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((n/m)d+nd/2). 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.Comment: 22 pages, 2 figures; v3: further minor changes, essentially published
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