Quantum Computing and Phase Transitions in Combinatorial Search

We introduce an algorithm for combinatorial search on quantum computers that is capable of signi cantly 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 nd 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. Speci cally, di cult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems.