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This means that the number of agents moving from bin [MATH] to [MATH] is upper-bounded by [MATH] The bound value is assumed to be very small with consideration of mission environments such as the number of agents, the number of bins, and their topology. Otherwise, all the agents can be distributed over the bins very so... |
Regarding the convergence rate of a Markov chain, there are respective analytical methods depending on whether it is time-homogeneous or time-inhomogeneous. For a time-homogeneous Markov chain, if the matrix is irreducible, the second largest eigenvalue of the matrix is used as an index indicating its asymptotic conver... |
Particularly, this paper uses the following proper coefficient of ergodicity, amongst others: Definition 8 Coefficient of Ergodicity |
33 , pp. 136–137] ). Given a stochastic matrix [MATH] , a (proper) coefficient of ergodicity [MATH] can be defined as: [EQUATION] |
A coefficient of ergodicity is said to be proper if [MATH] if and only if [MATH] , where [MATH] is a row-stochastic vector. The convergence rate of a time-inhomogeneous Markov chain [MATH] [MATH] can be maximised by minimising [MATH] at each time instant [MATH] , thanks to 33 , Theorem 4.8, p.137] |
[MATH] Hence, the objective of the specific problem considered in this subsection can be defined as: find [MATH] such that [EQUATION] |
subject to ( R1 )-( R5 ) and ( 21 ). Remark 5 Advantages of the coefficient of ergodicity in ( 22 Other proper coefficients in 33 , p. 137] such as |
[EQUATION] or [EQUATION] may have the trivial case such that [MATH] (or [MATH] ) for some time instant [MATH] , when they are applied to this problem. This is because, given a strongly-connected topology [MATH] , there may exist a pair of bins [MATH] and [MATH] such that [MATH] or [MATH] [MATH] To avoid this trivial ca... |
instead utilises [MATH] as the proper coefficient of ergodicity, where [MATH] denotes the diameter of the underlying graph of [MATH] However, this implies that agents in bin [MATH] are required to additionally access the information from other bins beside [MATH] , causing additional communicational costs. The coefficie... |
Finding the optimal solution for the problem ( 23 ) is another challenging issue, which can be called fastest mixing Markov chain problem . Since the purpose of this section is to show an example of how to implement our proposed framework, we heuristically address this problem at this moment. |
Suppose that matrix [MATH] satisfying ( R1 )-( R5 ) is given, and the topology of bins is not fully-connected. Since the matrix is non-negative and there exists at least one zero-value entry in each column, the coefficient of ergodicity can be said as [MATH] Assuming that [MATH] , which is generally true due to the sma... |
because minimising the maximum diagonal element of a stochastic matrix is equivalent to maximising the minimum row-sum of its off-diagonal elements. |
We turn now to the constraints ( R1 )-( R5 ) and ( 21 ). In order to comply with ( R3 ), we initially set [MATH] , where [MATH] is a symmetric matrix that we will design now. The constraint ( 21 ), ( R4 ), and the symmetricity of [MATH] are integrated into the following constraint: [MATH] [MATH] |
[EQUATION] For ( R2 ) and ( R5 ), we set the diagonal entries of [MATH] as [EQUATION] This can be rewritten, with consideration of ( R1 ) (i.e., [MATH] ), as |
[EQUATION] Then, the reduced problem can be defined as: find [MATH] such that [EQUATION] subject to ( 24 ) and ( 25 ). The algorithm for this problem is shown in Algorithm If we neglect ( 25 ), an optimal solution can be obtained by making [MATH] equal to its upper bound of ( 24 ) (Line ). However, this solution may no... |
Algorithm 3 Subroutine of Algorithm (Line ) for P1 1: // Initialise [MATH] 2: [MATH] [MATH] 3: // Compute [MATH] satisfying ( 24 |
12: // Compute [MATH] 13: [MATH] [MATH] 14: [MATH] IV-C Example III: Local-information-based Quorum Model This subsection shows that the proposed framework is able to incorporate a quorum model, which is introduced in |
In this model, if a bin is overpopulated above a certain level of predefined threshold called quorum , the probabilities that agents in the bin move to neighbour bins are temporarily increased, rather than following given [MATH] This feature eventually brings an advantage to the convergence performance of the swarm. |
