Sublinear time quantum algorithms have been established for many fundamental problems on strings. This work demonstrates that new, faster quantum algorithms can be designed when the string is highly compressible. We focus on two popular and theoretically significant compression algorithms -- the Lempel-Ziv77 algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT), and obtain the results below. We first provide a quantum algorithm running in $\tilde{O}(\sqrt{zn})$ time for finding the LZ77 factorization of an input string $T[1..n]$ with $z$ factors. Combined with multiple existing results, this yields an $\tilde{O}(\sqrt{rn})$ time quantum algorithm for finding the RL-BWT encoding with $r$ BWT runs. Note that $r = \tilde{\Theta}(z)$. We complement these results with lower bounds proving that our algorithms are optimal (up to polylog factors). Next, we study the problem of compressed indexing, where we provide a $\tilde{O}(\sqrt{rn})$ time quantum algorithm for constructing a recently designed $\tilde{O}(r)$ space structure with equivalent capabilities as the suffix tree. This data structure is then applied to numerous problems to obtain sublinear time quantum algorithms when the input is highly compressible. For example, we show that the longest common substring of two strings of total length $n$ can be computed in $\tilde{O}(\sqrt{zn})$ time, where $z$ is the number of factors in the LZ77 factorization of their concatenation. This beats the best known $\tilde{O}(n^\frac{2}{3})$ time quantum algorithm when $z$ is sufficiently small.

翻译：已经为字符串上的许多根本性问题建立了子线内时间算法。 这项工作表明, 当字符串高度压缩时, 可以设计出新的、 更快的量算法。 我们关注两个流行和理论上重要的压缩算法 -- Lempel- Ziv77 算法( LZ77 ) 和 Run- encod Burrows- Wheeler 变换( RL- BWT), 并获得以下结果。 我们首先提供以$tilde{ O} (sqrt{z} 美元运行的量算法。 我们用较低的界限来补充这些结果, 以证明我们的算法是最佳的 $T[1. n] 美元和 $z$。 结合多种现有结果, 这会产生 $Otreqr\ translation 结构的美元- BWT 编码。 注意, 当我们用 $rqrdal_ transal=