Volume 14 (2018) Article 15 pp. 1-24
Quantum-Walk Speedup of Backtracking Algorithms
Received: December 2, 2016
Revised: May 4, 2017
Published: December 3, 2018

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.