Degenerate Quantum LDPC Codes With Good Finite Length Performance (1904.02703v3)

Abstract: We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

• The paper introduces an OSD-inspired post-processing step that enhances the decoding performance of degenerate quantum LDPC codes.
• Simulations show that the improved BP-OSD decoder achieves orders of magnitude better error rates than conventional belief propagation for various QLDPC families.
• The study presents new generalized bicycle and hypergraph product codes that offer superior dimensions and error-correcting properties compared to surface codes.

An In-depth Analysis of Degenerate Quantum LDPC Codes with Enhanced Decoding Strategies

The paper, "Degenerate Quantum LDPC Codes with Good Finite Length Performance" by Pavel Panteleev and Gleb Kalachev, addresses significant challenges associated with quantum low-density parity-check (QLDPC) codes, particularly focusing on their practical error-correcting capabilities over depolarizing channels. Grounded in the field of quantum error correction, the authors primarily explore degenerate QLDPC codes where stabilizer weights are substantially smaller than the minimum distance, a property observed to complicate decoding performance with traditional belief propagation (BP) techniques.

A pivotal contribution of the paper is the enhancement of the BP decoder via an Ordered Statistics Decoding (OSD) inspired post-processing step. By leveraging an OSD-like algorithm, the authors demonstrate an impressive improvement in decoding efficacy, achieving orders of magnitude better performance compared to conventional BP for a range of QLDPC codes, including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. Through simulations, these advancements are compared against classical surface codes, showing that even with near-optimal decoders, the performance of some newly constructed codes with BP-OSD post-processing can surpass surface codes of similar lengths.

The introduction of generalized bicycle codes and generalized hypergraph product (GHP) codes forms another cornerstone of this research. By exploiting a specific structure within circulant matrices, these new classes yield notable increases in code dimension with favorable error correction properties. The work extends the notion of hypergraph product codes and elucidates the capacity to maintain high error rates while also improving the minimum distance, particularly when one of the parity-check matrices is taken as square.

Quantitative analysis reveals that the proposed BP-OSD post-processing has a transformative impact on previously challenging to decode degenerate codes, realizing a situation where, for example, a 10-limited generalized hypergraph product [[1270, 28]] code shows more robust performance against a similar-length surface code under standard conditions when decoded using the BP-OSD mechanism. The findings support a broader speculation that adjusting decoding strategies can extract enhanced performance from otherwise inadequately-performing degenerate constructions.

This investigation opens several theoretical avenues, such as the necessity for refined estimations of minimum distances for these codes and potential generalizable decoding algorithms applicable in higher-dimensional quantum error correction scenarios. Practically, the work underscores the feasibility of implementing such refined decoding approaches in quantum computing architectures, potentially smoothing the transition toward error-tolerant quantum systems.

Future research directions could extend to examining the applicability of the BP-OSD strategy across other quantum noise models, refining the algorithm for different classes of stabilizer codes, and investigating the upper bounds of performance improvements achievable through continued decoding innovation. This paper marks an insightful journey into optimizing QLDPC codes, bridging the gap between theoretical potential and practical realization.