发表机构
IBM Research; QuICS; Silicon Valley Lab; University of Maryland（IBM研究; 量子信息与计算科学中心; 硅谷实验室; 马里兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次系统研究量子平滑分析，证明平滑量子查询复杂度可指数级优于经典，并给出对称布尔函数近紧刻画及字符串问题的量子加速，揭示平滑分析下更大的量子优势。
AI 中文摘要
平滑分析是经典算法中解释算法超越最坏情况性能的核心框架，常用于解释算法在实践中表现良好的原因。我们开创了其量子对应物的系统性研究，并展示了以下结果。（1）我们证明存在一个全函数，其平滑量子查询复杂度比其经典查询复杂度指数级更小。（2）我们给出了对称布尔函数的平滑随机和量子查询复杂度的近紧刻画，统一了[Beals等人，FOCS'98]的最坏情况复杂度结果和[Ambainis和de Wolf，STACS'00]的平均情况复杂度结果。（3）我们研究了诸如模式匹配和编辑距离等字符串问题，并在各种情形下给出了从多项式到超多项式的量子加速。我们的主要技术成分包括一个用于ε-近似计算两个非重复字符串之间碰撞次数的近紧量子算法，改进了Le Gall和Ng [QIC'22]的结果。总的来说，我们的结果表明，平滑分析可以揭示比最坏情况分析所暗示的更大的量子加速，为在更现实的输入上实现量子优势开辟了道路。
英文摘要
Smoothed analysis is a central framework in classical algorithms for explaining the performance of algorithms beyond the worst case, often explaining why algorithms perform well in practice. We initiate a systematic study of its quantum counterpart and show the following results. $(1)$ We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. $(2)$ We give near-tight characterizations of smoothed randomized and quantum query complexities for symmetric Boolean functions, unifying the worst-case complexity results of [Beals et al, FOCS'98] and average-case complexity results of [Ambainis and de Wolf, STACS'00]. $(3)$ We study string problems such as pattern matching and edit distance and, in various regimes, give polynomial to superpolynomial quantum speedups. Our main technical ingredients include a near-tight quantum algorithm for $\varepsilon$-approximating the number of collisions between two non-repetitive strings, improving the result of Le Gall and Ng [QIC'22]. Together, our results show that smoothing can reveal larger quantum speedups than worst-case analysis suggests, opening a path towards quantum advantage on more realistic inputs.
