The Optimal Polynomial Intersection (OPI) problem is the following: Given sets $S_1, \ldots, S_m \subseteq \mathbb{F}$ and evaluation points $a_1, \ldots, a_m \in \mathbb{F}$, find a polynomial $Q \in \mathbb{F}[x]$ of degree less than $n$ so that $Q(a_i) \in S_i$ for as many $i \in \{1, 2, \ldots, m\}$ as possible. Decoded Quantum Interferometry (DQI) is a quantum algorithm that efficiently returns good solutions to the problem, even on worst-case instances (Jordan et. al., 2025). The quality of the solutions returned follows a semicircle law, which outperforms known efficient classical algorithms. But does DQI obtain the best possible solutions? That is, are there solutions better than the semicircle law for worst-case OPI instances? Surprisingly, before this work, the best existential results coincide with (and follow from) the best algorithmic results. In this work, we show that there are better solutions for worst-case OPI instances over prime fields. In particular, DQI and the semicircle law are not optimal. For example, when the lists $S_i$ have size $ρp$ for $ρ\sim 1/2$, our results imply the existence of a solution that asymptotically beats the semicircle law whenever $n/m \geq 0.6225$, and we show that an asymptotically perfect solution exists whenever $n/m \geq 0.7496$. Our results generalize to Max-LINSAT problems derived from any Maximum Distance Separable (MDS) code, and to any $ρ\in (0,1)$. The key insight to our improvement is a connection to local leakage resilience of secret sharing schemes. Along the way, we recover several re-proofs of the existence of solutions achieving the semicircle law.

翻译：最优多项式交集（OPI）问题定义如下：给定集合 $S_1, \ldots, S_m \subseteq \mathbb{F}$ 和评估点 $a_1, \ldots, a_m \in \mathbb{F}$，寻找一个次数小于 $n$ 的多项式 $Q \in \mathbb{F}[x]$，使得对于尽可能多的 $i \in \{1, 2, \ldots, m\}$，满足 $Q(a_i) \in S_i$。解码量子干涉测量（DQI）是一种量子算法，它能在最坏情况实例上也高效返回该问题的良好解（Jordan 等人，2025 年）。解的质量遵循半圆律，这优于已知的高效经典算法。但 DQI 是否获得了可能的最优解？也就是说，对于最坏情况 OPI 实例，是否存在优于半圆律的解？令人惊讶的是，在本文工作之前，最佳存在性结果与最佳算法结果一致（并由此推得）。本文证明，在素域上针对最坏情况 OPI 实例存在更优的解。特别地，DQI 和半圆律并非最优。例如，当列表 $S_i$ 的大小为 $ρp$ 且 $ρ\sim 1/2$ 时，我们的结果暗示：只要 $n/m \geq 0.6225$，就存在一个渐近地优于半圆律的解；并且我们证明，当 $n/m \geq 0.7496$ 时，存在一个渐近完美的解。我们的结果可推广到由任意最大距离可分（MDS）码导出的 Max-LINSAT 问题，并适用于任意 $ρ\in (0,1)$。改进的关键洞见在于与秘密共享方案的局部泄漏弹性的联系。在此过程中，我们恢复了若干关于存在达到半圆律的解的重新证明。