Quantum walk speedup of backtracking algorithms

TL;DR

This paper introduces a quantum algorithm that accelerates classical backtracking algorithms for constraint satisfaction problems, notably improving the efficiency of SAT solvers by leveraging quantum walks.

Contribution

The paper presents a general quantum speedup method for classical backtracking algorithms using quantum walks, reducing runtime from T to approximately sqrt(T) times polynomial factors.

Findings

Quantum algorithms can speed up backtracking-based CSP solving.

The method applies to SAT solvers like DPLL, offering practical improvements.

Potential exponential runtime reductions for certain input distributions.

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.