Quantum Pattern Matching Fast on Average

The d-dimensional pattern matching problem is to find an occurrence of a pattern of length m×⋯×m within a text of length n×⋯×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˜((n/m)d/22O(d3/2logm√)). For large m this is super-polynomially faster than the best possible classical algorithm, which requires time Ω˜(nd/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.