The scalability and interpretability of message-passing (MP) decoding, such as (quaternary) Belief Propagation, remain open challenges in quantum error correction. Even for surface codes, arguably the first testbed for decoding methods, studies of improved MP decoders have mostly been restricted to small distances ($d \lesssim 19$). Moreover, the mismatch with established message-passing theory limits the decoder's interpretability, making it unclear whether MP decoding can sustain its effectiveness at large system sizes. This work takes a step toward a more principled and interpretable MP decoding framework, with the goal of making MP-based decoding more reliable and bridging theory and practice. We introduce a dilution method, which allows a quaternary Min-Sum (MS) decoder to exhibit an apparent depolarizing threshold of $16\%$ up to distance $20$, outperforming Minimum-Weight Perfect Matching in finite-length regimes. Notably, for $X$-noise, the standard MS decoder under dilution has worst-case complexity $O(N \log^2 d)$ and outperforms BP-OSD at $d=65$. The observed $\sim 9\%$ threshold may correspond to a true asymptotic threshold. Finally, we give a graph-dilution argument that interprets the success of the dilution method and offers insight into when MP algorithms can genuinely scale. Taken together, these results provide encouraging progress toward scalable and interpretable MP decoding in quantum error correction.

翻译：消息传递（MP）译码（如四进制置信传播）的可扩展性和可解释性仍然是量子纠错中未解决的挑战。即使对于表面码（可以说是译码方法的第一个测试平台），对改进的MP译码器的研究也大多局限于小距离（$d \lesssim 19$）。此外，与已建立的消息传递理论的不匹配限制了译码器的可解释性，使得人们不清楚MP译码是否能在大规模系统下保持其有效性。本文朝着更规范、更具可解释性的MP译码框架迈出了一步，旨在使基于MP的译码更可靠，并弥合理论与实践之间的差距。我们引入了一种稀释方法，该方法使得四进制最小和（MS）译码器在距离达到20时表现出明显的去极化阈值为$16\%$，在有限长度区间内优于最小权重完美匹配。值得注意的是，对于$X$噪声，标准MS译码器在稀释下的最坏情况复杂度为$O(N \log^2 d)$，并在$d=65$时优于BP-OSD。观察到的约$9\%$的阈值可能对应于一个真实的渐近阈值。最后，我们给出了一个图稀释论点，该论点解释了稀释方法的成功，并提供了关于MP算法何时能够真正实现可扩展性的见解。综合来看，这些结果为量子纠错中可扩展且可解释的MP译码提供了令人鼓舞的进展。