Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor $w$, where $w$ is the computer word size. For example, edit distance of two strings of length $n$ can be solved in $O(n^2/w)$ time. In a reasonable classical model of computation, one can assume $w=\Theta(\log n)$. There are conditional lower bounds for such problems stating that speed-ups with factor $n^\epsilon$ for any $\epsilon>0$ would lead to breakthroughs in complexity theory. However, these conditional lower bounds do not cover quantum models of computing. Indeed, Boroujeni et al. (J. ACM, 2021) showed that edit distance can be approximated within a factor $3$ in sub-quadratic time $O(n^{1.81})$ using quantum computing. They also showed that, in their chosen model of quantum computing, the approximation factor cannot be improved using sub-quadractic time. To break through the aforementioned classical conditional lower bounds and this latest quantum lower bound, we enrich the model of computation with a quantum random access memory (QRAM), obtaining what we call the word QRAM model. Under this model, we show how to convert the bit-parallelism of quadratic time solvable problems into quantum algorithms that attain speed-ups with factor $n$. The technique we use is simple and general enough to apply to many bit-parallel algorithms that use Boolean logics and bit-shifts. To apply it to edit distance, we first show that the famous $O(n^2/w)$ time bit-parallel algorithm of Myers (J. ACM, 1999) can be adjusted to work without arithmetic + operations. As a direct consequence of applying our technique to this variant, we obtain linear time edit distance algorithm under the word QRAM model for constant alphabet. We give further results on a restricted variant of the word QRAM model to give more insights to the limits of the model.

翻译：许多可在二次时间内解决的问题，通过字并行加速可获得因子$w$的提升，其中$w$为计算机字长。例如，两个长度为$n$的字符串的编辑距离可在$O(n^2/w)$时间内求解。在合理的经典计算模型中，可假设$w=\Theta(\log n)$。此类问题存在条件性下界，表明若存在因子$n^\epsilon$（对任意$\epsilon>0$）的加速，将引发复杂度理论的突破。然而这些条件性下界不涵盖量子计算模型。事实上，Boroujeni等人（J. ACM, 2021）证明，利用量子计算可在亚二次时间$O(n^{1.81})$内将编辑距离近似到因子$3$以内。他们还指出，在其所选量子计算模型中，无法用亚二次时间改进近似因子。为突破前述经典条件性下界及最新量子下界，我们引入量子随机存取存储器（QRAM）对计算模型进行扩充，形成所谓的词QRAM模型。在此模型下，我们展示了如何将二次时间可解问题的字并行性转化为量子算法，从而实现因子$n$的加速。所用方法简洁通用，可适用于众多基于布尔逻辑和位移运算的字并行算法。为将其应用于编辑距离，我们首先证明Myers（J. ACM, 1999）著名的$O(n^2/w)$时间字并行算法可调整至无需算术加法运算。将此技术直接应用于该变体后，我们在词QRAM模型下针对恒定字母表得到了线性时间编辑距离算法。此外，我们针对词QRAM模型的受限变体给出了进一步结果，以更深入地揭示该模型的能力边界。