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 active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. 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撑。众所周知，在经典场景中，在保持正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）译码性能与较大的最小距离。然而在量子场景中，这一原理无法直接适用，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。当以直接方式同时强制正交性与正则性时，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定于构造中的活跃部分，同时保持正则的校验矩阵结构，从而突破了这一限制。该设计规避了由母矩阵正交性引起的传统结构距离限制，使得在避免潜在低权重逻辑算子的同时，能够构造具有大围长的量子LDPC码。作为具体示例，我们构造了一个围长为8的(3,12)-正则$[[9216,4612, \leq 48]]$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。