For any positive integer $g \ge 2$, we derive general conditions for the existence of a $2g$-cycle in the Tanner graph of two-dimensional ($2$-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Based on these conditions, we construct a family of $2$-D classical QC-LDPC codes with girth greater than $4$ by stacking $p \times p \times p$ tensors, where $p$ is an odd prime. Furthermore, for composite values of $p$, we propose two additional families of $2$-D classical LDPC codes obtained via similar tensor stacking. In this case, one family achieves girth greater than $4$, while the other attains girth greater than $6$. All the proposed $2$-D classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$. Based on the constructed classical $2$-D QC-LDPC codes, we derive two families of $2$-D entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2$-D EA-QLDPC codes is obtained from a pair of binary $2$-D classical LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of cycles of length four, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2$-D classical LDPC code whose Tanner graph is free from $4$-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of $p \times p$, as the underlying classical codes possess the same erasure correction property.

翻译：对于任意正整数 $g \ge 2$，我们推导了二维经典准循环低密度奇偶校验码的 Tanner 图中存在 $2g$-环的一般条件。基于这些条件，我们通过堆叠 $p \times p \times p$ 张量构造了一族围长大于 $4$ 的二维经典 QC-LDPC 码，其中 $p$ 为奇素数。此外，对于 $p$ 为合数的情况，我们通过类似的张量堆叠方法提出了另外两个二维经典 LDPC 码族。在这种情况下，其中一个码族实现了围长大于 $4$，而另一个码族达到了围长大于 $6$。所有提出的二维经典 QC-LDPC 码均展现出至少 $p \times p$ 的擦除纠正能力。基于所构造的经典二维 QC-LDPC 码，我们推导了两个二维纠缠辅助量子低密度奇偶校验码族。第一个二维 EA-QLDPC 码族由一对二进制二维经典 LDPC 码生成，其设计使得所得 EA-QLDPC 码的 Tanner 图中非辅助部分不含长度为四的环，且仅需在量子收发器之间共享单个纠缠比特。第二个码族由单个 Tanner 图不含 $4$-环的二维经典 LDPC 码构造而成。此外，由于底层经典码具有相同的擦除纠正特性，所构造的 EA-QLDPC 码继承了 $p \times p$ 的擦除纠正能力。