1 Introduction
Quantum computation (QC) aims to profit from the quantum properties of elementary particles to devise new, more efficient ways for representing and manipulating information. A number of paradigms have been proposed as promising candidates for QC. One of these paradigms is adiabatic quantum computing [23, 4] and perhaps the best known implementation of this approach for practical computation is the DWave quantum machine [6].
In very simple terms, the DWave architecture consists of an array of qubits coupled following a predefined topology (Chimera graph). Qubits have associated weights and each coupler has an associated strength. Current DWave quantum annealing processors are designed to minimize the energy of an Ising spin configuration whose pairwise interactions lie on the edges of a Chimera graph. DWave computers have served as an excellent testbed for investigating whether QC is actually possible. In particular, many studies have focused on the behavior of the quantum annealing (QA) optimization method
[13, 5]. QA is similar to simulated annealing (SA) but transitions between states (solutions) do not depend on thermodynamic fluctuations. These transitions are determined by quantum fluctuations.Although other optimization problems of arbitrary pairwise interaction structure and manageable size can be embedded into the DWave topology, a more straightforward approach to study the performance of this QC paradigm is to address problems whose structure has been defined on the Chimera graph. Understanding which are the characteristics of problems defined on Chimera graphs is thus a very important issue. The behavior of a number of algorithms have been investigated for problems with Chimera graph structure. On these problems, QA has been compared to optimization methods such as Tabu search [17], SA [14] and parallel tempering (PT) [28].
In this paper we propose a through investigation of evolutionary algorithms (EAs) for Ising spin glass problems defined on Chimera graphs. Our goal is to determine to what extent the search principles in which EAs are based on can be valuable for the solution of these problems. We argue that EAs can and should be used not only as solvers but also as a tool to better understand the characteristics of relevant problems. In particular, identifying sources of problem difficulty that are common to optimization approaches with completely different search strategies can advance the understanding of the problems and of the optimizers. We show how EAs can be used with these goals in mind and provide evidence of the benefits of this approach.
To study the target problems, we apply hybrid genetic algorithms (GAs) and variants of estimation of distribution algorithms (EDAs) based on factorizations [3, 18, 26]. This strategy allows us to investigate EAs that are blind to the problem structure and other variants that exploit the information about this structure for a more efficient search. In addition to elucidate which is the difficulty that Chimeragraph based spin glass instances pose to EAs, we expect to obtain insights from the behavior of EAs on these instances. To the knowledge of the authors problems defined on Chimera graphs have not been previously addressed using EAs.
The paper is organized as follows: The next section presents some necessary background on Chimera graphs and the Ising model. Section 3 discusses related work on the relevant questions treated in the paper. Section 4 presents the EAs used to optimize the instances. Section 5 introduces the experimental framework used to investigate our hypothesis, presents the numerical results and discusses our findings. Section 6 concludes the paper.
2 Background
2.1 Ising model
A Hamiltonian function serves to describe how the energy of a physical process depends on the state of the system’s particles and their interactions. The generalized Ising model is described by the Hamiltonian shown in Equation (1) where is the set of sites called a lattice. Each spin variable at site either takes the value or . One specific choice of values for the spin variables is called a configuration. The constants are the interaction coefficients or couplings. The ground state is the configuration with minimum energy.
(1) 
where the first sum is over pairs of spins adjacent in the lattice .
Spin couplings can take values from an a priori defined set of values and both the choice of the set and the distribution with which the couplings are sampled from it influence the characteristics of the problem and its difficulty for the optimizers. In this paper we use couplings selected from Sidon sets of the type as advocated in [14]. The rationale behind this choice of the couplings is explained in Section 5.1. We consider instances for which .
2.2 DWave architecture and Chimera graphs
The DWave architecture follows a regular repeating pattern to tile out a processor [6]. The building blocks of the Chimera graph are symmetric subgraphs referred to as graphs. The eight qubits included in the graph are connected as bipartite graphs. One example of two connected graphs is shown in Figure 1. In addition to bipartite connections, in each group, the four left nodes are also connected to their respective north/south grid neighbors and the right nodes are connected to their east/west neighbors. Thus internal nodes have degree and boundary nodes have degree . The Chimera graph is formed by subgraphs.
3 Related work
The questions treated in this paper are related to different lines of research in the fields EAs and also to studies on the behavior of different types of optimizers for Chimera graphs. Therefore, we organize the review of previous work according to these relevant topics.
3.1 Previous EA approaches to Ising and other problems from Physics
Ising spin glasses are not only relevant because of the role they play in Physics. They represent a formidable problem to investigate the capabilities of different EAs and as such they have been investigated in a number of papers.
