prepared a list of 4 measurement calibra-
+tion circuits for the full Hilbert space. Then we immedi-
+ately measured them to obtain the probability distribu-
+tions. Then we applied the calibration matrix to correct
+the measured results. The average measurement fidelity
+when using each quantum computer is summarized in
+Table III in Appendix C. The histograms of the observed
+states showed similar tendencies for all other quantum
+computers we used. It can be seen that the histograms
+obtained by the measurement of H agree with the sim-
+ulator results with good accuracy. The improvement of
+the values due to measurement error mitigation is also
+confirmed by the results in Table I. The observation of V
+is of utmost importance in this study. Although the raw
+data from quantum computers deviated from the simula-
+tor results, in some cases error mitigation improved them
+enough to observe negative energy expectation values.
+It should also be emphasized that we observed negative
+⟨V ⟩ for all parameter (k, h) combinations in all quantum
+computers used. As emphasized in Sec. II C, the amount
+of energy available to Bob is greater if only V is observed,
+since ⟨H1⟩ is always positive (Fig. 3).
+This would be
+enough for practical purposes.. Note that the energy that
+Bob gains becomes smaller when he observes H1.
+
+iloa mal lima Error Map
+lomi cairo Emor Map)
+Keidaut: Hror tw
+Heridoun: Hror ta?
+Featour: :Error tw?
+1
+2
+12
+2.41
+H arrar te ( [avg. : : DuE4s)
+Cl aror hie h avg.
+Distriloution of states when measuring V (ilom cairo, k = 1, h = 1)
+Distriloution of states when measuring H (ilom cairo, k = 1, h = 1)
+raw
+raw
+mitigated
+0.4
+0.474
+mitigated
+0.4702473
+0.350.349
+0.362
+0.330367362
+0.457
+simullator
+simullator
+0.45
+0.3
+Probabilities
+0.30
+Probal
+0.2
+0.139130139
+0.15
+0.1
+0.059
+04.8
+0.049
+.040
+0.027
+0.026
+0.00
+0.0
+15
+Backend
+(h, k) = (1, 0.2)
+(h, k) = (1, 0.5)
+(h, k) = (1, 1)
+(h, k) = (1.5, 1)
+Analytical value
+⟨E0⟩
+0.9806
+0.8944
+0.7071
+1.2481
+qasm simulator
+0.9827 ± 0.0031
+0.8941 ± 0.0001
+0.7088 ± 0.0001
+1.2437 ± 0.0047
+ibmq lima
+error mitigated
+0.9423 ± 0.0032
+0.8169 ± 0.0032
+0.6560 ± 0.0031
+1.2480 ± 0.0047
+unmitigated
+0.9049 ± 0.0017
+0.8550 ± 0.0032
+0.6874 ± 0.0031
+1.4066 ± 0.0047
+ibmq jakarta
+error mitigated
+0.9299 ± 0.0056
+0.8888 ± 0.0056
+0.7039 ± 0.0056
+1.2318 ± 0.0084
+unmitigated
+0.9542 ± 0.0056
+0.9089 ± 0.0056
+0.7232 ± 0.0056
+1.2624 ± 0.0083
+ibm cairo
+error mitigated
+0.9571 ± 0.0032
+0.8626 ± 0.0031
+0.7277 ± 0.0031
+1.2072 ± 0.0047
+unmitigated
+0.9578 ± 0.0031
+0.8735 ± 0.0031
+0.7362 ± 0.0031
+1.2236 ± 0.0047
+Analytical value
+⟨H1⟩
+0.0521
+0.1873
+0.2598
+0.3480
+qasm simulator
+0.0547 ± 0.0012
+0.1857 ± 0.0022
+0.2550 ± 0.0028
+0.3487 ± 0.0038
+ibmq lima
+error mitigated
+0.0733 ± 0.0032
+0.1934 ± 0.0032
+0.2526 ± 0.0032
+0.3590 ± 0.0047
+unmitigated
+0.1295 ± 0.0053
+0.2422 ± 0.0024
+0.2949 ± 0.0028
+0.4302 ± 0.0039
+ibmq jakarta
+error mitigated
+0.0736 ± 0.0055
+0.2018 ± 0.0056
+0.2491 ± 0.0056
+0.3390 ± 0.0084
+unmitigated
+0.0852 ± 0.0022
+0.2975 ± 0.0045
+0.3365 ± 0.0052
+0.4871 ± 0.0073
+ibm cairo
+error mitigated
+0.0674 ± 0.0032
+0.1653 ± 0.0031
+0.2579 ± 0.0031
+0.3559 ± 0.0047
+unmitigated
+0.0905 ± 0.0014
+0.1825 ± 0.0022
+0.2630 ± 0.0027
+0.3737 ± 0.0037
+Analytical value
+⟨V ⟩
+-0.0701
+-0.2598
+-0.3746
+-0.4905
+qasm simulator
+−0.0708 ± 0.0012 −0.2608 ± 0.0032 −0.3729 ± 0.0063 −0.4921 ± 0.0038
+ibmq lima
+error mitigated −0.0655 ± 0.0012 −0.2041 ± 0.0031 −0.2744 ± 0.0063 −0.4091 ± 0.0063
+unmitigated
+−0.0538 ± 0.0011 −0.1471 ± 0.0025 −0.1233 ± 0.0041 −0.2737 ± 0.0046
+ibmq jakarta
+error mitigated −0.0515 ± 0.0022 −0.2348 ± 0.0056 −0.3255 ± 0.0112 −0.4469 ± 0.0112
+unmitigated
+−0.0338 ± 0.0021 −0.1371 ± 0.0046 −0.0750 ± 0.0075 −0.2229 ± 0.0083
+ibm cairo
+error mitigated −0.0497 ± 0.0013 −0.1968 ± 0.0031 −0.2569 ± 0.0063 −0.3804 ± 0.0063
+unmitigated
+−0.0471 ± 0.0012 −0.1682 ± 0.0026 −0.1733 ± 0.0038 −0.3089 ± 0.0045
+Analytical value
+⟨E1⟩
+-0.0180
+-0.0726
+-0.1147
+-0.1425
+qasm simulator
+−0.0161 ± 0.0017 −0.0751 ± 0.00398 −0.1179 ± 0.0069 −0.1433 ± 0.0054
+ibmq lima
+error mitigated
+0.0078 ± 0.0034
+−0.0107 ± 0.0045 −0.0217 ± 0.0071 −0.0501 ± 0.0079
+unmitigated
+0.0757 ± 0.0054
+0.0950 ± 0.0035
+0.1715 ± 0.0050
+0.1565 ± 0.0060
+ibmq jakarta
+error mitigated
+0.0221 ± 0.0059
+−0.0330 ± 0.0079 −0.0764 ± 0.0125 −0.1079 ± 0.0140
+unmitigated
+0.0514 ± 0.0030
+0.1604 ± 0.0064
+0.2615 ± 0.0091
+0.2642 ± 0.00111
+ibm cairo
+error mitigated
+0.0177 ± 0.0035
+−0.0315 ± 0.0044
+0.0010 ± 0.0070
+−0.0245 ± 0.0079
+unmitigated
+0.0433 ± 0.0018
+0.0143 ± 0.0034
+0.0897 ± 0.0047
+0.0648 ± 0.0058
+TABLE I: Comparison between analytical values of ⟨E0⟩, ⟨H1⟩, ⟨V ⟩, ⟨E1⟩ and results from IBM’s real quantum computers,
+ibmq lima, ibmq jakarta and ibm cairo. We evaluate ⟨E1⟩ = ⟨H1⟩ + ⟨V ⟩. ”error mitigated” means results using measurement
+error mitigation and ”unmitigated” corresponds to results without measurement error mitigation.
