Near-Optimal Quantum Algorithms for String Problems

Abstract

We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with nearoptimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in Õ(n 2/3 ) time, improving upon the recent Õ(n 5/6

A Note on Quantum Divide and Conquer for Minimal String Rotation

“…Theorem 1.1 also implies that the quantum query complexity of minimal suffix, maximal suffix, and maximal string rotation are also improved to the same complexity as in Theorem 1.1 because they can be easily reduced to each other as discussed in [AJ22].…”

“…Inspired by the quantum algorithms for LMSR given in [WY20], [AJ22] and [CKKD + 22], we obtain better quantum query complexity of LMSR by a more refined design of divide and conquer.…”

“…Recently, Wang and Ying [WY20] achieved a quantum speedup on LMSR based on an observation, called "exclusion rule" or "Ricochet Property" in the literature, borrowed from parallel string processing algorithms [Vis91,IS92]. Immediately after, Akmal and Jin [AJ22] improved the exclusion rule given by [WY20] and developed a quantum algorithm with a divide-and-conquer structure. The (worst-case) query complexity √ n • 2 (log n) 1/2+ε of their quantum algorithm 1 is nearoptimal up to n o(1) factors, improving the result of O n 3/4 by [WY20].…”

“…The (worst-case) query complexity √ n • 2 (log n) 1/2+ε of their quantum algorithm 1 is nearoptimal up to n o(1) factors, improving the result of O n 3/4 by [WY20]. Very recently, inspired by the structures of quantum algorithms in [AGS19] and [AJ22], Childs, Kothari, Kovacs-Deak, Sundaram, and Wang [CKKD + 22] further investigated a general framework of quantum divide and conquer. Moreover, they applied the framework to various string problems and obtained near-optimal quantum algorithms for them, including recognizing regular languages, k-increasing subsequence, k-common subsequence, and LMSR.…”

“…Moreover, they applied the framework to various string problems and obtained near-optimal quantum algorithms for them, including recognizing regular languages, k-increasing subsequence, k-common subsequence, and LMSR. Especially, they considered the decision problem of LMSR, and showed that its quantum query complexity is O n log 5 n 2 , improving the result of √ n • 2 (log n) 1/2+ε induced by [AJ22]. However, the result of [CKKD + 22] does not imply better quantum query complexity of function problems.…”