发表机构
Pennsylvania State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次给出从3SUM到在线SetDisjointness的次线性时间量子归约，并推广到3XOR及一般3OA问题，从而在量子3SUM猜想下建立量子SetDisjointness算法的p+2q≥1权衡下界。
AI 中文摘要
在经典细粒度复杂度理论中，3SUM猜想被用于通过初始归约到SetDisjointness问题，来证明数据结构和图问题的一系列条件下界。然而，存在一个运行时间为$\tilde{O}(n)$的3SUM量子算法，并且将Grover算法直接应用于SetDisjointness查询，优于Kopelowitz、Pettie和Porat（SODA 2016）提出的最先进的经典条件界；这表明这些经典界在量子环境下不成立。因此，在量子环境中建立类似的条件下界，需要将量子3SUM猜想应用于从3SUM到SetDisjointness的量子细粒度归约。我们首次给出了从3SUM到在线SetDisjointness的次线性时间量子归约。通过我们的归约，量子3SUM猜想蕴含了量子SetDisjointness算法的一个$p + 2q \geqslant 1$的权衡界，其中预处理时间为$O(N^p)$，查询时间为$O(N^q)$。我们还给出了从3XOR的类似归约。这些结果源于一个适用于SetDisjointness的细粒度归约通用框架，该框架适用于任何具有合适近线性哈希函数的阿贝尔3-正交阵列（3OA）问题。
英文摘要
In classical fine-grained complexity, the 3SUM Conjecture is used to prove a variety of conditional lower bounds on data structure and graph problems via an initial reduction to the SetDisjointness problem. However, there is an $\tilde{O}(n)$-time quantum algorithm for 3SUM and a direct application of Grover's algorithm to SetDisjointness queries beats the state-of-the-art classical conditional bound by Kopelowitz, Pettie, and Porat (SODA 2016); this shows that these classical bounds do not apply in the quantum setting. Thus establishing analogous conditional lower bounds in the quantum setting requires applying the quantum 3SUM Conjecture to a \emph{quantum} fine-grained reduction from 3SUM to SetDisjointness. We give the first sub-linear time quantum reductions from 3SUM to online SetDisjointness. Via our reduction, the quantum 3SUM conjecture implies a $p + 2q \geqslant 1$ tradeoff bound for quantum SetDisjointness algorithms with $O(N^p)$ preprocessing time and $O(N^q)$ query time. We also give an analogous reduction from 3XOR. These results are derived from a general framework for fine-grained reductions to SetDisjointness which applies to any Abelian 3-Orthogonal Array (3OA) problem with suitable almost-linear hash functions.
