Given an array $a[1..n]$, the Range Minimum Query (RMQ) problem is to maintain a data structure that supports RMQ queries: given a range $[l, r]$, find the index of the minimum element among $a[l..r]$, i.e., $\operatorname{argmin}_{i \in [l, r]} a[i]$. In this paper, we propose a quantum data structure that supports RMQ queries and range updates, with an optimal time complexity $\widetilde Θ(\sqrt{nq})$ for performing $q = O(n)$ operations without preprocessing, compared to the classical $\widetildeΘ(n+q)$. As an application, we obtain a time-efficient quantum algorithm for $k$-minimum finding without the use of quantum random access memory.

翻译：给定数组 $a[1..n]$，区间最小值查询（RMQ）问题旨在维护一个支持 RMQ 查询的数据结构：给定区间 $[l, r]$，找出 $a[l..r]$ 中最小元素的下标，即 $\operatorname{argmin}_{i \in [l, r]} a[i]$。本文提出一种支持 RMQ 查询与区间更新的量子数据结构，在无预处理情况下执行 $q = O(n)$ 次操作时，其时间复杂度为最优的 $\widetilde Θ(\sqrt{nq})$，而经典算法的时间复杂度为 $\widetildeΘ(n+q)$。作为应用，我们获得了一种无需量子随机存取存储器的高效量子算法，用于求解 $k$ 最小值查找问题。