Building scalable quantum computers requires quantum error-correcting codes that enable reliable operations in the presence of noise. Motivated by such need, this paper introduces two constructions of high-rate, quantum dual-containing (DC) Calderbank-Shor-Steane (CSS) low-density parity-check (LDPC) codes based on quasi-dyadic matrices. Their DC structure enables the transversal implementation of the Hadamard gate, and, jointly with the sparsity of their parity-check matrices enable low-complexity decoding via a standard binary belief-propagation algorithm. We provide several theoretical results concerning the cycle properties of these CSS codes. We also investigate their automorphism groups as well as their minimum distance. Furthermore, through numerical simulations, we show that the quantum CSS LDPC codes obtained through these constructions achieve better finite-length error rate performance than existing DC codes across different block lengths and code rates.

翻译：构建可扩展量子计算机需要能够抵抗噪声实现可靠操作的量子纠错码。受此需求驱动，本文介绍了两种基于拟对偶矩阵的高码率量子双包含(dual-containing, DC) Calderbank-Shor-Steane (CSS)低密度奇偶校验(LDPC)码的构造方法。其双包含结构使Hadamard门的横向实现成为可能，并且结合其奇偶校验矩阵的稀疏性，可通过标准二元置信传播算法实现低复杂度译码。我们针对这些CSS码的环特性给出若干理论结果，同时研究其自同构群及最小距离。此外，数值仿真表明，通过上述构造方法得到的量子CSS LDPC码在不同码长和码率下均比现有双包含码具有更优的有限长误码率性能。