Quantum pattern matching fast on average

TL;DR

This paper introduces a quantum algorithm for high-dimensional pattern matching that significantly outperforms classical algorithms for large pattern sizes, leveraging quantum hidden shift techniques.

Contribution

It presents a novel quantum algorithm for $d$-dimensional pattern matching with super-polynomial speedup over classical methods for large patterns.

Findings

Quantum algorithm solves pattern matching faster for large $m$

Achieves super-polynomial speedup over classical algorithms

Uses quantum hidden shift subroutines in $d$ dimensions

Abstract

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