Two-dimensional Entanglement-assisted Quantum Quasi-cyclic Low-density Parity-check Codes

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.