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[MATH] For [MATH] , we define [MATH] to be the closed disc consisting of all complex numbers [MATH] , such that [MATH] Finally, we require the following lemma, which arises from the error analysis of Barvinok’s interpolation method Barvinok ( 2015 2016a (see Barvinok (Barvinok, 2016b , Lemma 2.2.1) ). The lemma states ... |
Lemma 4 (name=Barvinok Barvinok ( 2015 2016a 2016b , restate=[name=restatement]ApproximatePolynomialZeroFreeRegionInversePowerSums) |
Fix [MATH] . Let [MATH] be the roots of the polynomial [MATH] . Suppose that, for some [MATH] , the roots of [MATH] lie in the exterior of the closed disc [MATH] . Suppose further that we can compute [MATH] and the inverse power sums [MATH] in time [MATH] . Then, for any [MATH] in the interior of the closed disc [MATH]... |
We prove Lemma in Appendix III Ising Model Partition Functions The Ising model is described by a graph [MATH] , with the vertices representing spins and the edges representing interactions between them. A set of edge weights [MATH] characterise the interactions and a set of vertex weights [MATH] characterise the extern... |
Definition 7 (Ising model partition function) Let [MATH] be a graph with the weights [MATH] assigned to its edges and the weights [MATH] assigned to its vertices. Then the Ising model partition function is defined by |
[EQUATION] where [EQUATION] The model is called ferromagnetic if [MATH] for all [MATH] and anti-ferromagnetic if [MATH] for all [MATH] . Otherwise, the model is called non-ferromagnetic |
A classic result of Jerrum and Sinclair Jerrum and Sinclair ( 1993 establishes a fully polynomial-time randomised approximation scheme for the Ising model partition function for all graphs in the ferromagnetic regime with real vertex weights. In contrast, they showed that no such scheme could exists in the anti-ferroma... |
We shall extend the result of Theorem to the Ising model partition function. This is achieved by an approximation-preserving polynomial-time reduction from the Ising model partition function to the restricted multivariate graph homomorphism partition function. |
Proposition 5 (restate=[name=restatement]IsingModelGraphHomomorphismPartitionFunctionReduction) There is an approximation-preserving polynomial-time reduction from the Ising model partition function to the restricted multivariate graph homomorphism partition function. |
We prove Proposition in Appendix . Let us define the following closed polyregion, which arises naturally from applying Proposition to Theorem |
Definition 8 [MATH] For a graph [MATH] and [MATH] , we define [MATH] to be the closed polyregion consisting of all sets of weights [MATH] and [MATH] , such that [MATH] for all [MATH] and [MATH] for all [MATH] |
We have the following corollary of Theorem and Proposition Corollary 6 Fix [MATH] and [MATH] . There is a deterministic polynomial-time approximation scheme for the Ising model partition function [MATH] for all graphs [MATH] of maximum degree at most [MATH] and all [MATH] and all [MATH] in the closed polyregion [MATH] |
Proof. The proof follows directly from Theorem and Proposition , while noting that the reduction from the Ising model partition to the restricted multivariate graph homomorphism partition function increases the maximum vertex degree by one. |
Remark It is possible to marginally increase the size of the polyregion by applying the [MATH] -thickening technique of Jaeger, Vertigan, and Welsh Jaeger et al. 1990 |
It is important to mention that the bounds of Corollary are not sharp in general. To see this, let us compare the results in the anti-ferromagnetic regime with no external field, to those of Sinclair, Srivastava, and Thurley Sinclair et al. 2014 . In this case, Corollary tells us that there is a deterministic polynomia... |
We also have the following corollary concerning the location of the complex zeros of the Ising model partition function on bounded degree graphs. |
Corollary 7 Fix [MATH] . For any graph [MATH] of degree at most [MATH] and any [MATH] and [MATH] in the closed polyregion [MATH] , the Ising model partition function does not vanish, i.e., [MATH] |