+III.
+IMPLICATIONS FOR OUR REAL WORLD
+Our results provide implications for new quantum com-
+munication technologies with respect to different phases
+in the short, medium and long term. It is important to
+note that, like quantum teleportation, energy can also
+be teleported only by LOCC. Reproducing the minimal
+QET model we used in our demonstration in a labora-
+tory system is something that can be tackled in the short
+term with current quantum computing and communica-
+tion technology. A quantum device with 2 qubits and a
+gate depth of 10 would be ready for immediate experi-
+mentation. This is expected to lead to new developments
+in the use of quantum memory [18–20]. Furthermore, ver-
+ifying QET in a variety of quantum systems and materi-
+als beyond the minimal model is an important challenge
+for future applications.
+Quantum energy teleportation without limit of dis-
+tance is also provided [21]. The ability to transfer quan-
+tum energy over long distances will bring about a new
+revolution in quantum communication technology.
+In
+other words, a world in which physical quantities are
+freely and instantaneously transmitted to remote loca-
+tions connected by a large-scale Quantum Internet (Net-
+work) can be realized in the near future. For example
+there is a long-distance (∼158km) SBU/BNL quantum
+network in Long Island, New York [22]. Various quan-
+tum networks have been developed [23–25].
+Realizing
+QET on a quantum network, which is expected to be
+
+6
+in practical use around the 2030s, would be a milestone
+toward realizing QET on a worldwide quantum network.
+The realization of a long-range QET will have impor-
+tant implications beyond the development of information
+and communication technology and quantum physics. In-
+formation and energy are physical, but also economic.
+Allowing physical quantities to be traded concretely on
+the quantum network means that a new economic mar-
+ket will be born [26]. Quantum teleportation is an es-
+tablished technology and is being developed for practical
+use. In addition to this, if QET is put to practical use, it
+will mean that various quantum resources will be at the
+disposal of us. The expected value of the Hermite op-
+erator is called energy, but it need not literally be used
+only as energy.
+Teleported energy can be used as en-
+ergy, as well as for other uses. The ability to teleport a
+concrete physical quantity, energy, means that quantum
+information will have added value. In a quantum market
+where Alice, Bob, and Charlie exist, if Bob can get more
+energy from Charlie than from Alice, Bob may prefer to
+do business with Charlie rather than Alice, and he may
+prefer an entangle state with Charlie. However, depend-
+ing on transaction costs, Bob may choose Alice. A lot
+of such game-theoretic situations can be created [27–31].
+This implies that quantum information economics (which
+does not yet exist) will become a meaningful idea in the
+future.
+Acknowledgement
+I thank David Frenklakh,
+Adrien Florio,
+Sebas-
+tian Grieninger, Fangcheng He, Dmitri Kharzeev, Yuta
+Kikuchi, Vladimir Korepin, Qiang Li, Adam Lowe,
+Shuzhe Shi, Hiroki Sukeno, Tzu-Chieh Wei, Kwangmin
+Yu and Ismail Zahed for fruitful communication and col-
+laboration. I thank Megumi Ikeda for providing the car-
+toons. I acknowledge the use of IBM quantum comput-
+ers. I was supported by the U.S. Department of Energy,
+Office of Science, National Quantum Information Science
+Research Centers, Co-design Center for Quantum Advan-
+tage (C2QA) under Contract No.DESC0012704.
+Author contribution
+All work was performed by the author.
+competing interests
+The author declares that there is no competing finan-
+cial interests.
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+
+8
+Appendix A: Description of the Model
+1.
+Quantum Gates and Measurement
+Here we give a self-contained description of the back-
+ground knowledge of the text. We use the following one-
+qubit operators whose matrix representations are given
+as
+X =
+�
+0 1
+1 0
+�
+, Y =
+�
+0 −i
+i
+0
+�
+, Z =
+�
+1
+0
+0 −1
+�
+,
+H =
+1
+√
+2
+�
+1
+1
+1 −1
+�
+, S =
+�
+1 0
+0 i
+�
+.
+(A1)
+We use |0⟩ =
+�1
+0
+�
+, |1⟩ =
+�0
+1
+�
+for the computational
+basis states, which are eigenstates of Z:
+Z |0⟩
+=
+|0⟩ , Z |1⟩ = − |1⟩.
+We also work with another ba-
+sis vectors |±⟩ =
+|0⟩±|1⟩
+√
+2
+.
+They are eignestates of X:
+X |−⟩ = − |−⟩ , X |+⟩ = − |+⟩. Note that |±⟩ are created
+by applying H to |0⟩ and |1⟩; H |0⟩ = |+⟩ , H |1⟩ = |−⟩.
+Those are used for measuring Hn, V (n = 1, 2) in the
+QET protocol. For example, Alice finds µ = ±1 by ob-
+serving the eigenvalues ±1 of her local Pauli X operator.
+The rotation of X, Y, Z is defined by
+RX(α) = e−i α
+2 X, RY (α) = e−i α
+2 Y , RZ(α) = e−i α
+2 Z.
+(A2)
+Note that X and Y gates are related in a way that Y =
+SXS†.
+Using those representations, it will be easy to
+check the matrix representation
+exp(−iαX ⊗ Y ) = (I ⊗ S) exp(−iαX ⊗ X)(I ⊗ S†)
+=
+�
+�
+�
+�
+�
+cos α
+0
+0
+− sin α
+0
+cos α
+sin α
+0
+0
+− sin α cos α
+0
+sin α
+0
+0
+cos α
+�
+�
+�
+�
+�
+(A3)
+and the exact form of the ground state eq. (8):
+|g⟩ = exp(−iαX ⊗ Y ) |Φ−⟩
+=
+1
+√
+2(cos α + sin α) |00⟩ − 1
+√
+2(cos α − sin α) |11⟩ .
+We use two-qubit gate operations. In general, a control
+U operation Λ(U) is defined by
+Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U
+(A4)
+and the corresponding diagram is drwan as
+control U=
+U
+One of the most frequently used controlled gates is a
+CNOT gate CNOT = Λ(X), whose diagram is especially
+drawn as
+CNOT=
+It is convenient to define an anti-control gate, which is
+activated when the control bit is in state |0⟩: |1⟩ ⟨1|⊗I +
+|0⟩ ⟨0| ⊗ U, whose diagram is drawn as
+Anti-control U=
+U
+=
+X
+X
+U
+Now we describe measurement of quantum operators.
+We measure Z1 and X0X1. Measurement of Z1 is done
+by the following circuit
+The output of the measurement is a bit string b0b1 ∈
+{00, 01, 10, 11}. Since the eigenvalues of Z are −1, 1, we
+convert the bit string into 1 − 2b1.
+Let nshot be the
+number of repetitions of the circuit, and countsb0b1 be
+the number of times b0 and b1 are detected. Therefore
+countsb0b1
+nshots
+is the probability that a bit string b0b1 is ob-
+tained. Then the expectation value of Z1 is computed by
+the formula
+⟨Z1⟩ =
+�
+b0,b1
+(1 − 2b1)countsb0b1
+nshots
+.
+(A5)
+Measurement of X0X1 is done by the following circuit
+H
+H
+As we described previously,
+H
+maps |0⟩ , |1⟩ to
+|+⟩ , |−⟩, which are eigenvectors of X. The output of the
+measurement is again a bit string b0b1 ∈ {00, 01, 10, 11}.
+They are converted to the eigenvalues of X0X1 by (1 −
+2b0)(1 − 2b1).
+Then the expectation value of X0X1 is
+computed by the formula
+⟨X0X1⟩ =
+�
+b0,b1
+(1 − 2b0)(1 − 2b1)countsb0b1
+nshots
+.
+(A6)
+
+9
+FIG. 3: Heat maps visualizing expectation values ⟨V ⟩ = Tr[ρQETV ] and ⟨H1⟩ = Tr[ρQETH1] by (k, h).
+2.
+Some details of the model
+Here we describe details of the model we used. For
+more information please refer to Hotta’s original papers.
+First it is important to note that the ground state of
+the total Hamiltonian H is not the ground state of lo-
+cal operators.
+For example, V has three degenerated
+ground states |−+⟩ , |+−⟩ , |−+⟩+|+−⟩
+√
+2
+, and the ground
+state energy of V is −2k +
+2k2
+√
+h2+k2 .
+It is important
+that V ’s ground state energy is negative for all k > 0.
+This is also true for Hn, whose ground state energy is
+−h +
+h2
+√
+h2+k2 . The expected values of ⟨V ⟩ = Tr[ρQETV ]
+and ⟨H1⟩ = Tr[ρQETH1] obtained by QET are shown in
+Fig. 3.
+To understand the non-triviality of the QET protocol,
+it is important to note that after Alice’s measurement,
+no matter what unitary operation W1 is performed on
+Bob’s qubit, no energy can be extracted. This can be
+confirmed by
+Tr[ρW Htot] − ⟨E0⟩ = ⟨g| W †
+1 HtotW1 |g⟩ ≥ 0,
+(A7)
+where
+ρW = W †
+1
+�
+�
+�
+µ∈{−1,1}
+P0(µ) |g⟩ ⟨g| P0(µ)
+�
+� W1.
+(A8)
+The inequality in eq. (A7) is guaranteed by eq. (5).
+If Bob does not perform any operations on his own
+system after Alice’s measurement, the time evolution of
+Bob’s local system is as follows
+⟨H1(t)⟩ = Tr[ρMeitHH1e−itH] = h2(1 − cos(4kt))
+2
+√
+h2 + k2
+⟨V (t)⟩ = Tr[ρMeitHV e−itH] = 0,
+(A9)
+where ρM = �
+µ∈{±1} P0(µ) |g⟩ ⟨g| P0(µ).
+As a result of the natural time evolution of the sys-
+tem, energy is indeed transferred to Bob’s local system,
+but this is no more than energy propagation in the usual
+sense. In QET, energy is not obtained through the nat-
+ural time evolution of the system, but instantaneously
+as a result of communication. Since we consider a non-
+relativistic quantum many-body system, the speed of en-
+ergy propagation is sufficiently slower than the speed of
+light. Classical communication, realized by optical com-
+munication, can convey information to remote locations
+much faster than the time evolution of physical systems.
+Hence, QET can be described as a fast energy propaga-
+tion protocol.
+It is known that the change in entropy before and after
+the measurement can be evaluated as follows
+∆SAB = SAB −
+�
+µ∈{±1}
+pµSAB(µ)
+(A10)
+≥ 1 + sin2 ξ
+2 cos3 ξ
+ln 1 + cos ξ
+1 − cos ξ
+EB
+√
+h2 + k2
+(A11)
+where pµ is the probability distribution of µ, SAB(µ) is
+the entanglement entropy after the measurement, ξ =
+arctan
+� k
+h
+�
+and EB is the amount of energy that Bob can
+receive (EB = −⟨E1⟩ > 0) [16]. Moreover the maximal
+energy that Bob would receive is bounded below by the
+difference of entropy:
+max
+U1(µ) EB ≥
+2
+√
+h2 + k2(
+�
+4 − 3 cos2 ξ − 2 + cos2 ξ)∆SAB
+(1 + cos ξ) ln
+�
+2
+1+cos ξ
+�
++ (1 − cos ξ) ln
+�
+2
+1−cos ξ
+�.
+(A12)
+Appendix B: Simulation of Hotta’s original QET
+protocol
+Hotta’s original QET protocol, which can be imple-
+mented by Fig. 1 (B) in the main text, does require the
+conditional operations based on a signal µ ∈ {−1, +1}
+
+(V)
+(H1)
+ 0.0 
+2.0
+ 0.5
+18
+-
+0.1
+18
+1.6
+-
+16
+ 0.4
+0.2
+E'O-
+ 0.3
+10
+0.4
+-
+8°0
+0.8
+-
+ 0.2
+0.5
+0.6
+-
+0.6
+0.4
+ 0.1 
+0.7
+-
+ 0.0
+00.0
+0.4
+0.6
+8:0
+10
+12
+141618
+00.0
+20
+0.4
+9'0
+8:0
+10
+12
+14
+16
+18
+2.0
+k
+k10
+(h, k)
+(1,0.1)
+(1,0.2)
+(1,0.5)
+(1,1)
+(1.5,1)
+Analytical ⟨E0⟩
+0.9950
+0.9806
+0.8944
+0.7071
+1.2481
+qasm simulator ⟨E0⟩
+0.9929 ± 0.0010
+0.9807 ± 0.0010
+0.8948 ± 0.0010
+0.7067 ± 0.0010
+1.2492 ± 0.0015
+Analytical ⟨V ⟩
+-0.0193
+-0.0701
+-0.2598
+-0.3746
+-0.4905
+qasm simulator ⟨V ⟩ −0.0194 ± 0.0057 −0.0682 ± 0.0011 −0.2625 ± 0.0061 −0.3729 ± 0.0063 −0.4860 ± 0.0061
+Analytical ⟨H1⟩
+0.0144
+0.0521
+0.1873
+0.2598
+0.3480
+qasm simulator ⟨H1⟩
+0.0136 ± 0.0006
+0.0501 ± 0.0011
+0.1857 ± 0.0022
+0.2550 ± 0.0028
+0.3493 ± 0.0038
+Analytical ⟨E1⟩
+-0.0049
+-0.0180
+-0.0726
+−0.1147
+-0.1425
+qasm simulator ⟨E1⟩ −0.0058 ± 0.0057 −0.0181 ± 0.016 −0.0768 ± 0.0064 −0.1179 ± 0.0068 −0.1367 ± 0.0072
+TABLE II: Comparison between analytical values and numerical values from the quantum circuits with conditional opera-
+tion (Fig. 1 (B)). Each error corresponds to statistical error of 105 shots. We evaluate ⟨E1⟩ = ⟨H1⟩ + ⟨V ⟩.
