Optimal constructions of classical LDPC codes can be obtained by choosing the Tanner graph uniformly at random among biregular graphs. We introduce a class of codes that we call ``diffusion codes'', defined by placing each edge connecting bits and checks on some graph, and acting on that graph with a random SWAP network. By tuning the depth of the SWAP network, we can tune a tradeoff between the amount of randomness -- and hence the optimality of code parameters -- and locality with respect to the underlying graph. For diffusion codes defined on the cycle graph, if the SWAP network has depth $\sim Tn$ with $T> n^{2\beta}$ for arbitrary $\beta>0$, then we prove that almost surely the Tanner graph is a lossless ``smaller set'' vertex expander for small sets up size $\delta \sim \sqrt T \sim n^{\beta}$, with bounded bit and check degree. At the same time, the geometric size of the largest stabilizer is bounded by $\sqrt T$ in graph distance. We argue, based on physical intuition, that this result should hold more generally on arbitrary graphs. By taking hypergraph products of these classical codes we obtain quantum LDPC codes defined on the torus with smaller-set boundary and co-boundary expansion and the same expansion/locality tradeoffs as for the classical codes. These codes are self-correcting and admit single-shot decoding, while having the geometric size of the stabilizer growing as an arbitrarily small power law. Our proof technique establishes mixing of a random SWAP network on small subsystems at times scaling with only the subsystem size, which may be of independent interest.

翻译：经典LDPC码的最优构造可通过在双正则图中均匀随机选择Tanner图实现。本文提出一类称为“扩散码”的编码方案，其通过在特定图上放置连接比特节点与校验节点的边，并对此图施加随机SWAP网络操作而定义。通过调节SWAP网络的深度，我们可在随机性程度（进而影响码参数的最优性）与相对于底层图的局域性之间实现可调节的权衡。对于在循环图上定义的扩散码，若SWAP网络深度满足$\sim Tn$且$T> n^{2\beta}$（$\beta>0$为任意值），我们证明其Tanner图几乎必然成为尺寸不超过$\delta \sim \sqrt T \sim n^{\beta}$的小集合的无损顶点扩展器，同时保持有界的比特度与校验度。与此同时，最大稳定子的几何尺寸在图距离上受$\sqrt T$限制。基于物理直观，我们认为该结论应可推广至任意图结构。通过对这些经典码进行超图积操作，我们获得了定义在环面上的量子LDPC码，其具备小集合边界与共边界扩展特性，且保持与经典码相同的扩展性/局域性权衡关系。这类编码具有自校正特性并支持单次解码，同时其稳定子的几何尺寸可按任意小幂律增长。我们的证明技术建立了随机SWAP网络在子系统尺寸相关的时间尺度上对小系统的混合性质，该结论可能具有独立的研究价值。