Proof. The proof follows directly from Lemma and Proposition This may be of independent interest in statistical physics as the possible points of physical phase transitions are exactly the real limit points of such complex zeros Sokal et al. 2005 |
IV Quantum Simulation Complex-valued Ising model partition functions arise naturally in the output probability amplitudes of quantum circuits De las Cuevas et al. 2011 ); Iblisdir et al. 2014 . In particular, for the class of commuting quantum circuits, known as Instantaneous Quantum Polynomial-time (IQP) circuits Shep... |
Definition 9 (X-program) An X-program is a pair [MATH] , where [MATH] is a binary matrix and [MATH] is a real angle. The matrix [MATH] is used to construct a Hamiltonian of [MATH] commuting terms acting on [MATH] qubits, where each term in the Hamiltonian is a product of Pauli-X operators, |
[EQUATION] Thus, the columns of [MATH] correspond to qubits and the rows of [MATH] correspond to interactions in the Hamiltonian. |
An X-program induces a probability distribution [MATH] known as an IQP distribution Definition 10 [MATH] For an X-program [MATH] with [MATH] , we define [MATH] to be the probability distribution over binary strings [MATH] , given by |
[EQUATION] We shall consider X-programs that are induced by a weighted graph. Definition 11 (Graph-induced X-program) For a graph [MATH] with the weights [MATH] assigned to its edges and the weights [MATH] assigned to its vertices, we define the X-program induced by [MATH] to be an X-program [MATH] such that |
[EQUATION] It will be convenient for us to define [MATH] as a specific probability amplitude induced by a weighted graph [MATH] Definition 12 |
[MATH] For a graph [MATH] with the weights [MATH] assigned to its edges and the weights [MATH] assigned to its vertices, we define [MATH] to be the probability amplitude given by |
[EQUATION] We note that any X-program can be efficiently represented by a graph-induced X-program Shepherd and Bremner ( 2009 . Moreover, X-programs are known to become universal for quantum computation under postselection Bremner et al. 2010 . Therefore, any quantum amplitude can be expressed in the form of [MATH] . T... |
Proposition 8 (restate=[name=restatement]IQPIsingModelPartitionFunctionRelation) Let [MATH] be a graph with the weights [MATH] assigned to its edges and the weights [MATH] assigned to its vertices, then, |
[EQUATION] We prove Proposition in Appendix . We now apply Corollary to Proposition to achieve a deterministic polynomial-time approximation scheme for computing [MATH] for all graphs of bounded maximum degree with weights absolutely bounded sufficiently close to zero. |
Corollary 9 Fix [MATH] and [MATH] . There is a deterministic polynomial-time approximation scheme for the probability amplitude [MATH] for all graphs [MATH] of maximum degree at most [MATH] with the edge weights [MATH] satisfying [MATH] for all [MATH] and the vertex weights [MATH] satisfying [MATH] for all [MATH] |
Proof. It follows from Corollary and Proposition that we have a deterministic polynomial-time approximation scheme for computing [MATH] for all graphs of maximum degree at most [MATH] with [MATH] and [MATH] in the closed polyregion [MATH] . For weights in the range [MATH] , this is achieved when [MATH] for all [MATH] a... |
It is known that approximating [MATH] up to a multiplicative factor for bounded degree graphs with arbitrary weights in [MATH] is #P -hard |
Fujii and Morimae ( 2017 , and so it seems unlikely that Corollary can be extended to hold in this case. We note that Corollary applies to graph-induced X-programs with weights absolutely bounded by a constant that depends only on the maximum degree of the underlying graph. This corresponds to Hamiltonians that have be... |
Conclusion & Outlook We have established a deterministic polynomial-time approximation scheme for the Ising model partition function with complex parameters on bounded degree graphs when the interactions and external fields are absolutely bounded by a constant depending on the maximum degree of the graph. Furthermore, ... |
