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 spatially-coupled quantum LDPC (SC-QLDPC) 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（SC）码是一类卷积LDPC码，因其高性能以及与低时延译码器的兼容性而在经典编码理论中得到了广泛研究。我们将toric码描述为经典二维空间耦合（2D-SC）码的量子对应物，并引入空间耦合量子LDPC（SC-QLDPC）码作为其推广。利用卷积结构，我们将2D-SC码的校验矩阵表示为两个不定元的多项式，并推导出2D-SC码成为稳定子码的充要代数条件。这一代数框架有助于构造新的码族。尽管本文不重点讨论，但值得指出的是，小存储器有利于量子比特的物理连接，并可实现本地编码和低时延窗口译码。本文利用该代数框架优化2D-SC HGP码Tanner图中的短环，这些短环源于其分量码中的短环。先前的研究聚焦于码率低于1/10的QLDPC码，而本文构造了具有小存储器、更高码率（约1/3）以及优越译码阈值的2D-SC HGP码。