+that Bob receives from Alice. We performed quantum
+computation on the equivalent circuit (right quantum cir-
+cuit in Fig. 1) (C) that yielded exactly the same results.
+Let Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U be a controlled U
+gate. Note that Λ(U(−1)) and (X ⊗ I)Λ(U(+1))(X ⊗ I)
+commute:
+U(−1)
+U(+1)
+=
+U(+1)
+U(−1)
+Of course, the equivalence of these circuits is theoreti-
+cally trivial, we used qasm simulator and executed our
+simulation based on the left quantum circuit in Fig. 1
+(C), in order to confirm the consistency between them.
+Table II summarizes the numerical results and shows per-
+fect agreement with the analytical results as well as re-
+sults (Table IV) with the right circuit in Fig. 1 (C).
+Appendix C: Properties of Quantum Hardware
+Here we describe more on our experiments with IBM
+quantum computers. Graphs of IBM quantum computers
+we used are displayed in Fig 4. For example, the layout
+of ibmq lima corresponds to (A) in Fig. 4 and we used
+the pair of qubits in [0,1] that had the smallest readout
+assignment error among all pairs (Fig. 2 (A) [Left]). We
+can perform a direct CNOT operation between qubits
+connected at the edge. For ibmq lima, the CNOT error
+between [1,2] qubits were 0.00510 (Table. III).
+Appendix D: Additional results with 6 different
+quantum hardware
+Here we describe additional results obtained by some
+other IBM quantum computers. In the main text we fo-
+cused on best results with ibmq lima and ibmq jakarta,
+but in fact we also experimented with ibmq hanoi,
+ibm cairo, ibm auckland, ibmq montreal.
+Table IV
+summarizes the complete lists of the best data we ob-
+FIG. 4:
+Configurations of qubits on graphs:
+(A) the lay-
+out of ibmq lima which has 5 qubits; (B) the layout of
+ibmq jakarta which has 7 qubits; (C) the layout of 27-qubit
+hardware including ibmq hanoi, ibm cairo, ibm auckland
+and ibmq montreal. A direct CNOT gate can be applied to
+two qubits connected at the edge.
+tained and Table III summarizes the experimental con-
+ditions used for each hardware. In the entire circuit, the
+total number N of qubits is 2 and the circuit depth d(N)
+that can be executed is 9 (excluding measurement of V )
+and 10 (including measurement of V ). The quantum vol-
+ume is defined by QV =
+�
+arg maxn≤N min{n, d(n)}
+�2.
+Therefore quantum computers with QV
+=
+128 are
+enough for this work. Here QV is a metric that quantifies
+the largest random circuit of equal width and depth that
+a quantum computer can successfully implement. How-
+ever, QV may not be a crucial metric in this study, since
+we are only dealing with 2-qubit, relatively simple quan-
+tum circuits. Errors in quantum computers result from a
+combination of various factors, including readout error,
+CNOT error, etc.. Table IV shows that Alice’s measure-
+ments of X0 are relatively accurate in almost all cases.
+With respect to the observation of V , there is a deviation
+
+0
+2
+0
+3
+3
+4
+4
+5
+(6)
+17
+4
+10
+12
+15
+18
+21
+23
+13
+24
+5
+8
+11
+14
+16
+19
+22
+25
+26
+2011
+Backend
+ibmq lima
+ibmq jakarta
+ibm cairo
+ibm hanoi
+ibmq auckland ibmq montreal
+Ntot
+5
+7
+27
+27
+27
+27
+Quantum Volume
+8
+16
+64
+64
+64
+128
+shots
+105
+3.2 × 104
+105
+105
+105
+3.2 × 104
+Measurement fidelity
+0.961075
+0.924695
+0.961935
+0.979530
+0.979383
+0.957484
+qubits used
+[0,1]
+[3,5]
+[13,14]
+[14,16]
+[14,16]
+[14,16]
+CNOT error
+0.00510
+0.00665
+0.00439
+0.01996
+0.00570
+0.00739
+Gate time (ns)
+305.778
+291.556
+220.444
+472.889
+355.556
+355.556
+First qubit
+t1(µs)
+75.67
+93.53
+146.43
+219.15
+60.97
+129.56
+t2(µs)
+141.39
+41.09
+164.29
+25.75
+150.49
+168.53
+Frequency (GHz)
+5.030
+5.178
+5.282
+5.047
+5.167
+4.961
+Anharmonicity (GHz)
+-0.33574
+-0.34112
+-0.33874
+-0.34412
+-0.34196
+-0.32314
+Pauli X error
+2.781 × 10−4 2.140 × 10−4 1.630 × 10−4 2.305 × 10−4 2.4842 × 10−4
+1.942 × 10−4
+Readout assignment error 1.960 × 10−2 2.440 × 10−2 8.500 × 10−3 7.400 × 10−3
+8.100 × 10−3
+1.310 × 10−2
+Second qubit
+t1(µs)
+58.03
+143.52
+94.28
+190.07
+73.16
+83.73
+t2(µs)
+74.97
+59.33
+186.99
+253.46
+183.12
+39.92
+Frequency (GHz)
+5.128
+5.063
+5.044
+4.883
+4.970
+5.086
+Anharmonicity (GHz)
+-0.31835
+-0.34129
+-0.34289
+-0.34591
+-0.34389
+-0.33707
+Pauli X error
+1.469 × 10−4 1.708 × 10−4 1.732 × 10−4 4.708 × 10−4
+2.052 × 10−4
+2.221 × 10−4
+Readout assignment error 1.300 × 10−2 2.400 × 10−2 8.000 × 10−3 9.600 × 10−3
+7.700 × 10−3
+9.800 × 10−3
+TABLE III: Machine properties of IBM quantum computers and parameters we used.
+shots is the number of iterations
+we performed for sampling. Average measurement fidelity was computed when preparing a calibration matrix and used for
+measurement error mitigation. CNOT error corresponds to the direct CNOT error between two qubits [q0, q1] used. Gate time
+corresponds to the gate time between [q0, q1]. First and second qubits corresponds to q0 and q1, respectively. t1 is relaxation
+time and t2 is dephasing time.
+from the analytical value. It was confirmed that the error
+mitigation improved the results. In this study, what is
+important is that negative expectation values ⟨V ⟩ were
+observed for all cases. It is a noteworthy achievement
+that negative energy expectation values ⟨E⟩ < 0 were
+observed by error mitigation. In fact, the histograms of
+states (Fig. 2 (B)) have improved to approach the ex-
+act values, indicating that all operations were performed
+correctly.
