  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.