There are a number of interesting problems that remain to be solved, the most obvious of which is to sharpen the bounds of Corollary . One approach would be to improve Lemma , i.e., prove that the restricted multivariate graph homomorphism partition function does not vanish on a polydisc of a greater radius. It may als... |
Acknowledgements We thank Gavin Brennen, Jacob Bridgeman, Christopher Chubb, David Gosset, Richard Jozsa, and Hakop Pashayan for helpful discussions. This research was conducted by the ARC Centre of Excellence for Quantum Computation and Communication Technology (CQC2T), project number CE170100012. |
Appendix A Proof of Theorem We shall now prove Theorem \GraphHomomorphismPartitionFunctionBoundedDegreeGraphs Proof. Define [MATH] , with [MATH] and note that [MATH] . By Lemma , we have that [MATH] does not vanish whenever [MATH] lies in the closed polydisc [MATH] . Since [MATH] lies in the closed polydisc [MATH] [MAT... |
Appendix B Proof of Lemma We shall now prove Lemma . Our proof follows from a generalisation of a result due to Patel and Regts Patel and Regts ( 2017 |
(Lemma 11 and an additional lemma (Lemma 12 , which we prove in the remainder of the section. \GraphHomomorphismPartitionFunctionComputeConstantTermAndInversePowerSums |
Proof. The proof follows from combining Lemma 11 and Lemma 12 We shall begin with the following definitions. Definition 13 [MATH] |
For [MATH] , define [MATH] to be the collection of all edge-coloured graphs on at most [MATH] vertices. Definition 14 [MATH] For a graph [MATH] and a subset of vertices [MATH] , define [MATH] to be the subgraph induced by [MATH] |
Definition 15 [MATH] For two edge-coloured graphs [MATH] and [MATH] , define [MATH] to be the number of induced subgraphs of [MATH] that are edge-colour isomorphic to [MATH] |
Definition 16 (Multiplicative graph polynomial) A graph polynomial [MATH] is said to be multiplicative if [MATH] and [MATH] for any two graphs [MATH] and [MATH] |
Definition 17 (Edge-coloured bounded induced graph counting polynomial Patel and Regts ( 2017 Let [MATH] be a multiplicative graph polynomial defined by [MATH] with [MATH] . We say that [MATH] is an edge-coloured bounded induced graph counting polynomial if there exists constants [MATH] , such that (1) the coefficients... |
Patel and Regts (Patel and Regts, 2017 , Theorem 3.10) proved that, for any edge-coloured bounded induced graph counting polynomial, there is an efficient algorithm for computing the constant term and inverse power sums of its roots. |
Lemma 10 (Patel and Regts Patel and Regts ( 2017 Fix [MATH] [MATH] , and [MATH] . Let [MATH] be an edge-coloured graph of maximum degree at most [MATH] . Further let [MATH] be an edge-coloured bounded induced graph counting polynomial with roots [MATH] . Then there is a deterministic [MATH] -time algorithm for computin... |
We shall now generalise the result of Patel and Regts Patel and Regts ( 2017 to the restricted case, that is, where the graph polynomial is restricted to map a subset of vertices to a fixed index. We begin by extending the previous definitions. |
Definition 18 (Restricted graph) A restricted graph is a pair [MATH] , where [MATH] is a graph and [MATH] is a subset of [MATH] Definition 19 |
[MATH] For [MATH] , define [MATH] to be the collection of all edge-coloured restricted graphs on at most [MATH] vertices. Definition 20 |
(Induced restricted subgraph) For a restricted graph [MATH] and a subset of vertices [MATH] , the restricted subgraph induced by [MATH] is given by [MATH] |
Definition 21 (Isomorphic restricted graphs) Two restricted graphs [MATH] and [MATH] are said to be isomorphic if and only if there is an isomorphism [MATH] from [MATH] to [MATH] and [MATH] is the image of [MATH] under [MATH] |
Definition 22 [MATH] For two edge-coloured restricted graphs [MATH] and [MATH] , define [MATH] to be the number of induced restricted subgraphs of [MATH] that are edge-colour isomorphic to [MATH] |
Definition 23 (Multiplicative restricted graph polynomial) A restricted graph polynomial [MATH] is said to be multiplicative if [MATH] and [MATH] for any two restricted graphs [MATH] and [MATH] and integer [MATH] |