+
+12
+Backend
+(h, k) = (1, 0.2)
+(h, k) = (1, 0.5)
+(h, k) = (1, 1)
+(h, k) = (1.5, 1)
+Analytical value
+⟨E0⟩
+0.9806
+0.894
+0.7071
+1.2481
+ibmq lima
+error mitigated
+0.9423 ± 0.0032
+0.8169 ± 0.0032
+0.6560 ± 0.0031
+1.2480 ± 0.0047
+unmitigated
+0.9049 ± 0.0017
+0.8550 ± 0.0032
+0.6874 ± 0.0031
+1.4066 ± 0.0047
+ibmq jakarta
+error mitigated
+0.9299 ± 0.0056
+0.8888 ± 0.0056
+0.7039 ± 0.0056
+1.2318 ± 0.0084
+unmitigated
+0.9542 ± 0.0056
+0.9089 ± 0.0056
+0.7232 ± 0.0056
+1.2624 ± 0.0083
+ibm hanoi
+error mitigated
+1.0685 ± 0.0032
+0.9534 ± 0.0032
+0.7852 ± 0.0031
+1.3728 ± 0.0047
+unmitigated
+1.0670 ± 0.0031
+0.9524 ± 0.0031
+0.7809 ± 0.0031
+1.3663 ± 0.0047
+ibm cairo
+error mitigated
+0.9571 ± 0.0032
+0.8626 ± 0.0031
+0.7277 ± 0.0031
+1.2072 ± 0.0047
+unmitigated
+0.9578 ± 0.0031
+0.8735 ± 0.0031
+0.7362 ± 0.0031
+1.2236 ± 0.0047
+ibm auckland
+error mitigated
+0.9766 ± 0.0032
+0.8703 ± 0.0032
+0.6925 ± 0.0032
+1.2482 ± 0.0047
+unmitigated
+0.9771 ± 0.0032
+0.8712 ± 0.0032
+0.6931 ± 0.0032
+1.2487 ± 0.0047
+ibmq montreal
+error mitigated
+0.8774 ± 0.0056
+0.8084 ± 0.0056
+0.6315 ± 0.0056
+1.1449 ± 0.0084
+unmitigated
+0.9036 ± 0.0056
+0.8338 ± 0.0056
+0.6564 ± 0.0056
+1.1819 ± 0.0084
+Analytical value
+⟨H1⟩
+0.0521
+0.1873
+0.2598
+0.3480
+ibmq lima
+error mitigated
+0.0733 ± 0.0032
+0.1934 ± 0.0032
+0.2526 ± 0.0032
+0.3590 ± 0.0047
+unmitigated
+0.1295 ± 0.0053
+0.2422 ± 0.0024
+0.2949 ± 0.0028
+0.4302 ± 0.0039
+ibmq jakarta
+error mitigated
+0.0736 ± 0.0055
+0.2018 ± 0.0056
+0.2491 ± 0.0056
+0.3390 ± 0.0084
+unmitigated
+0.0852 ± 0.0022
+0.2975 ± 0.0045
+0.3365 ± 0.0052
+0.4871 ± 0.0073
+ibm hanoi
+error mitigated
+0.1786 ± 0.0032
+0.3256 ± 0.0032
+0.4276 ± 0.0032
+0.5890 ± 0.0047
+unmitigated
+0.2012 ± 0.0019
+0.3427 ± 0.0026
+0.4378 ± 0.0031
+0.6104 ± 0.0042
+ibm cairo
+error mitigated
+0.0674 ± 0.0032
+0.1653 ± 0.0031
+0.2579 ± 0.0031
+0.3559 ± 0.0047
+unmitigated
+0.0905 ± 0.0014
+0.1825 ± 0.0022
+0.2630 ± 0.0027
+0.3737 ± 0.0037
+ibm auckland
+error mitigated
+0.1218 ± 0.0032
+0.2004 ± 0.0031
+0.2181 ± 0.0032
+0.3215 ± 0.0047
+unmitigated
+0.1455 ± 0.0017
+0.2205 ± 0.0023
+0.2337 ± 0.0027
+0.3493 ± 0.0038
+ibmq montreal
+error mitigated
+0.0897 ± 0.0056
+0.1618 ± 0.0056
+0.1921 ± 0.0056
+0.2816 ± 0.0084
+unmitigated
+0.1603 ± 0.0032
+0.2251 ± 0.0041
+0.2454 ± 0.0049
+0.3704 ± 0.0068
+Analytical value
+⟨V ⟩
+-0.0701
+-0.2598
+-0.3746
+-0.4905
+ibmq lima
+error mitigated −0.0655 ± 0.0012 −0.2041 ± 0.0031 −0.2744 ± 0.0063 −0.4091 ± 0.0063
+unmitigated
+−0.0538 ± 0.0011 −0.1471 ± 0.0025 −0.1233 ± 0.0041 −0.2737 ± 0.0046
+ibmq jakarta
+error mitigated −0.0515 ± 0.0022 −0.2348 ± 0.0056 −0.3255 ± 0.0112 −0.4469 ± 0.0112
+unmitigated
+−0.0338 ± 0.0021 −0.1371 ± 0.0046 −0.0750 ± 0.0075 −0.2229 ± 0.0083
+ibm hanoi
+error mitigated −0.1136 ± 0.0013 −0.2820 ± 0.0031 −0.3497 ± 0.0063 −0.5512 ± 0.0063
+unmitigated
+−0.1061 ± 0.0011 −0.2494 ± 0.0022 −0.2704 ± 0.0034 −0.4767 ± 0.0038
+ibm cairo
+error mitigated −0.0497 ± 0.0013 −0.1968 ± 0.0031 −0.2569 ± 0.0063 −0.3804 ± 0.0063
+unmitigated
+−0.0471 ± 0.0012 −0.1682 ± 0.0026 −0.1733 ± 0.0038 −0.3089 ± 0.0045
+ibm auckland
+error mitigated −0.0138 ± 0.0012 −0.0854 ± 0.0032 −0.0591 ± 0.0063 −0.1887 ± 0.0063
+unmitigated
+−0.0113 ± 0.0012 −0.0665 ± 0.0027 −0.0046 ± 0.0044 −0.1412 ± 0.0049
+ibmq montreal error mitigated −0.0157 ± 0.0022 −0.1207 ± 0.0056 −0.1275 ± 0.0112 −0.1967 ± 0.0112
+unmitigated
+−0.0091 ± 0.0021 −0.0764 ± 0.0048 −0.0043 ± 0.0079 −0.0926 ± 0.0089
+Analytical value
+⟨E1⟩
+-0.0180
+-0.0726
+-0.1147
+-0.1425
+ibmq lima
+error mitigated
+0.0078 ± 0.0034
+−0.0107 ± 0.0045 −0.0217 ± 0.0071 −0.0501 ± 0.0079
+unmitigated
+0.0757 ± 0.0054
+0.0950 ± 0.0035
+0.1715 ± 0.0050
+0.1565 ± 0.0060
+ibmq jakarta
+error mitigated
+0.0221 ± 0.0059
+−0.0330 ± 0.0079 −0.0764 ± 0.0125 −0.1079 ± 0.0140
+unmitigated
+0.0514 ± 0.0030
+0.1604 ± 0.0064
+0.2615 ± 0.0091
+0.2642 ± 0.00111
+ibm hanoi
+error mitigated
+0.065 ± 0.0034
+0.0436 ± 0.0044
+0.0779 ± 0.0071
+1.2481 ± 0.015
+unmitigated
+0.0950 ± 0.0022
+0.0933 ± 0.0021
+0.1674 ± 0.0046
+1.0566 ± 0.015
+ibm cairo
+error mitigated
+0.0177 ± 0.0035
+−0.0315 ± 0.0044
+0.0010 ± 0.0070
+−0.0245 ± 0.0079
+unmitigated
+0.0433 ± 0.0018
+0.0143 ± 0.0034
+0.0897 ± 0.0047
+0.0648 ± 0.0058
+ibm auckland
+error mitigated
+0.1080 ± 0.0034
+0.1149 ± 0.0045
+0.5877 ± 0.0031
+1.2072 ± 0.0047
+unmitigated
+0.1341 ± 0.0021
+0.154 ± 0.0035
+0.6364 ± 0.0031
+1.2236 ± 0.0047
+ibmq montreal
+error mitigated
+0.0740 ± 0.0060
+0.0411 ± 0.0079
+0.0645 ± 0.0057
+0.0849 ± 0.0140
+unmitigated
+0.1512 ± 0.0038
+0.1487 ± 0.0063
+0.2411 ± 0.0093
+0.2778 ± 0.0112
+TABLE
+IV:
+Results
+by
+ibmq lima,
+ibmq jakarta,
+ibmq hanoi, ibm cairo, ibm auckland, ibmq montreal.
