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 hypergraph product (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 memories, higher rates (about 1/3), and superior thresholds.

翻译：空间耦合（SC）码是一类卷积LDPC码，因其高性能及与低时延解码器的兼容性而在经典编码理论中受到广泛研究。我们将环面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）码作为其泛化形式。利用卷积结构将二维空间耦合码的校验矩阵表示为两个未定元的多项式，并推导出二维空间耦合码成为稳定子码的充要代数条件。该代数框架有助于构建新型码族。虽然非本文重点，但需指出小内存有利于量子比特的物理连接，且支持本地编码与低时窗解码。本文利用该代数框架优化由分量码短环产生的二维空间耦合超图乘积（HGP）码的Tanner图短环。不同于先前聚焦于码率低于1/10的QLDPC码的研究，我们构建了具有小内存、高码率（约1/3）及优异门限值的二维空间耦合超图乘积码。