Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has been well investigated in classical coding theory thanks to their high performance and compatibility with low-latency decoders. We describe toric codes as quantum counterparts of classical two-dimensional spatially-coupled (2D-SC) codes, and introduce quantum spatially-coupled (QSC) codes as a generalization. We use the convolutional structure to represent the parity check matrix of a 2D-SC code as a polynomial in two indeterminates, and derive an algebraic condition that is both necessary and sufficient for a 2D-SC code to be a stabilizer code. This algebraic framework facilitates the construction of new code families. While not the focus of this paper, we note that small memory facilitates physical connectivity of qubits, and it enables local encoding and low-latency windowed decoding. In this paper, we use the algebraic framework to optimize short cycles in the Tanner graph of 2D-SC HGP codes that arise from short cycles in either component code. While prior work focuses on QLDPC codes with rate less than 1/10, we construct 2D-SC HGP codes with small memory, higher rates (about 1/3), and superior thresholds.

翻译：空间耦合码（SC）是一类卷积LDPC码，在经典编码理论中得到了很好的研究，因为它们具有高性能和兼容性低延迟译码器。我们将托里克码描述为二维空间耦合（2D-SC）码的量子对应物，介绍了量子空间耦合（QSC）码作为一种概括。我们使用卷积结构将2D-SC码的奇偶校验矩阵表示为两个不定元中的多项式，并导出一个代数条件，既必要又充分，使2D-SC码成为稳定子码。这个代数框架有助于构建新的代码族。虽然不是本文的重点，但我们注意到小内存有助于量子比特的物理连接，并且它能够进行本地编码和低延迟窗口译码。在本文中，我们使用代数框架来优化2D-SC HGP码的Tanner图中的短循环，这些短循环来自于任何组成分代码的短循环。虽然之前的工作集中于速率小于1/10的QLDPC码，但我们构造了小内存、更高速率（约1/3）和更优阈值的2D-SC HGP码。