GAs were early applied to Ising models [1, 15, 21]. Some of these works used knowledge about the topology of the problem to design the genetic operators. In [1], a blockcrossover, able to exploit the 2dimensional structure of the instances was proposed. EDAs have been also used to solve Ising models [19, 22, 25]. In [19], the Bayesian optimization algorithm (BOA) was applied to instances with and Gaussian couplings. A deterministic hill climber based on singlebit flips was used to solve problems of up to spins. To solve bigger instances of and spins, a more sophisticated cluster exact approximation method was required. In general, successful EDAs applications to large Ising models are in fact hybrid algorithms that incorporate local search methods.
While Ising problems addressed with EAs have been generally defined on regular 2d and 3d grids, other topologies have been also used. In [20], the behavior of EAs was investigated on onedimensional spin glass with powerlaw interactions. In [9], the impact of different network topologies (e.g. grids, smallworld networks and random graphs) of the underlying problem on the models learned by EDAs was investigated. We did not find previous reports on the application of EAs to Ising models defined on Chimera graphs. It is important to notice that the Chimera topology considerably departs from regular grids.
3.2 Exploiting the problem structure
Using problem information to improve EAs has been always a relevant issue in evolutionary computation (EC). One relevant question is whether and how knowledge about the underlying topology of the optimization problem can be used to increase the efficiency of the EA approaches to this problem. In this paper, we investigate this issue in three different ways: 1) Using the Chimera structure to design buildingblock wise crossover operators where the building blocks corresponds to groups of variables in the same
subgraph. 2) Using factorized EDAs in which the factors comprise variables in each subgraph. 3) Biasing the search for treebased EDAs to include dependencies between variables that are connected in the Chimera structure.Buildingblock wise crossover is at the core of successful probabilistic model building GAs like the extended compact GA [11]. EDAs that explicitly use to different extent the regular grid structure of the Ising problems have been previously proposed in [22, 25]. EDAs that bias the learning of probabilistic trees using a priori information about the problem has shown to be more efficient [2].
Other methods that use information about the structure of the graphs to solve Ising problems defined on Chimera graphs have been also proposed in the field of Physics. In [29], a heuristic method that finds optimal configurations of local clusters of spins as a way to reach the ground state is presented. The target clusters are those that are strongly coupled to each other and more weakly coupled to the rest of the system. Although the factorized EDA we use in our comparisons use information about clusters of solutions, these clusters depend on the Chimera graph and not on the strength of the couplings.
The Hamzede FreitasSelby (HFS) algorithm [10, 24], uses a subgraphbased sampling method for Isingtype models with frustration. A collection of subgraphs induced by the original topology are used instead of single spins for a more efficient sampling with GS and PT. In Selby [24], experiments on Chimera graphs were done using trees as the induced subgraphs. In this paper, we use EDAs based on tree models. However, the trees are learned from the selected solutions by applying statistical methods.
3.3 Fitness landscape analysis and investigation of fitness difficulty
A number of papers [7, 17] have investigated the behavior of classical optimizers on Ising instances defined on Chimera graphs. In [17], three conventional software solvers: CPLEX, a variant of Tabu, and a branchandbound search, were compared to QA in Chimerastructured problems.
In [12], different variants of SA were evaluated on Ising problems defined on Chimera graphs. However, the focus of the paper was not on the comparison of SA with other algorithms but on the development of fast SA implementations. In [14], an extensive comparison between SA and QA is presented for Ising problems with Chimera topology. This work identified the important effect that the choice of the couplings for the Ising instances could have in the comparison between optimizers for problems defined on Chimeragraphs. Instances generated using Sidon sets^{1}^{1}1In a Sidon set, the sum of two members of the set gives a number that is not part of the set were proposed as harder to optimize. Also in [28], a particular class of Sidon instances () was used as a testbed for evaluating the behavior of QA which was compared to PT.
4 Evolutionary optimization approaches
Let
denote a vector of discrete random variables where
is the number of nodes of the Chimera graph. We useto denote an assignment to the variables. In our problem representation, each binary variable represents one spin configuration of a node in the Chimera graph. In the chromosome representation, variables are ordered according to the position of the nodes in the Chimera graph. For instance, the first
variables would correspond to the nodes shown in Figure 1. Therefore, the variables that represent nodes that are connected in the Chimera graph are, in most of the cases, relatively close in the chromosome. The fitness function used to evaluate the solutions is the one represented by Equation (1).4.1 Benchmarked algorithms
Algorithm 4.1 shows the general pseudocode of all the EAs tried in the paper. Population size and truncation selection with parameter were used.
EA
Set . Generate an initial population of random solutions.
For each solution, apply a greedy local search method and output local optimum.
Select from population a set of points using truncation selection.
Generate a new population from applying the variator operator of choice.
Apply random bitflip mutation to solutions in .
Termination criteria are met.
In the design of EAs, certain assumptions about the structure of the optimization problem are usually implicitly or explicitly made. The extent to which these assumptions agree with the characteristics of the problem usually determine the success of the EA. We have selected six EAs by considering how their mechanisms to explore the space of solutions may be related with the structure of the Chimera graphs.

