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arXiv · 2603.04540

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

Maximilian J. Kramer·Carsten Schubert·Jens Eisert
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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.

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Maximilian J. Kramer, Carsten Schubert, Jens Eisert. 2026-03-04. Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry. https://arxiv.org/abs/2603.04540

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