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.

翻译：对于一般的max-k-XORSAT（其中$k \geq 3$），在最坏情况下，除非$\mathsf{P} = \mathsf{NP}$，否则没有多项式时间算法能显著优于随机猜测：超越随机赋值值$1/2$的逼近是$\mathsf{NP}$难的。当每个变量最多出现在$D$个约束中时，情况发生变化。在这个有界度设置下，多项式时间算法可以以约$1/\sqrt{D}$的加性量可证明地击败随机基线。对于布尔实例，已知这种缩放是最优的：匹配的困难性结果归功于Trevisan，而相应的算法保证由Barak等人建立。相同的结论是否适用于一般有限域，以及这对量子算法意味着什么，尚未得到确定。我们使这一联系明确，并将困难性扩展到具有有界度$D$且定义在任意有限域$\mathbb{F}_q$上的max-E$k$-LINSAT$(q,r)$，证明超过$r/q + \mathcal{O}_{q,r}(1/\sqrt{D})$是$\mathsf{NP}$难的。这些结果为解码量子干涉测量（DQI）、QAOA和经典启发式算法所针对的有界度实例提供了复杂性理论基准。因此，有界度实例上的任何量子优势都仅限于常数前因子。我们进一步证明，在DQI的背景下，对于$(k,D)$-正则实例，该前因子对解码器的性质敏感：使用经典解码器的DQI面临信息论上的$1/\sqrt{D \log D}$障碍，使其无法匹配困难性缩放，而使用量子解码器的DQI则与$1/\sqrt{D}$缩放兼容——这表明量子解码是使DQI匹配复杂性理论缩放的关键要素。