Several Families of Entanglement-Assisted Quantum Quasi-Cyclic LDPC Codes

We introduce several families of entanglement-assisted (EA) Calderbank-Shor-Steane (CSS) codes derived from two distinct classes of low-density parity-check (LDPC) codes. We derive two families of EA quantum QC-LDPC codes, namely, the spatially coupled (SC) and the non-spatially coupled cases. These two families are constructed by tiling permutation matrices of prime and composite orders. We establish several code properties along with conditions for guaranteed girth for the proposed code families. The Tanner graphs of the proposed EA quantum QC-LDPC and EA quantum QC-SC-LDPC codes have girths greater than four, which is required for good error correction performance. Some of the proposed families of codes require only \textit{minimal} Bell pairs to be shared across the quantum transceiver. Furthermore, we construct two families of EA quantum QC-LDPC codes based on a single classical code, with Tanner graphs having girths greater than six, further improving the error correction performance. We evaluate the performance of these codes using both depolarizing and Markovian noise models to assess the random and burst error performance. Using a modified version of the sum-product algorithm over a quaternary alphabet, we show how correlated Pauli errors can be handled within the decoding setup. Simulation results show that nearly an order of improvement in the error correction performance can be achieved with quaternary decoder compared to binary decoder over the depolarizing and Markovian error channels, thereby generalizing the approach of EA quantum QC-LDPC code designs to work with both random and burst quantum error models, useful in practice.