max-LINSAT的紧近似界限及其对解码量子干涉仪的启示
Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry
AI总结：
本文研究了max-LINSAT问题的紧近似界限，并探讨了其对解码量子干涉仪潜在优势的影响，证明了在P≠NP假设下，任何多项式时间算法无法超越随机赋值比r/q的界限。
摘要：
我们建立了max-LINSAT问题的紧近似界限，该问题是在有限域F_q上最大化满足的线性约束数量的问题，其中每个约束接受r个值。具体来说，我们通过直接从Håstad定理的归约证明，假设P≠NP，任何多项式时间算法都无法超越随机赋值比r/q的界限。这个阈值与支配解码量子干涉仪（DQI）的半圆定律中的ℓ/m→0极限一致，其中ℓ是底层码的解码半径。这些观察结果界定了最坏情况难度与潜在量子优势之间的边界，表明任何超越r/q的算法必须利用实例结构，而这种结构在由PCP归约产生的硬实例中并不存在。
英文摘要：
We establish tight inapproximability bounds for max-LINSAT, the problem of maximizing the number of satisfied linear constraints over the finite field $\mathbb{F}_q$, where each constraint accepts $r$ values. Specifically, we prove by a direct reduction from Håstad's theorem that no polynomial-time algorithm can exceed the random-assignment ratio $r/q$ by any constant, assuming $\mathsf{P} \neq \mathsf{NP}$. This threshold coincides with the $\ell/m \to 0$ limit of the semicircle law governing decoded quantum interferometry (DQI), where $\ell$ is the decoding radius of the underlying code. Together, these observations delineate the boundary between worst-case hardness and potential quantum advantage, showing that any algorithm surpassing $r/q$ must exploit instance structure beyond what is present in the hard instances produced by PCP reductions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
