Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the necessary parts of the construction, while preserving regular check-matrix structures. This design breaks the conventional trade-off between orthogonality, regularity, girth, and minimum distance, allowing us to construct quantum LDPC codes with large girth and without the usual distance upper bounds. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 \%$.

翻译：经典低密度奇偶校验（LDPC）码是一项广泛应用且成熟的技术，构成了现代通信与存储系统的核心支柱。众所周知，在经典场景中，在保持规则度分布的同时增大Tanner图的围长，能够同时实现良好的置信传播（BP）译码性能和较大的最小距离。然而在量子场景中，这一原理并不直接适用，因为量子LDPC码必须满足其奇偶校验矩阵之间额外的正交性约束。当以简单直接的方式同时强制正交性和规则性时，围长通常会减小，且最小距离在结构上存在上界。本工作通过采用具有受控交换性的置换矩阵，并将正交性约束仅限制在构造的必要部分，同时保持校验矩阵的规则结构，从而突破了这一限制。该设计打破了正交性、规则性、围长和最小距离之间的传统权衡，使我们能够构造具有大围长且不受常规距离上界限制的量子LDPC码。作为具体示例，我们构造了一个围长为8、(3,12)-规则的$[[9216,4612, \leq 48]]$量子LDPC码，并证明在结合低复杂度后处理算法的BP译码下，该码在错误概率为$4\%$的退极化信道上可实现低至$10^{-8}$的帧错误率。