+

EdE2T4oBgHgl3EQfowg5/content/tmp_files/2301.04021v1.pdf.txt

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+Need for “special” states in a deterministic
+theory of quantum mechanics
+L. S. Schulman
+Clarkson University, Potsdam, New York 13699-5820, USA
+email: schulman<at>clarkson.edu
+January 11, 2023
+Abstract
+There are several theories or processes which may underlie quantum mechanics and
+make it deterministic. Some references are given in the main text. Any such theory, plus
+a number of reasonable assumptions, implies the existence of what I have called “special”
+states. The assumptions are conservation laws, obedience (up to a point) of Schr¨odinger’s
+equation, and a single world, in the sense of the many worlds interpretation (the last one a
+consequence of any deterministic theory). This article also, for clarity, gives an example of
+a “special” state. There is an experimental test of the “special” state theory.
+1
+Introduction
+Determinism is a loophole in Bell’s [1] ideas, which he was aware of. I unwittingly exploited it in
+1984 [18] with what I will call the “special” state theory of measurement. (In Sec. 2 an example
+of a “special” state is given.) The present article reports new motivation for “special” states.
+There have been quite a few attempts to find an underlying process that would make the
+Schr¨odinger equation deterministic. I am not referring to Bohm’s interpretation [2] or that of
+his followers. Rather I have in mind those theories which would restore determinism, such as
+(not exclusively) those of ’tHooft [22, 23], Palmer [16], De la Pena Auerbach and Cetto [13],
+Cavalleri et al. [5], Cufaro-Petroni and Vigier [6] and Marshall [15]. For at least some of these
+the Schr¨odinger equation is an approximation—a good approximation, but an approximation
+nevertheless. There has also been discussion about the consequences of determinism [22, 16, 12,
+4, 11].
+There is an experimental test of the “special” state theory, which, if successful, would lend
+credence to some of the theories advanced. If negative, it would be challenging to maintain
+determinism.
+2
+“Special” states
+Most of this section is review. It may be skipped by those familiar with the kind of “special”
+states that I have in mind.
+1
+arXiv:2301.04021v1  [quant-ph]  8 Jan 2023
+
+We take, as an example of a “special” state, a spin, initially pointing in the positive z
+direction with a 50% probability of overturning at some given time, say at 0.15 (since all is
+determined the time of observation is also fixed). Moreover, we don’t deal with “registration” of
+the measurement; that will be accomplished by additional degrees of freedom.1 The Hamiltonian
+is
+H = ε
+2 (1 + σz) + ωa†a + βσx(a† + a).
+(1)
+The Pauli matrices σx and σz are the operators for the 2-state spin system, a and a† are the
+boson operators and ε, β and ω are parameters.
+“Special” states are particular initial conditions of the bath such that the microscopic final
+state of the spin is (either) all up, (eiφ1
+0 ), or all down, (
+0
+eiφ2 ). “Final” refers to the time of
+measurement, namely when (even) more degrees of freedom are involved (we use parameter
+values ϵ = 0.5, ω = 0.1, β = 0.6 and a time of 0.15).
+The system begins in all up and ordinarily at time 0.15 has probability of half up, half down.
+As indicated, that is not the case for these “special” initial conditions. If the probabilities are
+as in Fig. 1a—with fixed phases (not shown)—then the system will be found in an up state. If
+the initial state is as shown in Fig. 1b (again with particular phases, not shown) then the system
+will be found in a down state.
+There are three points to be raised: the first is what about residual amplitudes? The ampli-
+tude for (say) the up state is not perfect and for the given cutoff of the bosons at 250 is about
+10−4; the same is true for the state which is “fully” decayed. The second question has to do with
+Schr¨odinger’s cat. And the third issue is how do you find these states?
+Now 10−4 is a big number, especially since the final state of one interaction is the initial
+state for the next. I could improve that number if I had better computer power, but I doubt
+if it could be zero. But it doesn’t have to be zero! It only needs to be accurate as far as the
+Schr¨odinger equation has been checked. And I don’t think it has been checked to 10−12 (which I
+am reasonably confident I could get the discrepancy down to).
+The second issue I mentioned is, what about Schr¨odinger cats? The (possibly) decaying spin
+could be the determinant of whether the poison is released.2 With “special” states the cat is
+either alive or dead. It should be noticed that there are only what ’tHooft calls “ontological”
+states [22, 23]. I believe “special” and “ontological” have the same meaning in this case.
+Finally there is question of how “special” states are found.
+You can define a projection
+operator (cf. [19]) on the spin: P ≡ (ψupψ†
+up)⊗1boson bath. Using this operator, the probability of
+being all up at time t is
+Pr(up) = ⟨ψup ⊗ ψbath∣U †PU∣ψup ⊗ ψbath⟩ = ⟨ψup ⊗ ψbath∣PU †PPUP∣ψup ⊗ ψbath⟩,
+(2)
+with U ≡ exp(−iHt/ℏ) and where PP = P is used. Defining A ≡ PUP and using P † = P, we have
+Pr(up) = ⟨ψup⊗ψbath∣A†A∣ψup⊗ψbath⟩. Defining B ≡ A†A, it follows that the issue of whether any
+initial state (of the bath) can lead to a measurement of up, using purely unitary time evolution
+1The irreversible “registration” of the result of a measurement by the observer has been studied in many
+contexts. For example, in [9] the “measurement” is accompanied by the bath’s (not the same as the bath in
+the current Eq. (1)) changing in an irreversible fashion. Other models of measurement (e.g., [3, 10]) show the
+same feature. As a result, our considerations in the present article do not pursue the registration issue once the
+observer is coupled to the system, that coupling taking place (in our forthcoming example) at 0.15 time units.
+2I assume familiarity with the Schr¨odinger cat paradox.
+2
+
+is the matter of whether B has eigenvalues equal to one. For any fully decayed states you must
+have an eigenvalue (of B) be zero. (Of course A and B are functions of t, since U is.)
+Remark:
+It also is true that the number of decay states and non-decay (“special”) states is
+roughly equal at time 0.15.
+Figure 1: “Special” time-0 oscillator states. Figure (a) shows the (initial) probability of excitation
+of oscillator states that contribute to the non-decay state. Only shown are even states, since there
+is total amplitude zero for the odd states. Phases of the states are not shown, but are also fixed
+by the non-decay condition. In Figure (b) are shown the probabilities for the state that decays;
+in this case (and for the same reason) only even oscillator states are shown. As in image a, the
+phases, though not shown are crucial to the “special” nature of the state.
