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

Maximilian J. Kramer, Carsten Schubert, Jens Eisert

TL;DR

It is proved 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 $\mathsf{P} \neq \mathsf{NP}$.

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.

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

TL;DR

It is proved by a direct reduction from H\r{a}stad's theorem that no polynomial-time algorithm can exceed the random-assignment ratio by any constant, assuming .

Abstract

We establish tight inapproximability bounds for max-LINSAT, the problem of maximizing the number of satisfied linear constraints over the finite field , where each constraint accepts values. Specifically, we prove by a direct reduction from Håstad's theorem that no polynomial-time algorithm can exceed the random-assignment ratio by any constant, assuming . This threshold coincides with the limit of the semicircle law governing decoded quantum interferometry (DQI), where 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 must exploit algebraic structure specific to the instance.
Paper Structure (4 sections, 2 theorems, 8 equations, 1 figure)

This paper contains 4 sections, 2 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Discussion

Key Result

Theorem 4

For every finite Abelian group $\Gamma$ and every $\varepsilon>0$, it is $\mathsf{NP}$-hard to approximate max-E$3$-LIN-$\Gamma$ within a factor $|\Gamma|-\varepsilon$. Equivalently, max-E$3$-LIN-$\Gamma$ is non-approximable beyond the random assignment threshold: it is $\mathsf{NP}$-hard to disting In particular, setting $\Gamma=(\mathbb{F}_q, +)$ for a finite field $\mathbb{F}_q$ yields the $(1-

Figures (1)

  • Figure 1: Landscape of approximability for $\mathrm{max\text{-}LINSAT}(q,r)$. The x-axis parametrizes the decodable structure $\ell/m$, where $\ell$ is the decoding radius of the underlying error-correcting code and $m$ is the number of constraints. The shaded region, spanning approximation ratios from $r/q$ to $1$ across all values of $\ell/m$, is $\mathsf{NP}$-hard to achieve on worst-case instances (\ref{['thm:inapproximability_of_max_linsat']}). The pink curve depicts DQI's approximation ratio as given by the semicircle law Jordan2024DQI, which surpasses the random-assignment bound $r/q$ whenever $\ell/m > 0$ and saturates at $\alpha_{\mathrm{DQI}} = 1$ for $\ell/m \geq 1 - r/q$. In the limit $\ell/m \to 0$, DQI's performance degrades to exactly $r/q$, matching the worst-case barrier.

Theorems & Definitions (8)

  • Definition 1: max-LINSAT over $\mathbb{F}_q$
  • Definition 2: max-LINSAT$(q,r)$
  • Definition 3: max-E$k$-LIN-$q$
  • Theorem 4: Inapproximability of max-E$3$-LIN-$\Gamma$
  • Theorem 5: Inapproximability of max-LINSAT$(q,r)$
  • proof
  • Remark 6: Tightness of \ref{['thm:inapproximability_of_max_linsat']}
  • Remark 7: max-XORSAT