Definition 24 (Edge-coloured bounded induced restricted graph counting polynomial) Let [MATH] be a multiplicative restricted graph polynomial defined by [MATH] with [MATH] . We say that [MATH] is an edge-coloured bounded induced restricted graph counting polynomial if there exists constants [MATH] , such that (1) the c... |
The restricted version of Lemma 10 is then obtained by following the proof of Patel and Regts Patel and Regts ( 2017 with the definitions extended in the natural way. We omit the proof for the sake of brevity. |
Lemma 11 Fix [MATH] [MATH] , and [MATH] . Let [MATH] be an edge-coloured graph of maximum degree at most [MATH] . Further let [MATH] be an edge-coloured bounded induced restricted graph counting polynomial with roots [MATH] . Then there is a deterministic [MATH] -time algorithm for computing [MATH] and the inverse powe... |
Lemma 12 Let [MATH] be a graph with the [MATH] symmetric matrices [MATH] assigned to its edges and let each edge [MATH] be assigned a distinct colour. Further let [MATH] be a subset of [MATH] and let [MATH] be an integer. Then, up to an efficiently computable factor, the restricted multivariate graph homomorphism parti... |
Proof. Define [MATH] by [EQUATION] Then, [EQUATION] where [MATH] is the subgraph of [MATH] induced by [MATH] . Since the number of vertices in [MATH] is at most [MATH] , we can write |
[EQUATION] Therefore, we have [EQUATION] with [EQUATION] and [EQUATION] It is clear that [MATH] is a multiplicative restricted graph polynomial with [MATH] . Furthermore, for any restricted graph [MATH] , the coefficients [MATH] can be computed in time [MATH] . Hence, [MATH] is an edge-coloured bounded induced restrict... |
Appendix C Proof of Lemma We shall now prove Lemma . The lemma is due to Barvinok Barvinok ( 2015 2016a 2016b , however, our proof closely follows that of Patel and Regts Patel and Regts ( 2017 |
\ApproximatePolynomialZeroFreeRegionInversePowerSums Proof. Define the function [MATH] on the closed disc [MATH] by [EQUATION] where the branch of the logarithm is chosen by taking the principal value at [MATH] . By Taylor’s Theorem about the point [MATH] , for each [MATH] in the interior of the closed disc [MATH] |
[EQUATION] Define the Taylor expansion truncated at order [MATH] by [EQUATION] Now, let us write [MATH] in terms of its roots. By the Factor Theorem, |
[EQUATION] Then, [EQUATION] Therefore, [EQUATION] Let [MATH] be the [MATH] inverse power sum given by [EQUATION] Then, by noting that [MATH] |
[EQUATION] We shall now show that, for any [MATH] , the Taylor expansion truncated at order [MATH] gives an additive [MATH] -approximation to [MATH] |
[EQUATION] Since the roots [MATH] lie in the exterior of the closed disc [MATH] , we have [MATH] . Therefore, [EQUATION] Since [MATH] , by the geometric series formula, |
[EQUATION] Taking [MATH] , it follows that [EQUATION] We shall now show that the truncated Taylor expansion is a multiplicative [MATH] -approximation to [MATH] . For the norm, we have |
[EQUATION] and [EQUATION] Now, for the argument, [EQUATION] This completes the proof. Appendix D Proof of Proposition We shall now prove Proposition |
\IsingModelGraphHomomorphismPartitionFunctionReduction Proof. Let [MATH] be a graph with the [MATH] symmetric matrices [MATH] assigned to its edges. Let us construct a new graph [MATH] from [MATH] by the following vertex gadget. For every vertex [MATH] , add a new vertex [MATH] and an edge [MATH] with a [MATH] symmetri... |
[EQUATION] Taking [MATH] and [MATH] [EQUATION] where [MATH] and [MATH] . Hence, we have an approximation-preserving polynomial-time reduction from the Ising model partition function to the restricted multivariate graph homomorphism partition function. This completes the proof. |
Appendix E Proof of Proposition We shall now prove Proposition \IQPIsingModelPartitionFunctionRelation Proof. By definition, [EQUATION] This completes the proof. |
# Source: arxiv 1806.11307 # Title: Definable Inapproximability: New Challenges for Duplicator # Sections: all # Downloaded: 2026-03-03T01:44:31.679534+00:00 |
Definable Inapproximability: New Challenges for Duplicator Abstract We consider the hardness of approximation of optimization problems from the point of view of definability. For many [MATH] -hard optimization problems it is known that, unless [MATH] , no polynomial-time algorithm can give an approximate solution guara... |
