Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry

TL;DR

This paper proves tight inapproximability bounds for max-LINSAT, showing no polynomial-time algorithm can do better than a certain ratio unless P=NP, with implications for decoded quantum interferometry and quantum advantage boundaries.

Contribution

It establishes the first tight inapproximability bounds for max-LINSAT and links these bounds to quantum interferometry, revealing fundamental hardness thresholds.

Findings

No polynomial-time algorithm can exceed the ratio r/q for max-LINSAT unless P=NP.

The inapproximability threshold matches the semicircle law limit in quantum interferometry.

Surpassing the threshold requires exploiting instance structures beyond standard PCP reductions.

Abstract

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 H\r{a}stad'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.