Simple Quantum Algorithm for Approximate $k$-Mismatch Problem

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 $\epsilon \in (0, 1]$. It returns an occurrence as a match only if it is a $\left(1+\epsilon\right)k$-mismatch. If it does not return any occurrence, then there is no $k$-mismatch. This algorithm has a time (size) complexity of $\tilde{O}\left( \epsilon^{-1} \sqrt{\frac{mn}{k}} \right)$.