  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.
