Quantum Computing and Phase Transitions in Combinatorial Search

TL;DR

This paper presents a quantum algorithm for combinatorial search that effectively concentrates amplitude into solutions, mimicking classical backtracking, and exhibits similar phase transition behavior in problem difficulty.

Contribution

It introduces a novel quantum search algorithm that leverages problem structure to improve solution probability and demonstrates classical-like phase transition phenomena.

Findings

Quantum algorithm outperforms naive quantum parallelism in solution finding.

Empirical results show phase transition behavior similar to classical methods.

Difficult instances cluster near the underconstrained-overconstrained boundary.

Abstract

We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems.