Quantum Pattern Matching Fast on Average

The d-dimensional pattern matching problem is to find an occurrence of a pattern of length $$m \times \dots \times m$$m×⋯×m within a text of length $$n \times \dots \times n$$n×⋯×n, with $$n \ge m$$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 $$\widetilde{O}((n/m)^{d/2} 2^{O(d^{3/2}\sqrt{\log m})})$$O~((n/m)d/22O(d3/2logm)). For large m this is super-polynomially faster than the best possible classical algorithm, which requires time $$\widetilde{\Omega }( n^{d/2} + (n/m)^d)$$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.