超越解码量子干涉测量的最优多项式交集的最坏情况量子算法
Worst-Case Quantum Algorithm for Optimal Polynomial Intersection Beyond Decoded Quantum Interferometry
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中文总结 AI 辅助
研究OPI问题，针对最坏情况设计量子算法，改进Sun和Wootters的存在性界限，在特定参数范围有更好结果，还将存在性结果扩展到Max-LINSAT问题，算法基于MDS设置中Brascamp-Lieb型不等式的新颖应用。
中文摘要 AI 辅助
最优多项式交集（OPI）问题要求在有限域上找到一个低次多项式，使其在尽可能多的给定输入上的值位于规定子集中。解码量子干涉测量（DQI）为OPI提供了一种量子算法，后续工作改进了参数范围，但限于平均情况。Sun和Wootters表明即使在最坏情况下，OPI在比DQI覆盖的更大参数范围内有解，但能否设计出超出DQI范围的最坏情况量子算法仍未解决。我们给出了这样一种量子算法，还在某些参数范围内改进了Sun和Wootters的存在性界限。特别地，当每个子集包含约一半域元素时，我们的算法在满足率\(R>0.75\)时找到满足率\(s = 1\)的解，匹配之前平均情况界限，而DQI除非\(R = 1\)否则无法达到\(s = 1\)。我们的存在性界限保证在\(R>0.7158\)时有解，优于之前阈值\(R>0.7495\)。更一般地，我们的存在性结果扩展到关于任意最大距离可分（MDS）码的最大线性可满足性（Max-LINSAT）问题。相应算法结果仅适用于对偶允许有效列表解码器的MDS码。我们的结果通过在MDS设置中新颖应用Brascamp-Lieb型不等式获得，可能有进一步应用。
英文摘要
The Optimal Polynomial Intersection (OPI) problem asks us to find a low-degree polynomial over a finite field whose values lie in prescribed subsets on as many given inputs as possible. Decoded quantum interferometry (DQI) gives a quantum algorithm for OPI in parameter regimes beyond those achieved by the best known classical heuristics. Follow-up works improve the parameter regimes, but their analyses are limited to average-case settings. Recently, Sun and Wootters showed that, even in the worst case, OPI has a solution in a larger parameter regime than the one covered by DQI. However, they left open whether one can design a quantum algorithm that solves OPI in the worst-case beyond the DQI regime. We give such a quantum algorithm. As a byproduct, we also improve the existential bound of Sun and Wootters in certain parameter regimes. In particular, when each subset contains roughly half of the field elements, our algorithm finds a solution with satisfaction rate $s=1$ whenever the rate satisfies $R>0.75$. This matches the previous average-case bound, whereas DQI cannot achieve $s=1$ unless $R=1$. Our existential bound guarantees the existence of a solution when $R> 0.7158$, improving over the previous threshold $R>0.7495$. More generally, our existential results extend to the Max-LINSAT problem with respect to arbitrary maximum distance separable (MDS) codes. The corresponding algorithmic results apply only to MDS codes whose dual admits an efficient list decoder. Our results are obtained through a novel application of a Brascamp--Lieb-type inequality in the MDS setting, which may have further applications.
