Quantum Pattern Matching Fast on Average

The d-dimensional pattern matching problem is to find an occurrence of a pattern of length m x ... x m within a text of length n x ... x n, with n >= m. This task models various problems in text and image processing, among other application areas. This work describes a quantum algorithm which solves the pattern matching problem for random patterns and texts in time (O) over tilde((n/m)(d/2)2(O)(d(3/2) root log m)). For large m this is super-polynomially faster than the best possible classical algorithm, which requires time (Omega) over tilde (n(d/2) + (n/m)(d)). The algorithm is based on the use of a quantum subroutine for finding hidden shifts in d dimensions, which is a variant of algorithms proposed by Kuperberg.