Introduction Twenty years ago, the PCP theorem transformed the landscape of complexity theory. It showed that if [MATH] then not only is it impossible to efficiently solve [MATH] -hard problems exactly but for some of them it is also impossible to approximate the solution to within a constant factor. Consider for insta... |
probabilistically checkable proofs that gave the theorem its name. Over the years, stronger results were proved, improving the constant [MATH] and, by reductions, proving inapproximability results for a host of other [MATH] -hard problems. |
A structural theory of hardness of approximation was introduced by Papadimitriou and Yannakakis who defined the class MAX SNP of approximation problems, with a definition rooted in descriptive complexity theory. They showed that for every problem in this class, there is a constant [MATH] and a polynomial-time algorithm... |
there is a constant [MATH] such that no polynomial-time algorithm can approximate solutions within a factor [MATH] . This makes it a challenge, for each MAX SNP -complete problem, to determine the exact approximation ratio that is achievable by an efficient algorithm. In some cases, this has been pinned down exactly. F... |
. Another interesting case is MAX 3XOR , where we are given a formula which is the conjunction of clauses, each of which is the XOR |
of three literals. Here, satisfiability is decidable in polynomial time as the problem is essentially that of solving a system of linear equations over the two-element field. However, determining, for an unsatisfiable system, how many of its clauses can be simultaneously satisfied is MAX SNP -hard, and the exact approx... |
To give a problem of another flavour, consider minimum vertex cover , the problem of finding, in a graph [MATH] , a minimum set [MATH] of vertices such that every edge is incident on a vertex in [MATH] Let [MATH] denote the size of a minimum size vertex cover in [MATH] There are algorithms that are guaranteed to find a... |
. Very recent results announced in improve this lower bound to [MATH] . It is conjectured that indeed no such algorithm could achieve a ratio of [MATH] for arbitrarily small [MATH] |
but, as of our current knowledge, the right threshold constant could be somewhere between [MATH] and [MATH] We approach these questions on the hardness of approximability from the point of view of definability. Our aim is to show that the tools of descriptive complexity can be brought to bear in showing lower bounds on... |
A reference logic in descriptive complexity is fixed-point logic with counting, [MATH] . The class of problems definable in this logic form a proper subclass of the complexity class [MATH] . However, [MATH] is very expressive and many natural problems in [MATH] are expressible in this logic. For instance, any polynomia... |
. Also, problems that can be formulated as linear programming or semidefinite programming problems are in [MATH] . At the same time, for many problems we are able to prove categorically, i.e., without complexity theoretic assumptions, that they are not definable in [MATH] . Among these are [MATH] -complete problems lik... |
). We can also prove that certain problems in [MATH] are not in [MATH] such as 3XOR A particularly interesting class of problems is the class of optimization problems known as MAX CSP or constraint maximization problems, where we are given a collection of constraints and the problem is to find the maximum number of con... |
and definable in [MATH] or it is [MATH] -complete and provably not definable in [MATH] . We would like to extend such results also to the approximability of such problems. This paper develops the methodology for doing so. |
For MAX 3SAT , we prove, without any complexity theoretic assumption, that no algorithm expressible in [MATH] can achieve an approximation ratio of [MATH] . The question seems ill-posed at first sight as [MATH] is a formalism for defining problems rather than expressing algorithms. We return to the precise formulation ... |
