This paper proposes the use of estimation of distribution algorithms to deal with the problem of finding an optimal product of braid generators in topological quantum computing. We investigate how the regularities of the braid optimization problem can be translated into statistical regularities by means of the Boltzmann distribution. The introduced algorithm obtains solutions with an accuracy in the order of 10-6, and lengths up to 9 times shorter than those expected from braids of the same accuracy obtained with other methods.
@cite_18: Several Estimation of Distribution Algorithms (EDAs) based on Markov networks have been recently proposed. The key idea behind these EDAs was to factorise the joint probability distribution of solution variables in terms of cliques in the undirected graph. As such, they made use of the global Markov property of the Markov network in one form or another. This paper presents a Markov Network based EDA that is based on the use of the local Markov property, the Markovianity, and does not directly model the joint distribution. We call it Markovianity based Optimisation Algorithm. The algorithm combines a novel method for extracting the neighbourhood structure from the mutual information between the variables, with a Gibbs sampler method to generate new points. We present an extensive empirical validation of the algorithm on problems with complex interactions, comparing its performance with other EDAs that use higher order interactions. We extend the analysis to other functions with discrete representation, where EDA results are scarce, comparing the algorithm with state of the art EDAs that use marginal product factorisations.
@cite_8: The question of finding feasible ways for estimating probability distributions is one of the main challenges for Estimation of Distribution Algorithms (EDAs). To estimate the distribution of the selected solutions, EDAs use factorizations constructed according to graphical models. The class of factorizations that can be obtained from these probability models is highly constrained. Expanding the class of factorizations that could be employed for probability approximation is a necessary step for the conception of more robust EDAs. In this paper we introduce a method for learning a more general class of probability factorizations. The method combines a reformulation of a probability approximation procedure known in statistical physics as the Kikuchi approximation of energy, with a novel approach for finding graph decompositions. We present the Markov Network Estimation of Distribution Algorithm (MN-EDA), an EDA that uses Kikuchi approximations to estimate the distribution, and Gibbs Sampling (GS) to generate new points. A systematic empirical evaluation of MN-EDA is done in comparison with different Bayesian network based EDAs. From our experiments we conclude that the algorithm can outperform other EDAs that use traditional methods of probability approximation in the optimization of functions with strong interactions among their variables.
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@cite_6: This chapter presents results on the application of the concept of entropy to estimation of distribution algorithms (EDAs). Firstly, the Boltzmann mutual information curves are introduced. They are shown to contain a lot of information about the difficulty of the functions. Next, a design method of discrete benchmark functions is presented. The newly developed approach allows the construction of both single and random classes of functions that obey a given collection of probabilistic constraints. This application and the next — the construction of low cost search distributions — are based on the principle of maximum entropy. The last proposal is the linear entropic mutation (LEM), an approach that measures the amount of mutation applied to a variable as the increase of its entropy. We argue that LEM is a natural operator for EDAs because it mutates distributions instead of single individuals.
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@cite_5: This paper provides an in-depth empirical analysis of several hybrid evolutionary algorithms on the one-dimensional spin glass model with power-law interactions. The considered spin glass model provides a mechanism for tuning the effective range of interactions, what makes the problem interesting as an algorithm benchmark. As algorithms, the paper considers the genetic algorithm (GA) with twopoint and uniform crossover, and the hierarchical Bayesian optimization algorithm (hBOA). hBOA is shown to outperform both variants of GA, whereas GA with uniform crossover is shown to perform worst. The differences between the compared algorithms become more significant as the problem size grows and as the range of interactions decreases. Unlike for GA with uniform crossover, for hBOA and GA with twopoint crossover, instances with short-range interactions are shown to be easier. The paper also points out interesting avenues for future research.
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@cite_12: In topologically-protected quantum computation, quantum gates can be carried out by adiabatically braiding two-dimensional quasiparticles, reminiscent of entangled world lines. [Phys. Rev. Lett. 95, 140503 (2005)], as well as Leijnse and Flensberg [Phys. Rev. B 86, 104511 (2012)] recently provided schemes for computing quantum gates from quasiparticle braids. Mathematically, the problem of executing a gate becomes that of finding a product of the generators (matrices) in that set that approximates the gate best, up to an error. To date, efficient methods to compute these gates only strive to optimize for accuracy. We explore the possibility of using a generic approach applicable to a variety of braiding problems based on evolutionary (genetic) algorithms. The method efficiently finds optimal braids while allowing the user to optimize for the relative utilities of accuracy and or length. Furthermore, when optimizing for error only, the method can quickly produce efficient braids.
@cite_17: In topological quantum computation, quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are carried out by dragging particlelike excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world lines in three-dimensional space-time, and the corresponding quantum gates depend only on the topology of the braids formed by these world lines. We show how to find braids that yield a universal set of quantum gates for qubits encoded using a specific kind of quasiparticle which is particularly promising for experimental realization.
In addition to the use of genetic algorithms @cite_12 , brute force @cite_17 has been proposed to search for a braids of manageable size (up to @math exchanges). Other methods such as the Solovay-Kitaev algorithm @cite_21 @cite_19 @cite_9 provide bounds on the accuracy and length of the braids. However, these methods do not allow the user to tune the balance between the accuracy and the length as pioneered in @cite_12 . The Boltzmann distribution has played an important role in the theoretical analysis of EDAs and other authors works have analyzed the relationship between the function structure and the dependencies in the distribution @cite_25 @cite_6 @cite_8 . Other problems from physics have been previously treated with EDAs. In particular, spin glass models with different types of interactions and topologies have been addressed @cite_27 @cite_5 @cite_18 . Two important differences between the braid problem and the spin glass models that makes it particularly challenging is that its fitness function is multiplicative and that the representation is non binary. In fact the cardinality of the variables can increase with the number of generators.