Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems
François Le Gall, Saeed Seddighin
Abstract
Abstract Longest common substring (), longest palindrome substring (), and Ulam distance () are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for and are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact $${{\tilde{O}}}(\sqrt{n})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>O</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mrow><mml:mo>(</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> time algorithm for , we prove that needs at least time $$\tilde{\Omega }(n^{2/3})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> even for 0/1 strings.