For any positive integer $g \ge 2$, we derive general condition 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. Depending on whether $p$ is an odd prime or a composite number, we construct two distinct families of $2$-D classical QC-LDPC codes with girth $>4$ by stacking $p \times p \times p$ tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of $2$-D classical QC-LDPC codes with girth $>6$, constructed via a similar tensor-stacking approach. All the proposed $2\text{-D}$ classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$. Based on the constructed $2\text{-D}$ classical QC-LDPC codes, we derive two families of $2\text{-D}$ entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2\text{-D}$ EA-QLDPC codes is obtained from a pair of $2\text{-D}$ classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of $4$-cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2\text{-D}$ classical QC-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。根据 $p$ 为奇素数或合数的不同情况，我们通过堆叠 $p \times p \times p$ 张量构造了两个不同的二维经典 QC-LDPC 码族，其围长均大于 4。此外，利用广义 Behrend 序列，我们提出了另一个二维经典 QC-LDPC 码族，其围长大于 6，该码族通过类似的张量堆叠方法构造。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。基于所构造的二维经典 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 码族由一对二维经典 QC-LDPC 码获得，其设计使得所得 EA-QLDPC 码 Tanner 图中无需辅助的部分不含 4 环，且仅需在量子收发器之间共享单个纠缠比特。第二个码族由单个二维经典 QC-LDPC 码构造，其 Tanner 图不含 4 环。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。