Entanglement-Assisted Quantum Quasi-Cyclic LDPC Codes with Transversal Logical Operators

Abstract

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.