Approximability limits for bounded-degree max-LINSAT and implications for decoded quantum interferometry

2026-06-11

AI Summary

This work investigates the approximation limits of the max-LINSAT problem with bounded variable degree over arbitrary finite fields. By integrating computational complexity theory, approximation algorithm analysis, linear algebra over finite fields, and quantum information theory, it establishes—for the first time—the NP-hardness of achieving an approximation ratio beyond the random assignment baseline of \(r/q + O(1/\sqrt{D})\) for general finite fields. Furthermore, the study reveals a fundamental information-theoretic bottleneck in decoding quantum interferometry (DQI): classical decoders are inherently limited to an error scaling of \(1/\sqrt{D \log D}\), whereas quantum decoders achieve the optimal \(1/\sqrt{D}\) scaling. This result underscores the essential role of quantum decoding in aligning with the intrinsic complexity-theoretic structure of DQI.

Abstract

For general max-k-XORSAT with $k \geq 3$, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless $\mathsf{P} = \mathsf{NP}$: approximating beyond the random-assignment value of $1/2$ is $\mathsf{NP}$-hard. The picture changes when each variable appears in at most $D$ constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order $1/\sqrt{D}$. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-E$k$-LINSAT$(q,r)$ with bounded degree $D$ and over arbitrary finite fields $\mathbb{F}_q$, proving that it is $\mathsf{NP}$-hard to exceed $r/q + \mathcal{O}_{q,r}(1/\sqrt{D})$. These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on $(k,D)$-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic $1/\sqrt{D \log D}$ barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the $1/\sqrt{D}$ scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.

Problem

• bounded-degree
• max-LINSAT
• approximability
• finite fields
• quantum interferometry

Innovation

• bounded-degree LINSAT
• approximability limits
• decoded quantum interferometry
• quantum decoding
• NP-hardness