Entanglement-Assisted Quantum Quasi-Cyclic Low-Density Parity-Check Codes

TL;DR

This paper presents a method for constructing entanglement-assisted quantum LDPC codes from classical quasi-cyclic LDPC codes, avoiding dual-containing constraints and improving decoding performance with minimal shared entanglement.

Contribution

It introduces a novel construction of quantum QC-LDPC codes that do not require dual-containing classical codes, leveraging entanglement to eliminate 4-cycles and enhance decoding.

Findings

Quantum codes constructed with minimal entanglement.

Avoidance of 4-cycles improves decoding performance.

Efficient decoding algorithms like SPA are applicable.

Abstract

We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.