Quantum Pattern Matching with Wildcards

TL;DR

This paper presents a quantum algorithm for pattern matching with wildcards that operates in sublinear time when the number of wildcards is sublinear, advancing quantum solutions for complex string matching problems.

Contribution

It introduces a novel quantum algorithm for wildcard pattern matching with sublinear runtime, extending quantum string matching capabilities.

Findings

Quantum algorithm runs in  O( oot n oot k) time for k wildcards.

Achieves sublinear time complexity when wildcards are sublinear in input size.

Extends quantum pattern matching to more complex wildcard scenarios.

Abstract

Pattern matching is one of the fundamental problems in Computer Science. Both the classic version of the problem as well as the more sophisticated version where wildcards can also appear in the input can be solved in almost linear time $\tilde O(n)$ using the KMP algorithm and Fast Fourier Transform, respectively. In 2000, Ramesh and Vinay~\cite{ramesh2003string} give a quantum algorithm that solves classic pattern matching in sublinear time and asked whether the wildcard problem can also be solved in sublinear time? In this work, we give a quantum algorithm for pattern matching with wildcards that runs in time $\tilde O(\sqrt{n}\sqrt{k})$ when the number of wildcards is bounded by $k$ for $k \geq \sqrt{n}$. This leads to an algorithm that runs in sublinear time as long as the number of wildcards is sublinear.