To this end, we set the secondary guidance matrix [MATH] as follows: [MATH] and [MATH] [EQUATION] This matrix makes agents in a bin equally disseminated over its neighbour bins. In addition, the secondary feedback gain [MATH] is defined as |
[EQUATION] where [MATH] is a design parameter, and [MATH] is the quorum for bin [MATH] The value of the gain is shown in Figure (b), varying depending on [MATH] and [MATH] As [MATH] becomes higher than the quorum, [MATH] gets close to [MATH] (i.e., [MATH] becomes more dominant than [MATH] ). The steepness of the functi... |
The existing quorum models in require each agent to know [MATH] , which implies that the total number of agents [MATH] should be tracked in real time. It could be possible that some agents in a swarm unexpectedly become faulted by internal or external effects during a mission, which hinders for other alive agents from ... |
Algorithm 4 Subroutine of Algorithm (Line 10 ) for the quorum-based method 1: Compute [MATH] [MATH] using ( 26 ); 2: Compute [MATH] using ( 27 ); |
Asynchronous Implementation A synchronous process induces extra time delays and inter-agent communications to make entire agents, who may have different timescales for obtaining new information and make decisions, remain in sync. Such unnecessary waiting time and communications may cause unfavourable effects on mission... |
In the previous sections, it was assumed that a swarm of agents act synchronously at every time instant. Here we show that the proposed framework allows agents to operate in an asynchronous manner, assuming that the union of underlying graphs of the corresponding Markov matrices across some time intervals is frequently... |
Algorithm 5 Asynchronous Construction of [MATH] (Substitute for Line of Algorithm 1: if [MATH] [MATH] then 2: Compute [MATH] as usual, [MATH] |
3: [MATH] [MATH] 4: [MATH] 5: else 6: [MATH] [MATH] [MATH] 7: end if Suppose that an algorithm to compute [MATH] that satisfies ( R1 )-( R5 ) in a synchronous environment is given (e.g., Algorithm or ). We propose an asynchronous implementation, as shown in Algorithm , which substitutes for Line in Algorithm We refer t... |
Hereafter, for the sake of differentiation from the original [MATH] generated in a synchronous environment, let us refer to the matrix resulted by Algorithm as asynchronous primary guidance matrix , denoted by [MATH] Accordingly, the asynchronous Markov matrix can be defined as: |
[EQUATION] Here, we show that this asynchronous Markov process also converges to the desired swarm distribution. Lemma 3 The matrix [MATH] , for every time instant [MATH] , satisfies the following properties: (1) row-stochastic; (2) all diagonal elements are positive, and all other elements are non-negative; and (3) [M... |
Proof. The matrix [MATH] is row-stochastic because of Line and in Algorithm Furthermore, given that [MATH] satisfies ( R2 ), the property (2) is valid for [MATH] because [MATH] for [MATH] |
Let us now turn to the property (3). For [MATH] it is trivial that [MATH] because of Line For [MATH] , it turns out from Algorithm that (i) [MATH] for [MATH] (ii) [MATH] for [MATH] |
and (iii) [MATH] We apply the findings into the following equation: [EQUATION] The first term of the right hand side becomes zero because of (i). Due to (ii) and the fact that [MATH] |
[MATH] , the second term becomes [MATH] The last term becomes [MATH] because of (iii). Putting all of them together, Equation ( 28 ) is equivalent to |
[MATH] Lemma 4 If the union of a set of underlying graphs of [MATH] is strongly-connected, then the matrix product [MATH] is irreducible. |
Proof. Since the union of a set of underlying graphs of [MATH] is strongly-connected, the underlying graph of [MATH] is also strongly-connected. Noting that every [MATH] [MATH] is a nonnegative [MATH] matrix and its diagonal elements are positive (by Lemma ), it follows from 36 , Lemma 2] that [MATH] , where [MATH] Thi... |
Theorem 3 Suppose that there exists an infinite sequence of non-overlapping time intervals [MATH] [MATH] , such that the union of underlying graphs of [MATH] in each interval is strongly-connected. Let the stochastic state of agent [MATH] at time instant [MATH] , governed by the corresponding Markov process from an arb... |
[MATH] Then, it holds that [MATH] pointwise for all agents, irrespective of the initial condition. Proof. Thanks to Lemma and the matrix product [MATH] for each time interval [MATH] satisfies ( R1 )-( R4 ). Therefore, one can prove this theorem by similarly following the proof of Theorem |
