A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes (2401.06874v3)

Abstract: Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.

• The paper proposes a joint optimization of QLDPC code construction and belief propagation decoding to overcome four-cycle limitations in Tanner graphs.
• It presents QC-based and finite geometry-based methods that achieve a minimum distance near the square root of the block length, balancing performance and complexity.
• Simulation results demonstrate significantly improved frame error rates under depolarizing channels, underscoring potential for efficient quantum error correction.

A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes

In this paper, the authors propose a novel approach to designing quantum low-density parity-check (QLDPC) codes and corresponding decoders to address the challenges faced by existing quantum error correction schemes. Despite the promising potential of QLDPC codes for quantum error correction (QEC), they often suffer from sub-optimal decoding performance due to the presence of short cycles in their Tanner graphs, particularly cycles of length 4, which hinder effective belief propagation (BP) decoding. This paper presents a methodology that jointly optimizes QLDPC code construction and BP decoder design, crucially mitigating the adverse effects of these short cycles.

The proposed framework involves constructing QLDPC codes with a minimum distance approximately equal to the square root of the block length, representing a balance between error correction capability and implementation complexity. Notably, this approach results in the first known family of QLDPC codes where a quaternary BP decoder is not significantly compromised by the presence of four-length cycles. The ensemble BP decoder introduced in the paper effectively mitigates the influence of these short cycles by assembling them on a single variable node and decimating a bit within each BP decoding path, akin to strategies employed with decimation in classical coding theory.

Two code construction methods are outlined—based on classical quasi-cyclic (QC) codes and finite geometry (FG) codes. The QC code construction utilized exploits properties of cyclic permutation matrices from tailored base matrices, ensuring that pairs of base matrix rows differ by vectors with specific properties that align with the requirements for cycle mitigation. The FG-based construction employs incidence vectors of finite geometries to form PCMs that inherently satisfy required conditions for four-cycle mitigation. Empirical results suggest that these methods yield QLDPC codes with superior depolarizing channel performance compared to existing approaches that require more sophisticated and computationally expensive decoders, such as BP-OSD.

The paper provides detailed simulation results that demonstrate the efficacy of the proposed joint construction and decoding scheme, showing significant improvements in the frame error rate (FER) over various depolarizing probabilities. Results underscore the importance of using the ensemble BP decoder in mitigating cycle-induced errors compared to single-path BP techniques.

The theoretical implications of this work suggest a reevaluation of traditional QLDPC decoding strategies by incorporating decoder-construction co-design, thus allowing QEC to benefit from latency improvements and enhanced error correction performance. Practically, the proposed methodology holds promise for real-world quantum communication systems requiring efficient, low-complexity decoding techniques suitable for near-term quantum devices.

Future research could explore further optimization of the ensemble decoder, potentially integrating machine learning techniques to improve the guessing of variable node assignments or adapting the method to other types of error models beyond depolarizing channels. As quantum technologies continue to advance, enhancing algorithms that underpin their error correction mechanisms will remain a pivotal area of paper, with this paper providing a meaningful contribution toward that effort.