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The European Physical Journal Special Topics

Performance of the quantum adiabatic algorithm on constraint satisfaction and spin glass problems

Itay Hen, A. P. Young

Published Feb 1, 2015• 2 citations
Publisher
DOI
10.1140/epjst/e2015-02343-3

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Quantum annealers promise to solve practical optimization problems potentially faster than conventional classical computers. One of the major ongoing debates in this context pertains to their robustness against the decohering effects of finite temperature and interactions with the environment. We argue that even in an ideal setting of very low temperatures and in the absence of a decohering environment, quantum annealers do not necessarily perform better than classical heuristic solvers. Here, we numerically study the performance of the quantum adiabatic algorithm (QAA) on a variety of constraint satisfaction problems and a spin glass problem by studying the size dependence of the minimum energy gap during the evolution of the QAA. We do so by employing Quantum Monte Carlo schemes as these allow us to study these problems at much larger scales than exact methods would allow. We find that in all cases a quantum phase transition occurs and the minimum gap decreases exponentially with system size, leading to an exponentially large running time for the QAA. Based on these and other results, we briefly discuss potential modifications to the QAA that may improve the scaling of the minimum gap, leading to faster quantum adiabatic algorithms.
In the past decade, Matrix Product State (MPS) algorithms have emerged as an efficient method of modeling some many-body quantum spin systems. Since spin system Hamiltonians can be considered constraint satisfaction problems (CSPs), it follows that MPS should provide a versatile framework for studying a variety of general CSPs. In this thesis, we apply MPS to two types of CSP. First, use MPS to simulate adiabatic quantum computation (AQC), where the target Hamiltonians are instances of a fully connected, random Ising spin glass. Results of the simulations help shed light on why AQC fails for some optimization problems. We then present the novel application of a modified MPS algorithm to classical Boolean satisfiability problems, specifically k-SAT and max k-SAT. By construction, the algorithm also counts solutions to a given Boolean formula (\#-SAT). For easy satisfiable instances, the method is more expensive than other existing algorithms; however, for hard and unsatisfiable instances, the method succeeds in finding satisfying assignments where other algorithms fail to converge.
This paper introduces Quantum-Enhanced Constraint Satisfaction (QECS), a novel algorithm designed to accelerate the solution of complex optimization problems using quantum annealing. Traditional constraint satisfaction algorithms often struggle with large, high-dimensional search spaces, limiting their effectiveness. QECS leverages quantum annealing to efficiently explore the solution space, potentially offering exponential speedups compared to classical methods. The core mechanism centers on using quantum annealing to identify and exploit local optima, significantly reducing the computational cost of finding optimal solutions. We present a detailed analysis of the algorithm's performance, demonstrating its potential for substantial improvements in optimization tasks across diverse problem domains.