VI Numerical Experiments VI-A Effects of Primary Local-feedback Gain [MATH] Depending on the shape of primary feedback gain [MATH] , the performance of the proposed framework changes, especially with respect to convergence rate, fraction of transitioning agents, and residual convergence error. Let us first investigate ... |
We consider a scenario where a set of [MATH] agents are supposed to be distributed over an arena consisting of [MATH] bins, as depicted in Figure There are vertical and horizontal paths between adjacent bins. Note that the agents are allowed to move at most [MATH] paths away within a unit time instant. All the agents s... |
For the rigorous validation, the performance of the proposed algorithm will be compared with that of the GICA-based algorithm . To this end, |
[MATH] is set to be the same as the corresponding coefficient in 13 , Corollary 1] [EQUATION] where [MATH] is the maximum element of the travelling expense matrix [MATH] , and [MATH] is a user-design parameter. |
[MATH] is defined as a linear function based on the distance between bin [MATH] and [MATH] [EQUATION] where [MATH] is the minimum required number of paths from [MATH] to [MATH] |
[MATH] and [MATH] are user-design parameters. The agents are assumed to follow any shortest route when they transition between two bins. The design parameters are set as follows: [MATH] and [MATH] in ( 30 ); |
[MATH] in ( 29 ); [MATH] in ( ); [MATH] in ( 20 ); [MATH] in ( 18 ); and [MATH] in ( ). As a performance index for the closeness between the current swarm distribution [MATH] and [MATH] , we use Hellinger Distance , i.e., |
[EQUATION] Hellinger Distance is known as a “concept of measuring similarity between two distributions” and is utilised as a feedback gain in the existing work |
More importantly, to examine the effects of the shape of [MATH] , we set [MATH] in ( ) as [MATH] and [MATH] Figure reveals that the convergence rate can be traded off against the fraction of transitioning agents and the residual convergence error. As [MATH] becomes more concave, i.e. the value of [MATH] decreases, the ... |
VI-B Comparison with a GICA-based Method Let us now compare the LICA-based method for ( P1 ) with the GICA-based method in The scenario considered is the same as the one in the previous subsection except for [MATH] Note that [MATH] in Remark can control convergence rate, but is not discussed in |
. For the fair comparison, [MATH] is applied to both the methods. We conduct 100 runs of Monte Carlo experiments. Figure presents the results of one representative scenario and the statistical results of the Monte Carlo experiments are shown in Figure According to Figure (a), the convergence rate of the proposed method... |
Figure (c) shows that the cumulative travel expenses are smaller in the proposed method. The expenses by the proposed method and those by the compared method are [MATH] and [MATH] , respectively, and their ratio is [MATH] This is also confirmed by the statistical result in Figure (b). A possible explanation is that whe... |
VI-C Robustness in Asynchronous Environments This subsection investigates the effects of asynchronous environments in the proposed LICA-based method for ( P1 ) and compares them with those in the GICA-based method in |
. Hence, a realistic scenario where an asynchronous process is required is considered: in the scenario, it is assumed that agents in some bins cannot communicate for some reason (such bins are called blocked ) and thus other agents in normal bins have to perform their own process without waiting them. |
The proportion of blocked bins to the entire bins is set to be different values, i.e. [MATH] [MATH] [MATH] and [MATH] At each time instant, the corresponding proportion of bins are randomly selected as blocked bins. For the proposed framework, the asynchronous implementation in Section is built upon Algorithm In the GI... |
Figure illustrates the performance of each method: convergence rate, fraction of transitioning agents, and cumulative travel expenses. As the proportion of the blocked bins increases, the GICA-based method tends to have faster convergence speed, whereas it loses Desired Feature and thus increases cumulative travelling ... |
