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 \(\Omega \left( \sqrt{n/\log n}\right) \) 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.

字典顺序最小字符串旋转（LMSR）是一个在字典顺序中找到字符串所有旋转中最小的一个问题，广泛应用于图、多边形、自动机和化学结构的相等性检查。在本文中，我们提出了一种用于 LMSR 的\(O(n^{3/4})\)量子查询算法。特别是，该算法具有平均情况查询复杂度\(O(\sqrt{n} \log n)\)，与它的\(\Omega \left( \sqrt{n/\log n}\right) \)下界。此外，我们表明，在最坏和平均情况下，我们的量子算法都优于任何（经典）随机算法。作为一种应用，它用于苯环识别和不相交循环自动机最小化。