Simple Quantum Algorithm for Approximate $k$-Mismatch Problem

TL;DR

The paper addresses the quantum approximate solution to the $k$-mismatch problem in strings, proposing a simple quantum algorithm that, given a pattern of length $m$ and text of length $n$, finds a substring with at most $k$ mismatches or reports no such occurrence, with a guaranteed $(1+\epsilon)k$-mismatch if any $k$-mismatch exists. The approach blends Weak Search through amplitude amplification and a decider for bounded Hamming distance using quantum counting, achieving a time complexity of $\tilde{O}(\epsilon^{-1}\sqrt{mn/k})$. This improves the practicality of quantum approaches in certain regimes (notably when $k$ is large) relative to prior quantum results, while aligning with the standard probabilistic success guarantees. The work also outlines possible preprocessing strategies to further reduce search space and motivates future exploration of efficiency under the $\epsilon$-approximation setting.

Abstract

In the $k$-mismatch problem, given a pattern and a text of length $n$ and $m$ respectively, we have to find if the text has a sub-string with a Hamming distance of at most $k$ from the pattern. This has been studied in the classical setting since 1982 and recently in the quantum computational setting by Jin and Nogler and Kociumaka, Nogler, and Wellnitz. We provide a simple quantum algorithm that solves the problem in an approximate manner, given a parameter $ε\in (0, 1]$. It returns an occurrence as a match only if it is a $\left(1+ε\right)k$-mismatch. If it does not return any occurrence, then there is no $k$-mismatch. This algorithm has a time (size) complexity of $\tilde{O}\left( ε^{-1} \sqrt{\frac{mn}{k}} \right)$.

Theorems & Definitions (12)

• definition 1: Intervals
• definition 2: Hamming Distance
• definition 3: $k$-mismatch Problem
• theorem 1
• theorem 2: Amplitude Amplification Brassard_2002
• theorem 3: Counting Brassard_2002
• theorem 4: Weak Search
• proof
• theorem 5
• proof