Quantum Algorithm for Lexicographically Minimal String Rotation
Qisheng Wang, Mingsheng Ying
TL;DR
This work addresses the LMSR problem by introducing a quantum algorithm with worst-case query complexity $O\left(n^{3/4}\right)$ and average-case complexity $O\left(\sqrt{n}\log n\right)$, surpassing classical randomized approaches in both regimes and approaching optimality up to polylog factors. The authors develop a robust framework for error reduction in nested quantum algorithms, introduce an exclusion rule that reduces candidate LMSR positions, and leverage string sensitivity to improve average-case performance. A key technical contribution is the deterministic sampling technique, including a quantum lexicographical comparator and a deterministic-sampling algorithm, enabling efficient string periodicity checks and pattern matching. The LMSR algorithm is designed as a multi-level nested quantum procedure, combining minimum-finding, search, and deterministic sampling across blocks to achieve $O\left(n^{3/4}\right)$ queries, with a practical average-case enhancement to $O\left(\sqrt{n}\log n\right)$. The paper also provides classical and quantum lower bounds, demonstrating a quantum separation from classical algorithms, and showcases concrete applications in benzenoid identification and disjoint-cycle automata minimization, underscoring the method’s potential impact in computational chemistry and automata theory.
Abstract
Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an $O(n^{3/4})$ quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity $O(\sqrt n \log n)$, which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its $Ω\left(\sqrt{n/\log n}\right)$ lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.