Quantum walk speedup of backtracking algorithms (1509.02374v2)

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.

• The paper introduces a novel quantum walk algorithm that achieves a quadratic speedup for backtracking in constraint satisfaction problems.
• It reduces the number of evaluations from T to O(√T n^(3/2) log n), significantly enhancing efficiency in traversing large search trees.
• The study highlights both theoretical innovation and practical benefits for SAT solving and AI, paving the way for future quantum algorithm research.

Quantum Walk Speedup of Backtracking Algorithms

The paper "Quantum walk speedup of backtracking algorithms," authored by Ashley Montanaro, presents a compelling approach to enhancing classical backtracking algorithms through quantum computation. Specifically, it explores how quantum walk algorithms can be applied to achieve polynomial speedups in solving constraint satisfaction problems (CSPs), a fundamental aspect of computational complexity and artificial intelligence.

Backtracking is a classic technique extensively employed in solving CSPs like boolean satisfiability (SAT) and graph coloring. It involves systematically exploring potential solutions in a search tree, where each node represents a partial solution to the given problem. Classically, the time complexity of these algorithms depends heavily on the size of the search tree, which can often be exponential in the number of variables.

Quantum Speedup

Montanaro introduces a quantum algorithm that leverages the quantum walk technique developed by Belovs. This algorithm can explore the backtracking tree with significantly fewer evaluations than classical algorithms. For a problem involving $n$ variables with a search tree containing $T$ vertices, the proposed quantum algorithm requires only $O(\sqrt{T} n^{3/2} \log n)$ tests, showcasing a quadratic speedup over its classical counterpart. The implication of this result is substantial, especially considering the widespread use of backtracking in various CSPs solving scenarios.

Key Results

The paper provides two main theoretical results:

1. Detection of a Solution: The proposed quantum algorithm can determine the existence of a solution in a CSP with bounded error using roughly $O(\sqrt{Tn} \log (1/\delta))$ evaluations.
2. Finding a Solution: When tasked to locate a specific solution, the quantum algorithm performs $O(\sqrt{T}n^{3/2} \log n \log (1/\delta))$ evaluations, offering an instance-dependent runtime benefit.

The novelty of these algorithms lies in their general applicability to any backtracking algorithm following the discussed framework, irrespective of the specific predicate $P$ or heuristic $h$ used to extend partial solutions.

Practical and Theoretical Implications

From a practical perspective, these quantum backtracking algorithms promise to improve the efficiency of CSP solvers substantially, including those used in SAT solving, which have broad uses in verification, planning, and artificial intelligence. Theoretically, the results underscore the potential of quantum computation to enhance classical algorithms significantly, particularly in problem settings involving large, complex spaces that are traditionally intractable.

The paper also explores techniques demonstrating that, under specific input distributions, quantum algorithms can achieve exponential reductions in expected runtime compared to classical methods. Such separations highlight the profound implications of quantum computational paradigms across average-case scenarios.

Future Research Directions

One of the intriguing suggestions for future research involves further reducing the quantum algorithm's complexity by exploring more efficient operational frameworks or possibly reducing the dependency on the depth of the search tree maintained by the classical algorithm. Additionally, addressing the challenges of searching with multiple marked vertices or finding all solutions with less overhead could unlock further efficiencies.

Moreover, the paper hints at exploring more natural input distributions and leveraging heavy-tailed runtime distributions to exploit quantum backtracking's efficiencies in exponential speedups.

In summary, Montanaro's work is a substantial addition to the quantum computing landscape, promising to reshape approaches to solving CSPs. Its implications extend to various domains that rely on efficient problem-solving capabilities, signaling a step forward in the application of quantum technologies to classical algorithmic challenges.