+This is the main idea of the “special” state theory: no macroscopic superpositions because
+of particular initial conditions. There is also no entanglement. At time-0.15 the spin state is
+wholly in one state or the other.
+3
+Determinism implies “special” states
+The title of this section needs a bit of enhancement: you need a few more concessions to reality.
+Besides determinism you need conservation laws and Schr¨odinger’s equation, at least to the
+extent that it’s been checked. It is also understood that there is just one world. These rules,
+together with determinism, imply “special” states.
+You start with a wave function describing some state, say a spin in a Stern-Gerlach experi-
+ment. Then it must go to some particular outcome, say spin up. Presumably there were involved
+other coordinates (such as the bosons in the above example) that fixed its outcome. The final
+state is definite. But the Schr¨odinger equation holds also. Therefore it could only have evolved
+to that final state. How can that be? There must have been a coordination of degrees of freedom
+on the initial state that forced it to its final form. That is, there must have been a “special”
+state.
+4
+Experiment
+Finally, there is the matter of experiment. In [20] and [21] we have described in detail experi-
+mental tests of the “special” state theory. An example is the double Stern-Gerlach experiment
+3
+
+0.2
+a
+0.15
+0.1
+0.05
+0
+0
+50
+100
+state label
+200
+2500.07
+b
+0.06
+0.05
+probability
+0.04
+0.03
+0.02
+0.01
+0
+0
+50
+100
+200
+250
+state label([17, 8, 14, 7]) which requires the detection of a magnetic field of 5×10−8 tesla in an environment
+of half a tesla, a challenging experiment. A firm absence of the small magnetic field would in my
+opinion spell the end of efforts to find a deterministic theory (but no-go theorems are made to
+be disproved).
+5
+Conclusions
+You don’t have to believe in any of the deterministic theories to reach the conclusion that
+“special” states are needed in any theory which is deterministic, goes from one “special” state
+into another, satisfies Schr¨odinger’s equations (as far as has been measured), has a single world
+and satisfies conservation laws. You only have to believe that it’s possible.
+Three points are worth mentioning. First—and this is new—you don’t need to eliminate
+“incorrect” choices (by “special” states) at the level of (say) 10−12, since the Schr¨odinger equa-
+tion has not been checked at that level. Second, there is an experimental test of the special
+state theory. Failure would eliminate deterministic theories (or leave people struggling for an
+explanation), while success would encourage attempts to find deterministic theories. Third, it
+may be that ’tHooft is right, and one should look to extremely small times and distances for
+theoretical support for determinism. However, given the fragmentary understanding of events at
+10−17 cm I’d be reluctant to make predictions about what happens at 10−33 cm.
+References
+[1] J. S. Bell.
+Speakable and unspeakable in quantum mechanics.
+Cambridge Univ. Press,
+Cambridge, 1987.
+[2] David Bohm. Quntum Theory. Prentice Hall, New Jersey, 1961.
+[3] P. B´ona. A solvable model of particle detection in quantum theory. Acta Facultatis Rerum
+Naturalium Uuniversitatis Comenianae Physica, XX:65–95, 1980.
+[4] C. H. Brans. Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theor.
+Phys., 27:219, 1988.
+[5] G. Cavalleri, E. Cesaroni, E. Tonni, and P. Di Sia. About the derivation of Planck’s black
+body spectrum from classical mechanics. in “Physical Interpretations of Relativity Theory
+VI” (Imperial College, London, 15-18 September 2000), Ed. by M.C.Duffy, School of Me-
+chanical Engineering, Sunderland Polytecnic, Chester Road, Sunderland, SR1 3SD, England
+(p. 78), 09 2000.
+[6] N. Cufaro-Petroni and J-P. Vigier. Single-particle trajectories and interferences in quantum
+mechanics. Found. Phys., 22:1–40, 1992.
+[7] R. Frisch, T. E. Phipps, E. Segre, and O. Stern. Process of space quantisation. Nature, 130
+(no. 3293):892–3, 1932.
+4
+
+[8] R. Frisch and E. Segr`e. ¨Uber die Einstellung der Richtungsquantelung. II. Zeitschrift f¨ur
+Physik, 80:610–616, 1933.
+[9] B. Gaveau and L. S. Schulman. Model apparatus for quantum measurements. J. Stat. Phys.,
+58:1209–1230, 1990.
+[10] H. S. Green. Observation in quantum mechanics. Nuov. Cim., 9:880–889, 1958.
+[11] M. J. W. Hall. Local deterministic model of singlet state correlations based on relaxing
+measurement independence. Phys. Rev. Lett., 105:250404, 2010.
+[12] S. Hossenfelder and T. Palmer. Rethinking superdeterminism. Front. Phys., 8:139, 2020.
+[13] L. De la Pe˜na Auerbach and A. M. Cetto. Stochastic electrodynamics as a foundation for
+quantum mechanics. Phys. Lett., 56A:253–254, 1976.
+[14] E. Majorana. Atomi orientati in campo magnetico variabile. Nuovo Cimento, 9:43–50, 1932.
+[15] T. W. Marshall. Statistical electrodynamcs. Proc. Camb. Phil. Soc., 61:537–546, 1965.
+[16] T. N. Palmer. The Invariant Set Postulate: a new geometric framework for the foundations
+of quantum theory and the role played by gravity. Proc. Roy. Soc. A, 465:3165–3185, 2009.
+[17] T. E. Phipps and O. Stern. ¨Uber die einstellung der richtungsquantelung. Zeitschrift f¨ur
+Physik, 73:185–191, 1932.
+[18] L. S. Schulman. Definite measurements and deterministic quantum evolution. Phys. Lett.
+A, 102:396–400, 1984.
+[19] L. S. Schulman. Time’s Arrows and Quantum Measurement. Cambridge Univ. Press, New
+York, 1997.
+[20] L. S. Schulman. Experimental test of the “special state” theory of quantum measurement.
+Entropy, 14:665–686, 2012.
+[21] L. S. Schulman and M. G. E. da Luz. Looking for the source of change. Found. Phys.,
+46:1495–1501, 2016.
+[22] G. ’tHooft. The Cellular Automaton Interpretation of Quantum Mechanics. Springer, Berlin,
+2016.
+[23] G. ’tHooft. Deterministic quantum mechanics: The mathematical equations. Front. Phys.,
+8:253, 2020.
