Quantum Data Structure for Range Minimum Query

TL;DR

The paper establishes tight quantum query bounds for dynamic RMQ without requiring QRAM, showing a near-optimal $\tilde{\Theta}(\sqrt{n q})$ time for $q$ online operations on an array of length $n$. It introduces two key techniques for the quantum data structure: top-down completion and lazy node creation within a segment-tree framework, enabling on-demand initialization with sublinear costs. A complementary lower bound via reduction from quantum $k$-minimum finding proves the upper bound is optimal up to polylogarithmic factors. As an application, the authors derive a time-efficient quantum algorithm for $k$-minimum finding without QRAM, expanding the practicality of quantum data-structure techniques. Overall, the work advances quantum algorithm design for dynamic data-structure problems and suggests further exploration of QRAM-free quantum speedups in other domains.

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 $\widetilde Θ(\sqrt{nq})$ 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.

Figures (1)

• Figure 1: Segment tree for RMQ. In the example, $n = 8$ and the query range is $\lbrack2, 5\rbrack$ which is covered by (and thus involves) the green nodes $5, 9, 12$. In addition to the green nodes, the pink nodes $1, 2, 3, 4, 6$ are also visited, while the blue nodes are not visited.

Theorems & Definitions (11)

• Theorem 1.1: Upper bound for RMQ, \ref{['thm:QDynamicRMQ']}
• Theorem 1.2: Lower bound for RMQ, \ref{['thm:QDynamicRMQ-lower-bound']}
• Corollary 1.3: Quantum $k$-minimum finding without QRAM, \ref{['corollary:qkmin']}
• Lemma 2.1: WY23
• Lemma 3.1: Classical RMQ
• Theorem 3.2: Quantum RMQ
• proof
• Corollary 3.3: Quantum $k$-minimum finding without QRAM
• proof
• Theorem 3.4: Lower bound for quantum RMQ