VI-D Demonstration of Example II and III This subsection demonstrates the LICA-based method for ( P2 ) (i.e., Algorithm ) and the quorum model (i.e., Algorithm ). For the former, we consider a scenario where [MATH] agents and an arena consisting of [MATH] bins are given. The arena is as depicted in Figure , where the a... |
For the latter, we build the quorum model upon the LICA-based method for ( P2 ). This can be a good strategy for a user who wants to achieve not only faster convergence rate but also lower unnecessary transitions after equilibrium, which are regulated by the upper flux bounds. Thus, in the same scenario described above... |
Figure (a) and (b) presents that the both approaches make the swarm converge to the desired swarm distribution. It is observed that the number of transitioning agents in the method for ( P2 ) are restricted because of the upper flux bound during the entire process. Meanwhile, the quorum-based method very quickly dissem... |
Figure (c) presents the maximum value amongst the number of transitioning agents via each (one-way) path. The red line indicates the actual result by the method for ( P2 ), while the green line indicates the corresponding probabilistic value (i.e., [MATH] ). It is shown that the stochastic decision policies reflect the... |
VII Conclusion This paper poposed a LICA-based closed-loop-type framework for probabilistic swarm distribution guidance. Since the feedback gains can be generated based on local information, agents have shorter and different timescales for using new information, and can incorporate an asynchronous decision-making proce... |
Future works include optimisation of [MATH] , which can mitigate the trade-off between convergence rate and residual error. In addition, it is expected that the communication cost required for the proposed framework can be reduced by incorporating a vision-based local density estimation |
Acknowledgement The authors gratefully acknowledge that this research was supported by International Joint Research Programme with Chungnam National University (No. EFA3004Z). Thanks to Sangjun Bae for supportive discussions. |
Appendix VII-A Regarding the Convergence Analysis in Theorem Definition 9 Irreducible A matrix is reducible if and only if its associated digraph is not strongly connected. A matrix that is not reducible is irreducible |
Definition 10 Primitive primitive matrix is a square nonnegative matrix [MATH] such that for every [MATH] [MATH] there exists [MATH] such that [MATH] |
Definition 11 Regular regular matrix is a stochastic matrix such that all the entries of some power of the matrix are positive. Definition 12 |
33 , pp.92, 149] Asymptotic Homogeneity “A sequence of stochastic matrices [MATH] [MATH] , is said to be asymptotically homogeneous (with respect to [MATH] if there exists a row-stochastic vector [MATH] such that [MATH] .” |
Definition 13 33 , pp.92, 149] Strong Ergodicity The matrix product [MATH] , formed from a sequence of stochastic matrices [MATH] [MATH] , is said to be |
strongly ergodic if for each [MATH] we get [MATH] , where [MATH] is a row-stochastic vector. Here, is called its unique limit vector (i.e., [MATH] ). |
Lemma 5 Given the requirements ( R1 )-( R4 ) are satisfied, [MATH] in Equation ( 11 ) has the following properties: 1. row-stochastic; |
2. irreducible; 3. all diagonal elements are positive, and all other elements are non-negative; 4. there is a positive lower bound [MATH] such that [MATH] (Note that [MATH] refers to the minimum of the positive elements); |
5. asymptotically homogeneous with respect to [MATH] In addition, [MATH] in Equation ( 12 ) has the following properties: 6. irreducible; |
7. all diagonal elements are positive, and all other elements are non-negative; 8. primitive 39 , Lemma 8.5.4, p.541] Proof. This lemma can be proved by similarly following the mathematical development for 13 , Theorem 4] |
[MATH] is row-stochastic because [MATH] [MATH] is irreducible and [MATH] is always less than 1, thus [MATH] is also irreducible (i.e., [MATH] if [MATH] ). The property 3) is true because |
[MATH] [MATH] , and [MATH] is also a non-negative matrix such that its diagonal elements are positive. The property 4) is implied by either the property 2) or 3). From the definition of [MATH] , it follows that [MATH] (because of [MATH] ), and thereby [MATH] Hence, [MATH] , and the property 5) is valid. |
