Quantum Data Structure for Range Minimum Query

TL;DR

This paper introduces a quantum data structure for Range Minimum Query that achieves faster query and update times than classical methods, enabling efficient quantum algorithms for related problems without quantum memory.

Contribution

It presents a novel quantum data structure for RMQ with optimal time complexity, improving over classical approaches and enabling efficient quantum solutions for k-minimum finding.

Findings

Quantum RMQ data structure supports faster queries and updates.

Achieves optimal time complexity $ ilde{ heta}( oot{nq})$ for $q=O(n)$ operations.

Enables time-efficient quantum algorithms for k-minimum finding without quantum RAM.

Abstract

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 $\widetilde \Theta(\sqrt{nq})$ for performing $q = O(n)$ operations without preprocessing, compared to the classical $\widetilde\Theta(n+q)$. As an application, we obtain a time-efficient quantum algorithm for $k$-minimum finding without the use of quantum random access memory.