A Note on Quantum Divide and Conquer for Minimal String Rotation

TL;DR

This paper improves the quantum query complexity for the minimal string rotation problem using a divide-and-conquer approach, achieving a quasi-polylogarithmic enhancement over previous results.

Contribution

It demonstrates a refined quantum algorithm with lower query complexity for minimal string rotation, utilizing logarithmic level-wise optimization in a divide-and-conquer framework.

Findings

Quantum query complexity is $\sqrt{n} imes 2^{O(\sqrt{\log n)}}$

Improvement over previous complexity bounds by quasi-polylogarithmic factors

Utilizes fault-tolerant quantum minimum finding in the algorithm

Abstract

Lexicographically minimal string rotation is a fundamental problem in string processing that has recently garnered significant attention in quantum computing. Near-optimal quantum algorithms have been proposed for solving this problem, utilizing a divide-and-conquer structure. In this note, we show that its quantum query complexity is $\sqrt{n} \cdot 2^{O(\sqrt{\log n})}$, improving the prior result of $\sqrt{n} \cdot 2^{(\log n)^{1/2+\varepsilon}}$ due to Akmal and Jin (SODA 2022). Notably, this improvement is quasi-polylogarithmic, which is achieved by only logarithmic level-wise optimization using fault-tolerant quantum minimum finding.