. This is the basis for showing that many commonly used algorithmic techniques for approximation problems, such as semidefinite programming relaxations, are also expressible in [MATH] . Thus, on the one hand, reductions from MAX SNP -hard problems show inapproximability by any |
polynomial-time algorithm, assuming [MATH] . On the other hand, our results show, without the assumption, inapproximability by the most commonly used polynomial-time methods. |
Undefinability of a class of structures [MATH] in [MATH] is typically established by showing that structures in [MATH] cannot be distinguished from structures not in [MATH] |
in [MATH] —first-order logic with counting and just [MATH] variables—for any fixed [MATH] . In the terminology of [MATH] has unbounded counting width On the other hand, hardness of approximation for a maximization problem is typically established by showing that every class that includes all instances with an optimum [... |
. This methodology poses new challenges for Spoiler-Duplicator games in finite model theory. Such games are typically played on pairs of structures that are minimally different. In the new setting, we need to show Duplicator winning strategies in games on pairs of structures that differ substantially, on some numeric p... |
The PCP theorem is the fons et origo of results on hardness of approximation. It established the first provably [MATH] -hard constant gap between the fully satisfiable instances of MAX 3SAT i.e., those in which all clauses can be satisfied, and the less satisfiable ones, those where no more than [MATH] can be satisfied... |
and [MATH] was then amplified and also transferred to other problems by means of reductions. For us, the starting point is the problem MAX 3XOR . We are able to establish a definability gap between the satisfiable instances of this and instances in which little more than [MATH] of the clauses can be satisified. The con... |
The methods for establishing this optimal initial gap are very different from that for the PCP theorem. We construct a [MATH] -locally satisfiable instance of MAX 3XOR which, by a random construction is at the same time highly unsatisfiable. We can then combine this with a construction adapted from |
to obtain an optimal gap that defeats any fixed counting width. This shows that no algorithm that is expressible in [MATH] can approximate MAX 3XOR within a constant above [MATH] , even on satisfiable instances. It should be pointed out that, although the inapproximability of MAX 3XOR above [MATH] matches algorithmic l... |
are distinguished from unsatisfiable ones by a polynomial-time algorithm. To show inapproximability for any constant greater than [MATH] one has to show that it is the almost satisifable ones that are indistinguishable from those that are highly unsatisfiable. This distinction supports our claim that our methods are ve... |
With such an optimal initial gap for MAX 3XOR in hand, we can then transfer it to other problems by means of reductions, just as in classical inapproximability. Our reductions have to preserve [MATH] |
definability and we mostly rely on first-order definable reductions. For one, the standard direct reduction from 3XOR to 3SAT is trivially first-order definable and gives an optimal undefinability gap for MAX 3SAT : no algorithm expressible in [MATH] can achieve an approximation ratio of [MATH] , even on satisfiable in... |
first-order reductions but this requires close examination and proof. We show that the long-code reductions from are definable in first-order logic. Such reductions have the merit of providing different constructions of optimal gaps for MAX 3XOR and MAX 3SAT starting at any initial gap whatsoever. In addition, the tech... |
, which is based on the same long-code reduction techniques as in , is first-order definable, showing that [MATH] cannot give an approximation better than [MATH] . It is possible that this could be improved to [MATH] using the recent breakthrough of |
but we leave this to future work. Preliminaries We use [MATH] to denote the [MATH] -element field. For any positive integer [MATH] , let [MATH] |
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