A note on quantum divide and conquer for minimal string rotation

Qisheng Wang

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 n⋅2O(log⁡n)<math><msqrt is="true"><mrow is="true"><mi is="true">n</mi></mrow></msqrt><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></mrow></msup></math>, improving the prior result of n⋅2(log⁡n)1/2+ε<math><msqrt is="true"><mrow is="true"><mi is="true">n</mi></mrow></msqrt><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">2</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">ε</mi></mrow></msup></mrow></msup></math> by Akmal and Jin (2022). Notably, this improvement is quasi-polylogarithmic, which is achieved by only logarithmic level-wise optimization using fault-tolerant quantum minimum finding.