+5
+

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+page_content='  1  Introduction  Determinism is a loophole in Bell’s [1] ideas, which he was aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' I unwittingly exploited it in  1984 [18] with what I will call the “special” state theory of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' (In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=') The present article reports new motivation for “special” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  There have been quite a few attempts to find an underlying process that would make the  Schr¨odinger equation deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' I am not referring to Bohm’s interpretation [2] or that of  his followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Rather I have in mind those theories which would restore determinism, such as  (not exclusively) those of ’tHooft [22, 23], Palmer [16], De la Pena Auerbach and Cetto [13],  Cavalleri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' [5], Cufaro-Petroni and Vigier [6] and Marshall [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' For at least some of these  the Schr¨odinger equation is an approximation—a good approximation, but an approximation  nevertheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' There has also been discussion about the consequences of determinism [22, 16, 12,  4, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  There is an experimental test of the “special” state theory, which, if successful, would lend  credence to some of the theories advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' If negative, it would be challenging to maintain  determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  2  “Special” states  Most of this section is review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' It may be skipped by those familiar with the kind of “special”  states that I have in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='04021v1  [quant-ph]  8 Jan 2023  We take, as an example of a “special” state, a spin, initially pointing in the positive z  direction with a 50% probability of overturning at some given time, say at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15 (since all is  determined the time of observation is also fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Moreover, we don’t deal with “registration” of  the measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' that will be accomplished by additional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='1 The Hamiltonian  is  H = ε  2 (1 + σz) + ωa†a + βσx(a† + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  (1)  The Pauli matrices σx and σz are the operators for the 2-state spin system, a and a† are the  boson operators and ε, β and ω are parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  “Special” states are particular initial conditions of the bath such that the microscopic final  state of the spin is (either) all up, (eiφ1  0 ), or all down, (  0  eiφ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' “Final” refers to the time of  measurement, namely when (even) more degrees of freedom are involved (we use parameter  values ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='5, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='6 and a time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  The system begins in all up and ordinarily at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15 has probability of half up, half down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  As indicated, that is not the case for these “special” initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' If the probabilities are  as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' 1a—with fixed phases (not shown)—then the system will be found in an up state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' If  the initial state is as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' 1b (again with particular phases, not shown) then the system  will be found in a down state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  There are three points to be raised: the first is what about residual amplitudes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' The ampli-  tude for (say) the up state is not perfect and for the given cutoff of the bosons at 250 is about  10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' the same is true for the state which is “fully” decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' The second question has to do with  Schr¨odinger’s cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' And the third issue is how do you find these states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Now 10−4 is a big number, especially since the final state of one interaction is the initial  state for the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' I could improve that number if I had better computer power, but I doubt  if it could be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' But it doesn’t have to be zero!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' It only needs to be accurate as far as the  Schr¨odinger equation has been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' And I don’t think it has been checked to 10−12 (which I  am reasonably confident I could get the discrepancy down to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  The second issue I mentioned is, what about Schr¨odinger cats?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' The (possibly) decaying spin  could be the determinant of whether the poison is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='2 With “special” states the cat is  either alive or dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' It should be noticed that there are only what ’tHooft calls “ontological”  states [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' I believe “special” and “ontological” have the same meaning in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Finally there is question of how “special” states are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  You can define a projection  operator (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' [19]) on the spin: P ≡ (ψupψ†  up)⊗1boson bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Using this operator, the probability of  being all up at time t is  Pr(up) = ⟨ψup ⊗ ψbath∣U †PU∣ψup ⊗ ψbath⟩ = ⟨ψup ⊗ ψbath∣PU †PPUP∣ψup ⊗ ψbath⟩,  (2)  with U ≡ exp(−iHt/ℏ) and where PP = P is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Defining A ≡ PUP and using P † = P, we have  Pr(up) = ⟨ψup⊗ψbath∣A†A∣ψup⊗ψbath⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Defining B ≡ A†A, it follows that the issue of whether any  initial state (of the bath) can lead to a measurement of up, using purely unitary time evolution  1The irreversible “registration” of the result of a measurement by the observer has been studied in many  contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' For example, in [9] the “measurement” is accompanied by the bath’s (not the same as the bath in  the current Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' (1)) changing in an irreversible fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Other models of measurement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=', [3, 10]) show the  same feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' As a result, our considerations in the present article do not pursue the registration issue once the  observer is coupled to the system, that coupling taking place (in our forthcoming example) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  2I assume familiarity with the Schr¨odinger cat paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  2  is the matter of whether B has eigenvalues equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' For any fully decayed states you must  have an eigenvalue (of B) be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' (Of course A and B are functions of t, since U is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=')  Remark:  It also is true that the number of decay states and non-decay (“special”) states is  roughly equal at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Figure 1: “Special” time-0 oscillator states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Figure (a) shows the (initial) probability of excitation  of oscillator states that contribute to the non-decay state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Only shown are even states, since there  is total amplitude zero for the odd states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Phases of the states are not shown, but are also fixed  by the non-decay condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' In Figure (b) are shown the probabilities for the state that decays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  in this case (and for the same reason) only even oscillator states are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' As in image a, the  phases, though not shown are crucial to the “special” nature of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  This is the main idea of the “special” state theory: no macroscopic superpositions because  of particular initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' There is also no entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' At time-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15 the spin state is  wholly in one state or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  3  Determinism implies “special” states  The title of this section needs a bit of enhancement: you need a few more concessions to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Besides determinism you need conservation laws and Schr¨odinger’s equation, at least to the  extent that it’s been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' It is also understood that there is just one world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' These rules,  together with determinism, imply “special” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  You start with a wave function describing some state, say a spin in a Stern-Gerlach experi-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Then it must go to some particular outcome, say spin up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Presumably there were involved  other coordinates (such as the bosons in the above example) that fixed its outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' The final  state is definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' But the Schr¨odinger equation holds also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Therefore it could only have evolved  to that final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' How can that be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' There must have been a coordination of degrees of freedom  on the initial state that forced it to its final form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' That is, there must have been a “special”  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  4  Experiment  Finally, there is the matter of experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' In [20] and [21] we have described in detail experi-  mental tests of the “special” state theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' An example is the double Stern-Gerlach experiment  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='2  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='05  0  0  50  100  state label  200  2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='07  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='05  probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='01  0  0  50  100  200  250  state label([17, 8, 14, 7]) which requires the detection of a magnetic field of 5×10−8 tesla in an environment  of half a tesla, a challenging experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' A firm absence of the small magnetic field would in my  opinion spell the end of efforts to find a deterministic theory (but no-go theorems are made to  be disproved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  5  Conclusions  You don’t have to believe in any of the deterministic theories to reach the conclusion that  “special” states are needed in any theory which is deterministic, goes from one “special” state  into another, satisfies Schr¨odinger’s equations (as far as has been measured), has a single world  and satisfies conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' You only have to believe that it’s possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Three points are worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' First—and this is new—you don’t need to eliminate  “incorrect” choices (by “special” states) at the level of (say) 10−12, since the Schr¨odinger equa-  tion has not been checked at that level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Second, there is an experimental test of the special  state theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Failure would eliminate deterministic theories (or leave people struggling for an  explanation), while success would encourage attempts to find deterministic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Third, it  may be that ’tHooft is right, and one should look to extremely small times and distances for  theoretical support for determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' However, given the fragmentary understanding of events at  10−17 cm I’d be reluctant to make predictions about what happens at 10−33 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Bell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Speakable and unspeakable in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Press,  Cambridge, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' Time’s Arrows and Quantum Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' Schulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Experimental test of the “special state” theory of quantum measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content='  Entropy, 14:665–686, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' Schulman and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' Looking for the source of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' ’tHooft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' The Cellular Automaton Interpretation of Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Springer, Berlin,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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+page_content=' Deterministic quantum mechanics: The mathematical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'}
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