Near-Optimal-Time Quantum Algorithms for Approximate Pattern Matching

TL;DR

This paper introduces near-optimal quantum algorithms for approximate pattern matching, significantly improving time complexity over classical methods for certain parameter ranges, and providing solutions for both mismatches and edits.

Contribution

It presents the first quantum algorithms with near-optimal time complexity for approximate pattern matching with mismatches and edits, surpassing previous quantum query bounds.

Findings

Quantum algorithms achieve near-optimal time complexity for pattern matching.

The algorithms outperform classical methods when mismatch threshold is small.

They can also report approximate occurrences with increased complexity.

Abstract

Approximate Pattern Matching is among the most fundamental string-processing tasks. Given a text $T$ of length $n$, a pattern $P$ of length $m$, and a threshold $k$, the task is to identify the fragments of $T$ that are at distance at most $k$ to $P$. We consider the two most common distances: Hamming distance (the number of character substitutions) in Pattern Matching with Mismatches and edit distance (the minimum number of character insertions, deletions, and substitutions) in Pattern Matching with Edits. We revisit the complexity of these two problems in the quantum setting. Our recent work [STOC'24] shows that $\hat{O}(\sqrt{nk})$ quantum queries are sufficient to solve (the decision version of) Pattern Matching with Edits. However, the quantum time complexity of the underlying solution does not provide any improvement over classical computation. On the other hand, the state-of-the-art algorithm for Pattern Matching with Mismatches [Jin and Nogler; SODA'23] achieves query complexity $\hat{O}(\sqrt{nk^{3/2}})$ and time complexity $\tilde{O}(\sqrt{nk^2})$, falling short of an unconditional lower bound of $\Omega(\sqrt{nk})$ queries. In this work, we present quantum algorithms with a time complexity of $\tilde{O}(\sqrt{nk}+\sqrt{n/m}\cdot k^2)$ for Pattern Matching with Mismatches and $\hat{O}(\sqrt{nk}+\sqrt{n/m}\cdot k^{3.5})$ for Pattern Matching with Edits; both solutions use $\hat{O}(\sqrt{nk})$ queries. The running times are near-optimal for $k\ll m^{1/3}$ and $k\ll m^{1/6}$, respectively, and offer advantage over classical algorithms for $k\ll (mn)^{1/4}$ and $k\ll (mn)^{1/7}$, respectively. Our solutions can also report the starting positions of approximate occurrences of $P$ in $T$ (represented as collections of arithmetic progressions); in this case, the unconditional lower bound and the complexities of our algorithms increase by a $\Theta(\sqrt{n/m})$ factor.