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 $\tilde{O}\left( ε^{-1} \sqrt{\frac{mn}{k}} \right)$.

翻译：在$k$失配问题中，给定长度分别为$n$和$m$的模式串与文本串，我们需要判断文本中是否存在某个子串与模式串的汉明距离至多为$k$。该问题自1982年起在经典计算领域得到研究，近期Jin与Nogler以及Kociumaka、Nogler和Wellnitz在量子计算背景下进行了探索。本文提出一种简易的近似求解量子算法，给定参数$ε∈(0,1]$，仅当存在$\left(1+ε\right)k$失配时返回匹配位置。若算法未返回任何匹配，则表明不存在$k$失配。该算法的时间（规模）复杂度为$\tilde{O}\left( ε^{-1} \sqrt{\frac{mn}{k}} \right)$。