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：空间耦合码是一类卷积LDPC码，因其高性能和低延迟解码器的兼容性而在经典编码理论中得到深入研究。我们提出将环面码视为经典二维空间耦合码的量子对应物，并引入量子空间耦合码作为其推广形式。通过利用卷积结构将二维空间耦合码的奇偶校验矩阵表示为两个不定元的多项式，我们推导出二维空间耦合码成为稳定子码的一个充要代数条件。该代数框架有助于构建新型码族。虽然编码器物理连接性、局部编码能力及低延迟窗口解码功能并非本文重点，但我们注意到小存储器可促进这些特性的实现。本文基于该代数框架，优化了由任意分量码中短环引发的二维空间耦合超图积码坦纳图中的短环。现有研究聚焦于码率低于1/10的量子LDPC码，而我们构建了具有小存储器、较高码率（约1/3）和优异阈值的二维空间耦合超图积码。