GA with 1point crossover (1PCXGA): Simple GA algorithm with a crossover probability of
and where a crossover point is randomly selected between positions and . The two offspring are created by taking one segment from each parent. 
GA with uniform CX (uCXGA): Similar to 1PCXGA but the alleles of the offspring are randomly taken from each of the two parents with probability .

GA with bitwise CX (BWCXGA): Similar to uCXGA but instead of single bits, blocks of variables are taken from each parent. These blocks correspond to variables that are related by the Chimera graph. In particular, there is one block for each subgraph involving variables and for each edge joining these subgraphs.

Factorized distribution algorithm (FDA): An EDA that uses the same structure as BWCXGA. Marginal probabilities for all configurations of the blocks are learned from the selected solutions and new solutions are generated sampling from a junction tree [18] constructed from this factorization.

Treebased EDA (TreeEDA): An EDA that learns the structure of the graphical model from the matrix of mutual information between all possible pairs of variables [3]. The pairs with the strongest mutual information are used to build a tree structure from which solutions are sampled.

Treebased EDA with a priori information (TreeEDA): Idem to TreeEDA but only pairs of variables connected in the Chimera topology are used to learn the probabilistic model.
4.2 Efficient local search
In step of Algorithm 4.1 a greedy local search method was added as part of the evaluation process. Previous works [19, 22] have shown that without the inclusion of local optimizers, EAs are not expected to be competitive algorithms for large Ising spin glass problems.
The details of the greedy local search implemented are shown in Algorithm 4.2. One main characteristic of the method is that the local fitness value of the current solution are stored in memory and only local computations are needed to evaluate the effect of bit flips. Therefore, only a fraction of computations are needed to evaluate the possible bitflips possible from the current solution. In addition, when the initial solution is a local optimum and no bitflip improves the current fitness value, the local optimizer allows a predefined number of transitions () to solutions with lower fitness. In these cases, the bitflip that decreases the fitness the least is selected. This mechanism, similar to one of the components of Tabu search was designed as a way to help the algorithm to escape from local optima. However, the local optimizer does not implement a memory or any other type of more sophisticated components. The local optimizer stops when the current solution can not be improved or the maximum number of transitions to solutions of lower fitness has been consumed.
Greedy search
Set current solution to initial solution.
Initialize number of accepted negative .
Evaluate the possible assignments for the variables of the initial solution keeping other variables fixed.
Select the variable whose bitflip improves the fitness the most
If improvement is negative, initial solution was already improved or
Return current best solution
Else if improvement is negative, initial solution has not been improved, and .
Bitflip in current solution.
Update local fitness values for variable and all its neighbors in the Chimera graph.
After solutions have been evaluated a random bitflip mutation operator was applied. Extensive preliminary experiments showed that this way of infusing diversity in the population was a required ingredient for avoiding early convergence to poor solutions. More remarkably, a high mutation rate of showed to produce the best results. Notice, that all the EAs are compared in the same conditions. The only difference between the implementation of the six EAs is in step of Algorithm 4.1. This step comprises the recombination operator for the GAs and the learning and sampling step for EDAs. The termination criteria for all algorithms was a maximum of generations or reaching a low diversity in the population ( or less genotypically different individuals).
