Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) 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 LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact $\tilde O(\sqrt{n})$ time algorithm for LPS, we prove that LCS needs at least time $\tilde \Omega(n^{2/3})$ even for 0/1 strings.

翻译：最长期常见子字符串( LCS ), 最长的平原子字符串( LPS ) 和 Ulam 距离( UL) 是三大基本字符串问题, 在近线性时间里可以典型地解决。 在这项工作中, 我们提出这些问题的亚线性时间量算法, 以及量子下限 。 我们的结果揭示了一个非常令人惊讶的事实: 虽然 LCS 和 LPS 的经典解决方案几乎相同( 通过fix 树), 但其量数计算复杂性是不同的。 虽然我们给出了精确的 $\ tilde O(\ sqrt{n}) 时间算法, 我们证明 LCS 需要至少时间 $\ tilde\ Omega (n\\\\\\\\\ 3} $, 即使 0/1 字符串 。