A Note on Quantum Divide and Conquer for Minimal String Rotation
Qisheng Wang
TL;DR
The paper addresses the problem of finding the lexicographically minimal string rotation (LMSR) in the quantum query model and embeds it in a quantum divide-and-conquer framework. It introduces per-level polylogarithm optimizations via fault-tolerant quantum minimum finding, showing that small improvements at each level accumulate into a quasi-polylog speedup. The main result establishes that both the worst-case and function versions of LMSR have quantum query complexity $\sqrt{n}\,2^{O(\sqrt{\log n})}$, improving the prior $\sqrt{n}\,2^{(\log n)^{1/2+\varepsilon}}$ bound. The work additionally yields faster quantum algorithms for benzenoid identification and advances understanding of quantum divide-and-conquer in non-Boolean settings, while outlining open questions about polynomial or exponential speedups and the limits of the Boolean/non-Boolean gap.
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.