4.3 Structural hypothesis and EAs
The choice of the EAs has been made with the aim of evaluating different questions about the best EA approach to problems with a Chimera structure. We briefly state these questions and their relation to the algorithms.

How relevant is the impact of potential building block disruption? 1PCXGA and uCXGA differ only in the type of crossover mechanism they use. Uniform crossover is a more disruptive operator. By evaluating the difference in the performance of these two algorithms we can have a rough idea of the importance of respecting the interactions of the problem.

Which is the added gain of considering the underlying structure of the Chimera graph in the design of the crossover operator? By comparing BWCXGA to 1PCXGA we can measure if a more informed choice of the information to exchange between the parents has a strong effect in the performance of the GAs.

Can structuredinformed factorized approximations of the selected solutions make a difference over “blind” and more intelligent crossover operators? By comparing the FDA, which uses the blocks of variables related in the Chimera graph as factors, to all other GAs we can determine if using problem information the way EDAs do makes a difference for this problem.

Should EDAs learn the structure of the interactions from the data instead of inferring them from the Chimera structure? Are pairwise interactions sufficient to solve the problem? By comparing TreeEDA to FDA and GAs we can find answers to these questions.

Are the pairwise interactions determined by the Chimera graph sufficient to solve the problem? The comparison between the performance of TreeEDA and TreeEDA will help to answer this question.
5 Experiments
The goal of the experiments is twofold. 1) We want to evaluate the difference in the behavior of the evolutionary algorithms and whether these differences offer clues about the characteristics of the Chimera instances. 2) We would like to identify which characteristics of the instances (descriptors) have an impact in the performance of EAs.
5.1 Experimental benchmark
To evaluate the behavior of the algorithms, Ising instances defined on the Chimera graph were used. These instances were proposed in [14, 28] and investigated to evaluate the behavior of QA in the DWave quantum annealer. Instances were generated with couplings defined in the Sidon set.
There are a number of reasons why the Sidon set instances have been particularly useful for studies in Physics [14, 28, 27] and relevant for us: 1) All Instances have unique (two if spin reversal symmetry is considered) ground states and all spins have nonzero local fields (by carefully selecting the combination of bonds), these features make instances harder than random bimodal instances. 2) With Sidon interactions, there is full control of the number of ground states and low lying excited sates, even energy gaps, so it is possible to study resilience of instances with more precision. 3) With full control of many parameters of the instances, it would be possible to identify the important ones that are related to the behavior of different algorithms.
For these instances, the ground states and first exited states with equal probabilities have been found by applying the isoenergetic cluster algorithm [27] and running millions of Monte Carlo Sweeps for each instance. Recent results [16] show that the isoenergetic cluster algorithm is one of the few algorithms that scales better than quantum annealing on Google instances [8].
We use a number of descriptors that characterize the instances. These descriptors are derived from an analysis of the landscape of the spinglass order parameter distribution for the instances. overlap distributions is the proxy to the complexity of energy landscape of an instance. It measures the average Hamming distance between two solutions randomly sampled from the problem’s lowtemperature Boltzmann distribution. This approach can be used to identify the instances with tall and thin energy barriers [16]. We considered the following descriptors:

