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arXiv:1509.02374

Quantum walk speedup of backtracking algorithms

Ashley Montanaro

Published Sep 8, 2015• 28 citations
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Abstract

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.

DOI
10.48550/arxiv.1509.02374

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Constraint Satisfaction Problems (CSPs) are a fundamental class of computational problems prevalent in various fields, including artificial intelligence, optimization, and cryptography. Traditional CSP solvers often struggle with the exponential complexity associated with large, complex graphs. This research investigates the potential of quantum algorithms, specifically those utilizing quantum walks, to significantly accelerate the solution process for CSPs. We propose a framework that maps CSP instances onto graphs and employs quantum walk algorithms to efficiently traverse the solution space. The core of our approach leverages the quantum superposition principle, enabling the simultaneous exploration of multiple potential solutions. We demonstrate, through theoretical analysis and algorithmic design, that quantum walks can offer a substantial speedup compared to classical algorithms for certain CSP instances, particularly those with high connectivity and large solution spaces. This work contributes a novel approach to tackling computationally intensive CSPs by integrating quantum computing principles with graph traversal techniques, representing a potentially transformative advancement in solving complex optimization challenges.
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Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over 105 relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over 103 . However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.