Let us now turn to [MATH] . It is irreducible due to the fact that if [MATH] for some [MATH] and [MATH] , then the corresponding element [MATH] is greater or equal to the product of positive diagonal elements and [MATH] and the property 2). The property 7) is true because [MATH] is a product of nonnegative matrices whe... |
# Source: arxiv 1711.09442 # Title: Quantum Artificial Life in an IBM Quantum Computer # Sections: all # Downloaded: 2026-03-03T01:55:24.621944+00:00 |
Quantum Artificial Life in an IBM Quantum Computer Abstract We present the first experimental realization of a quantum artificial life algorithm in a quantum computer. The quantum biomimetic protocol encodes tailored quantum behaviors belonging to living systems, namely, self-replication, mutation, interaction between ... |
Introduction As described by Deutsch, a quantum computer is a device that intends to fulfill the Deutsch-Church-Turing principle, namely, to efficiently simulate a finitely realizable physical system in the framework of quantum mechanics qc . In this context, quantum supremacy would be reached when a quantum processor ... |
The keyword “quantum” has overflowed the limits to which was initially constrained and, currently, incessantly spreads through the interdisciplinary scientific literature. Indeed, it is a source of inspiration for the breeding extensions of already existing models with their quantum counterparts qh1 BennettBrassard Eke... |
In the particular scenario of artificial life, simple models of organisms are able to undergo most common stages of life in a controlled virtual environment al1 al2 al3 . When extending this to the quantum realm, particularities of quantum physics, such as its limitation to linear dynamics, the no-cloning theorem, or t... |
In this article, we report the first experimental implementation of a model for quantum artificial life qbiom2 into a quantum computer. To this end, we make use of the facilities provided by the IBM ibmqx4 quantum computing chip ibm . This work should be aligned with the ramping developments in classical and quantum ma... |
II Results We begin with a brief description of the model for quantum artificial life qbiom2 , whose most important elements are the quantum living units or individuals. Each of them is expressed in terms of two qubits that we call genotype and phenotype. The genotype contains the information describing the type of liv... |
The goal of the proposed model is to reproduce the characteristic processes of Darwinian evolution, adapted to the language of quantum algorithms and quantum computing. The self-replication mechanism is based on two partial quantum cloning events, an operation that entangles either the genotype or the phenotype with a ... |
The first step for this implementation is to express each of the building blocks of the previous paragraph in terms of the quantum gates available in the superconducting circuit architecture of IBM cloud quantum computer ibm . Since we have selected [MATH] as the observable to clone, every partial quantum cloning event... |
[EQUATION] The gate [MATH] acting on genotype qubits can be used for the mutation events. Ideally, and in order to emulate their randomness both in the phase [MATH] and in the presence or absence of the event, we could design the experimental runs following a classical program. For making the procedure tractable, we co... |
This is a useful formula for designing the quantum circuit in an experimental platform that only allows a single direction for the implementation of the [MATH] |
II.1 Experiments Interaction between two individuals. We start with a quantum circuit designed for reproducing the dynamics of two interacting individuals. Two precursor genotypes are initialized in [MATH] and [MATH] with [MATH] . Afterwards, both individuals are completed by copying the genotype qubits in blank states... |
[EQUATION] We now apply the interaction gate [MATH] to conclude this building block, [EQUATION] Notice that the interaction fully exchanges the phenotypes, [MATH] and [MATH] , that are now equal to the opposite genotype, [MATH] and [MATH] |
The experiment is planned to reduce the total errors induced by the use of two-qubit gates. Consequently, we have reordered the initial Hilbert space [MATH] , where [MATH] is genotype and [MATH] is phenotype, as [MATH] and assigned each of these qubits to the experimental ones [MATH] . See Fig. for the remaining quantu... |
