AI 中文总结
研究通过极值集系统改善TSP等排列问题的时空权衡，提出量子类似物，利用量子最小值查找，给出TSP量子算法的时空复杂度，结合改进构造得到权衡曲线，在\(1 < S \leq 1.657\)时优于已知量子权衡。
AI 中文摘要
最近，阿梅利、内德洛夫和王以及达兰特和科兹马的工作引入了一个框架，通过具有许多极大链的极值集系统来改善旅行商问题（TSP）和相关排列问题的经典时空权衡。在本笔记中，我们观察到，对于外部聚合为最小值的所谓排列问题（如TSP），相同框架允许一个简单的量子类似物：不是遍历集系统的覆盖族，而是对该族应用量子最小值查找。更精确地说，设\(P_S\)表示归一化大小至多为\(S\)的集系统中的最优逆归一化链密度。那么TSP允许一个有界误差量子算法，使用\(\widetilde O(S^n)\)的量子随机存取存储器（QRAM）空间和\(\widetilde O((S\sqrt{P_S})^n)\)时间。相同的论证适用于与TSP结构相似的其他排列最小化问题。将这一观察结果与安多尼、达兰特、科兹马和于改进的极值集系统构造相结合，得到了一条明确的量子时空权衡曲线，在所有\(1 < S \leq 1.657\)时优于卡罗波等人已知的量子权衡。
英文摘要
We give a quantum analogue of the extremal set-system framework introduced by Ameli, Nederlof and Wang (FOCS 2026) and Dallant and Kozma (FOCS 2026) for space-time tradeoffs in TSP and related permutation problems. Let $1<S\leq 2$, and let $P_S$ denote the optimal inverse normalized chain density among set systems of normalized size at most $S$. We use quantum minimum finding over a covering family of set systems to obtain a bounded-error algorithm using $S^{n+o(n)}$ QRAM space and $\left(S\sqrt{P_S}\right)^{n+o(n)}$ time. Combined with the set-system constructions of Andoni, Dallant, Kozma and Yu, this gives an explicit tradeoff improving the Caroppo et al. (ESA 2026) TSP tradeoff for $1<S\le1.657$. The same method applies to a variety of permutation problems, including the Hypercube Path query problem introduced by Ambainis et al. (SODA 2019), for which it improves the previous bound of $1.817^{n+o(n)}$ for both time and QRAM space to $1.784^{n+o(n)}$ time using $1.430^{n+o(n)}$ QRAM space.
CommentsImproved exposition and added results about the Hypercube Path query problem