GRstate: Fitness value of the ground state solution.

dJ+dh_RES: Resilience of instance to coupler and qubit noise (upper bound success probability). Quantifies the robustness of groundstate configurations to noise in the DWave device.

PEAK_1, PEAK_2: Position of the highest and second highest peaks in the distribution plot.

HEIGHT_1,HEIGHT_2: Height of the highest and second highest peaks in the distribution plot.

H_SCORE: Ratio between the height of second highest peak and the height of highest peak.

NOISE_SCR: Ratio between the height of third highest peak and the height of second highest peak.
runs were executed for all the algorithms except for TreeEDA and TreeEDA for which, due to the high computational time spent by the algorithms, only runs were executed.
5.2 Behavior of EAs
Table 1 shows the number of instances for which the ground state was found in percentage of the runs. For , the table shows the number of instances whose optima were found at least once in all the runs. It can be seen that the best results were achieved by GAs over EDAs, with the best absolute results achieved by 1PCXGA. These results are better detailed in Figure 2 where instances are sorted according to the success rate reached by each algorithm. BWCXGA and uCXGA have a similar behavior, while these algorithms are more efficient than TreeEDA and TreeEDA for finding the ground state of the time or less frequently, the EDAs find more instances with and higher success rate, a fact that can be also observed in Table 1.
r (%)  1PCX  UCX  BWCX  FDA  Tree  Treer 

0.25  991  935  940  543  821  832 
1  982  881  878  322  821  832 
10  868  480  471  45  474  462 
25  645  188  199  4  218  212 
50  274  37  38  1  65  63 
75  54  6  4  0  12  13 
90  13  1  0  0  3  2 
The success rate of the algorithms does not provide the whole picture about their behavior since some EAs may exhibit a high variability being able to achieve high quality or poor solutions depending on the instances. Therefore, for each instance, we applied a multiple comparison statistical test to look for significant differences between algorithms using the best solutions reached in
runs (i.e. not the number of times that the optimum was found in these runs). The Kruskal Wallis test was applied first, and by applying a posthoc test we looked for statistical differences between each pair of algorithms. A Bonferroni correction was added to compensate for multiple comparisons. All tests used as pvalue
.Table 2 summarizes the results of the pairwise tests for all instances. In the table, cell indicates the number of instances for which algorithm in row was significantly better than algorithm in column . For example, algorithm 1PCXGA was significantly better than FDA for the instances. From the analysis of Table 2 it is clear that 1PCXGA significantly outperforms all other algorithms. There are not significant differences between the pair of algorithms (uCXGA,BWCXGA) and the pair (TreeEDA,TreeEDA). This seems to indicate that using information about the structure of the problem does not provide any advantage for the search, at least this is the case for GA with uniform crossover and TreeEDA. Also, a conclusion from the analysis is that FDA is not a good choice for this problem.
Alg.  1PCX  UCX  BWCX  FDA  Tree  Treer 

