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

• URL: http://arxiv.org/abs/2603.04540v1
• Date: Wed, 04 Mar 2026 19:26:26 GMT
• Title: Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry
• Authors: Maximilian J. Kramer, Carsten Schubert, Jens Eisert,
• Abstract summary: We prove by a direct reduction from Hstad's theorem that no-time algorithm can exceed the randomassignment ratio $r/q$ by any constant.<n>This threshold coincides with the $ell/m to 0$ limit of the semicircle law governing decoded quantum interferometry.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 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: as the decodable structure vanishes, DQI's approximation ratio degrades to exactly the worst-case bound established by our result. Together, these observations delineate the boundary between worst-case hardness and potential quantum advantage, showing that any algorithm surpassing $r/q$ must exploit algebraic structure specific to the instance.

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