We derive two families of EA-QC quantum LDPC (EA-QC-QLDPC) codes by tiling permutation matrices of prime and composite orders. The unassisted portion of the Tanner graphs corresponding to these codes, constructed from two distinct classical QC-LDPC codes, exhibits girth greater then 4 an essential property for effective error correction. We analytically derive the exact code rate of the proposed constructions. Remarkably, one of these families requires only a single Bell pair to be shared between the quantum transmitter and receiver. Furthermore, two additional families of EA-QC-QLDPC codes are constructed based on a single classical code, whose Tanner graphs exhibit girths exceeding six, thereby further enhancing the error-correction capability. For one of these families, we explicitly determine the transversal logical operators an aspect that is typically non-trivial for random quasi-cyclic codes. The performance of the proposed codes is assessed under both random and burst error models under the depolarizing and Markovian noise actions. Employing a modified sum-product decoding algorithm over a quaternary alphabet, we demonstrate that correlated Pauli errors can be effectively addressed within the decoding framework. Simulation results reveal nearly an order of improvement in error-correction performance with the quaternary decoder compared to the binary decoder over both depolarizing and Markovian channels. Further, the proposed codes are compared with existing ones, demonstrating significant improvement.

翻译：本文通过拼接素数阶和合数阶置换矩阵，推导出两个EA-QC-QLDPC码族。这些码对应的Tanner图中非辅助部分由两个不同的经典QC-LDPC码构建，其围长大于4，这是实现有效纠错的关键特性。我们通过解析方法推导出所提出构造的确切码率。值得注意的是，其中一个码族仅需在量子发射端与接收端之间共享一对贝尔态即可实现。此外，基于单一经典码构建了另外两个EA-QC-QLDPC码族，其Tanner图围长超过六，从而进一步提升了纠错能力。针对其中一个码族，我们显式确定了横向逻辑算子——这对于随机准循环码通常是非平凡的特性。所提出码的性能在退极化噪声和马可夫噪声作用下的随机与突发错误模型中进行评估。采用基于四元字母表的改进和积解码算法，我们证明相关泡利错误可在解码框架内得到有效处理。仿真结果表明，在退极化信道和马可夫信道上，四元解码器相比二元解码器的纠错性能提升近一个数量级。进一步地，所提出码与现有码进行比较，展现出显著改进。