1PCX    862  858  1000  671  707 
UCX  0    0  973  33  31 
BWCX  0  0    974  39  41 
FDA  0  0  0    0  0 
Tree  2  201  196  980    0 
Treer  2  182  173  971  0   
Figure 3 shows the average computational time of all the algorithms across all instances. It can be appreciated that 1PCXGA is also the fastest among all the EAs compared. It is slightly faster than the other two GAs because uniform crossover requires the systematic generation of random numbers during the crossover and in 1PCXGA only one random number has to be generated for each crossover. All EDAs require a higher computational time than any GA. In particular TreeEDA is approximately times slower than 1PCXGA. As expected, since TreeEDAs need to learn the structure of the model from data, they are more computationally costly than FDA. Also, there is a clear gain in efficiency in TreeEDA over TreeEDA. This gain is due to considering for the construction of the tree only the pairwise relationships that exist in the Chimera graph.
5.3 Relationship with instance descriptors
Another important question is to determine whether and how is the performance of the EAs related to the characteristics of the instances. This issue is particularly relevant since the Sidon instances were originally engineered to investigate the behavior of QA on the DWave computers. Unveiling this type of relationship could help to find links between the behavior of EAs and optimizers that use completely different search mechanisms. In addition, by investigating this relationship we can determine whether the instance descriptors have a similar impact on all EAs or some of the descriptors are better signatures of behavior for some algorithms than for others.
Descriptors  1PCX  UCX  BWCX  FDA  Tree  Treer 

GRstate  0.21  0.17  0.17  0.14  0.17  0.18 
dJ+dh_RES  0.30  0.22  0.22  0.07  0.16  0.16 
PEAK_1  0.16  0.11  0.12  0.03  0.12  0.12 
HEIGHT_1  0.20  0.17  0.17  0.08  0.19  0.18 
PEAK_2  0.19  0.13  0.12  0.00  0.07  0.07 
HEIGHT_2  0.05  0.02  0.02  0.00  0.06  0.04 
H_SCORE  0.15  0.13  0.13  0.07  0.17  0.15 
NOISE_SCR  0.32  0.27  0.27  0.09  0.20  0.19 
We computed the Pearson’s correlations between the success rates of EAs and the descriptors of the instances. This information is shown in Table 3 where only a small number of correlations were not statistically significant using . Most of them involve algorithm FDA or descriptor . The strongest correlations were found for descriptors dJ+dh_RES and NOISE_SCR. Figure 4 shows the patterns of the relationships between these descriptors and the success rate of 1PCXGA for all instances. Also included in Figure 4 is the relationship for descriptor , which showed significant correlations for all GAs but no significant correlations for any of the EDAs. The strong correlation between success rates and dJ+dh_RES seems to suggest that the more the first excited states (low resilience), then there will be more local minima where the algorithm will likely get trapped. Thus these instances are harder. Also the strong anticorrelation between success rates and NOISE_SCR suggests that instances with multiple peaks (high NOISE_SCR) are typically harder than the ones with one peak.
5.4 Comparison with Simulated Annealing
As a final step we compared the results of the EAs to results achieved by SA. We ran two variants of SA proposed in [12] and named and . Both variants are implemented for spin glasses with fixed number of neighbors but is conceived to take advantage of the structure bipartite graphs. The two variants were run with the same parameters: Number SA sweeps , , , where is the inverse of the temperature, and and are the initial and final parameters of the linear schedule used for annealing. As in the case of the EAs these parameters are not expected to be optimal for all instances but we checked that increasing the number sweeps did not improve the results significantly. For each Ising instance, repetitions of the algorithm were executed and from these runs we computed the success rate.