The results, in Table , agree with the ideal case with a [MATH] fidelity according to [MATH] , that compares the probability distribution obtained when measuring in the computational basis with the theoretical prediction. Therefore, this result is valid, but not equivalent to the one that is expected when the complete ... |
r1 . The expectation values extracted from the data show a reasonable overlap between [MATH] and [MATH] , as expected, and a considerable distance between [MATH] and [MATH] |
Interaction with the environment. In this round of experiments we test the combination of partial quantum cloning events and dissipation. A precursor genotype is initialized in [MATH] , and the individual completed with a first partial quantum cloning event via [MATH] and a blank state, [MATH] . Then, a single qubit ro... |
For the self-replication instance, there is an additional property of the model that only arises when measuring some purely quantum correlations of the system. The partial quantum cloning operation entangles the qubits which are involved on it, transmitting [MATH] of the original state into [MATH] . Note that this data... |
Even if this implementation does not coincide with the time evolution presented in the original model, it is able to emulate its results when only focusing on the [MATH] or [MATH] basis, but not to compare both measurements in general. Accordingly, if the lifetimes of each living qubits undergo a similar dynamics to th... |
[EQUATION] where [MATH] is the [MATH] component in the precursor genotype. Given that there is no solution for [MATH] and [MATH] which is independent of [MATH] , the method of single-qubit gates for mimicking the dissipation is not valid as a general protocol, because it has to be tuned for each case. Nevertheless, the... |
In more practical terms, the implementations summarized in Table II and Table III are realistic, but not compatible between them, because both can be associated to dissipative dynamics but with different representative parameters, as we have seen in Eq. ( ). Furthermore, we believe that the ideal realization of the exp... |
The implementation of mutations requires to combine the outcome of different designs of quantum circuit diagrams and, therefore, experimental runs. In this case, we consider that a mutation event, which can affect both individuals, is simulated with a [MATH] . The complete result is achieved when gathering data from [M... |
Realization of the complete model of quantum artificial life. The last round of experiments is devoted to the reproduction of the aggregate of properties in the quantum artificial life algorithm. In order to maintain the fidelity in values that allow us to claim that the experiment is indeed behaving according to the p... |
III Discussion III.1 Quantum vs Classical A natural method to evaluate the Quantum Artificial Life framework is to clearly describe on the similarities and differences between the quantum model and a classical analogue approach. On the one hand, all the indicators based in measurements on the [MATH] basis can be reprod... |
One can show that a classical counterpart of this model would not be able to store the information of the time correlations. The correlations we are interested in are originated in the initial nonzero value of [MATH] the precursor genotype, and afterwards propagated in each partial quantum cloning event because of the ... |
III.2 Experimental Errors Regarding errors in the experimental protocol, even if the fidelities achieved are satisfactory, they do not correspond to the fidelities of the complete quantum state. In this sense, the prediction for the number of events to measure is done by simply multiplying the probability distribution ... |
In parallel, the assignment between the simulated and the simulating Hilbert spaces is designed to maximize the fidelity according to the calibration parameters provided by IBM. Nevertheless, the recalibration of the circuit changes the gate and readout errors, so we reevaluate our circuit according to the new paramete... |
Despite the different factors degrading the implementation, the performed experiments reproduce the characteristic properties of the sought quantum natural selection scenario. We have observed how the partial quantum cloning events allow us to inherit the information of [MATH] from qubit to qubit, and use this property... |
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