Results for were very poor and therefore we present results here for . Figure 5 shows the correlation between the success rate of SA and 1PCXGA. SA achieved a success rate above for only instances (versus for 1PCXGA). However, it was able to find the optimum for all the instances at least once in repetitions. SA is also orders of magnitude faster than EAs. It is very difficult to compare both optimizers since the way they apply partial evaluation and combine it with full evaluation of the solutions differ between the algorithms. Our results show that GAs are at least competitive with SA. More importantly, the analysis of Figure 5 reveals that while there is a correlation between the hardness of the instances for both optimizers, 1PCXGA exhibits a wider variability in its behavior and the sources of difficulty for both methods are not completely the same. Further experimental work is needed, using other problem benchmarks to assess the differences in the behavior between SA and 1PCXGA.
6 Conclusions
In this paper we have investigated for the first time the behavior of EAs on problems defined of the Chimera instances used by DWave architectures. We have shown that a simple GA with onepoint crossover is able to solve of the instances considered although the success rate of the algorithm depends on the instances. Our results show that EAs that use probabilistic modeling of the solutions dot not produce an improvement over methods that do not incorporate any type of modeling. To some extent this was an unexpected result because different variants of EDAs had shown good results for Ising problems defined on other topologies. Using problem information does not provide improvements in terms of success rate, although by restricting the number pairwise dependencies to the edges of the Chimera graph important gains in terms of computational time were achieved by TreeEDAr over TreeEDA.
We have also identified a number of instance descriptors which are correlated with the behavior of the algorithms. This could serve as a first step for a more complete characterization of the impact that certain features of the Ising instances have in the performance of EAs. An analysis of the impact of the same features for other optimizers could help to understand how different methods explore the space of solutions to identify the optimum.
Finally, although constructing instances whose pattern of interactions match that of the DWave architecture is a sensible way to investigate the performance of QA on the spin glass problem, it is not completely clear how the topological restrictions can affect the complexity of the instances and bias the behavior of other optimization algorithms applied to these instances. Our work could be used to advance the understanding of this and related issues.
7 Acknowledgments
R. Santana’s work has received support by the IT60913 program (Basque Government) and TIN201341272P (Spanish Ministry of Science and Innovation). H.G.K acknowledges support from the National Science Foundation (Grant No. DMR1151387). The work of H.G.K. is supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via MIT Lincoln Laboratory Air Force Contract No. FA872105C0002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon.
References
 [1] C. A. Anderson, K. F. Jones, and J. Ryan. A twodimensional genetic algorithm for the Ising problem. Complex Systems, 5(3):327–334, 1991.
 [2] S. Baluja. Incorporating a priori knowledge in probabilisticmodel based optimization. In M. Pelikan, K. Sastry, and E. CantúPaz, editors, Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications, Studies in Computational Intelligence, pages 205–222. Springer, 2006.

[3]
S. Baluja and S. Davies.
Using optimal dependencytrees for combinatorial optimization:
Learning the structure of the search space.
In
Proceedings of the 14th International Conference on Machine Learning
, pages 30–38, 1997.  [4] J. Brooke, D. Bitko, R. T. F., and G. Aeppli. Quantum annealing of a disordered magnet. Science, 284(5415):779–781, 1999.
 [5] B. K. Chakrabarti and A. Das. Transverse Ising model, glass and quantum annealing. In Quantum Annealing and Other Optimization Methods, pages 1–36. Springer, 2005.
 [6] E. D. Dahl. Programming with DWave: Map coloring problem. White paper, DWave. The Quantum Computing Company, 2013.
 [7] S. Dash. A note on QUBO instances defined on Chimera graphs. arXiv preprint arXiv:1306.1202, 2013.
 [8] V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Babbush, V. Smelyanskiy, J. Martinis, and H. Neven. What is the computational value of finite range tunneling? arXiv preprint arXiv:1512.02206, 2015.

[9]
C. Echegoyen, A. Mendiburu, R. Santana, and J. A. Lozano.
Estimation of Bayesian networks algorithms in a class of complex networks.
In Proceedings of the 2010 Congress on Evolutionary Computation CEC2010, Barcelone, Spain, 2010. IEEE Press.  [10] F. Hamze and N. de Freitas. From fields to trees. In Uncertainty in Artificial Intelligence (UAI), pages 243–250, Arlington, Virginia, 2004. AUAI Press.
 [11] G. R. Harik, F. G. Lobo, and K. Sastry. Linkage learning via probabilistic modeling in the ECGA. In M. Pelikan, K. Sastry, and E. CantúPaz, editors, Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications, Studies in Computational Intelligence, pages 39–62. Springer, 2006.
 [12] S. V. Isakov, I. N. Zintchenko, T. F. Rønnow, and M. Troyer. Optimised simulated annealing for Ising spin glasses. Computer Physics Communications, 192:265–271, 2015.
 [13] T. Kadowaki and H. Nishimori. Quantum annealing in the transverse Ising model. Physical Review E, 58(5):5355, 1998.
 [14] H. G. Katzgraber, F. Hamze, Z. Zhu, A. J. Ochoa, and H. MunozBauza. Seeking quantum speedup through spin glasses: The good, the bad, and the ugly*. Phys. Rev. X, 5:031026, Sep 2015.
 [15] A. Maksymowicz, J. Galletly, M. Magdon, and I. Maksymowicz. Genetic algorithm approach for Ising model. Journal of magnetism and magnetic materials, 133(1):40–41, 1994.
 [16] S. Mandrà, Z. Zhu, W. Wang, A. PerdomoOrtiz, and H. G. Katzgraber. Strengths and weaknesses of weakstrong cluster problems: A detailed overview of stateoftheart classical heuristics vs quantum approaches. arXiv preprint arXiv:1604.01746, 2016.
 [17] C. C. McGeoch and C. Wang. Experimental evaluation of an adiabiatic quantum system for combinatorial optimization. In Proceedings of the ACM International Conference on Computing Frontiers, page 23. ACM, 2013.
 [18] H. Mühlenbein, T. Mahnig, and A. Ochoa. Schemata, distributions and graphical models in evolutionary optimization. Journal of Heuristics, 5(2):213–247, 1999.
 [19] M. Pelikan and A. K. Hartmann. Searching for ground states of Ising spin glasses with hierarchical BOA and cluster exact approximation. In M. Pelikan, K. Sastry, and E. CantúPaz, editors, Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications, Studies in Computational Intelligence, pages 333–349. Springer, 2006.
 [20] M. Pelikan and H. G. Katzgraber. Analysis of evolutionary algorithms on the onedimensional spin glass with powerlaw interactions. In Proceedings of the 11th Annual conference on Genetic and evolutionary computation, pages 843–850. ACM, 2009.
 [21] A. PrügelBennett and J. L. Shapiro. The dynamics of a genetic algorithm for simple random Ising systems. Physica D: Nonlinear Phenomena, 104(1):75–114, 1997.
 [22] R. Santana. Estimation of distribution algorithms with Kikuchi approximations. Evolutionary Computation, 13(1):67–97, 2005.
 [23] G. E. Santoro, R. Martoňák, E. Tosatti, and R. Car. Theory of quantum annealing of an Ising spin glass. Science, 295(5564):2427–2430, 2002.
 [24] A. Selby. Efficient subgraphbased sampling of Isingtype models with frustration. arXiv preprint arXiv:1409.3934, 2014.
 [25] S. Shakya, J. McCall, and D. Brown. Solving the Ising spin glass problem using a bivariate EDA based on Markov random fields. In Proceedings of the IEEE Congress on Evolutionary Computation. CEC2006, pages 908–915. IEEE, 2006.
 [26] S. Shakya and R. Santana, editors. Markov Networks in Evolutionary Computation. Springer, 2012.
 [27] Z. Zhu, A. J. Ochoa, and H. G. Katzgraber. Efficient cluster algorithm for spin glasses in any space dimension. Phys. Rev. Lett., 115:077201, 2015.
 [28] Z. Zhu, A. J. Ochoa, S. Schnabel, F. Hamze, and H. G. Katzgraber. Bestcase performance of quantum annealers on native spinglass benchmarks: How chaos can affect success probabilities. arXiv preprint arXiv:1505.02278, 2015.
 [29] I. Zintchenko, M. B. Hastings, and M. Troyer. From local to global ground states in Ising spin glasses. Physical Review B, 91(2):024201, 2015.
Comments
There are no comments yet.