Fully Parallelized BP Decoding for Quantum LDPC Codes Can Outperform BP-OSD

TL;DR

This work tackles the challenge of real-time decoding for quantum LDPC codes by introducing BP-SF, a fully parallelizable belief-propagation decoder with speculative syndrome flips guided by oscillation statistics. By avoiding costly Gaussian elimination and exploiting parallelism, BP-SF achieves logical error rates comparable to BP-OSD while delivering substantial latency reductions across code families and noise models. The approach is supported by thorough analyses of BP dynamics, a novel oscillation-guided candidate selection, and extensive simulations on code-capacity and circuit-level noise, including bivariate bicycle codes and SHYPS variants. The results indicate strong potential for hardware-friendly, real-time quantum error correction in scalable architectures, with the main limitations arising in circuit-level regimes that require more trial decoding, and future work focusing on candidate selection, sampling strategies, and inner BP improvements for further gains.

Abstract

This work presents a hardware-efficient and fully parallelizable decoder for quantum LDPC codes that leverages belief propagation (BP) with a speculative post-processing strategy inspired by classical Chase decoding algorithm. By monitoring bit-level oscillation patterns during BP, our method identifies unreliable bits and generates multiple candidate vectors to selectively flip syndromes. Each modified syndrome is then decoded independently using short-depth BP, a process we refer to as BP-SF (syndrome flip). This design eliminates the need for costly Gaussian elimination used in the current BP-OSD approaches. Our implementation achieves logical error rates comparable to or better than BP-OSD while offering significantly lower latency due to its high degree of parallelism for a variety of bivariate bicycle codes. Evaluation on the [[144,12,12]] bivariate bicycle code shows that the proposed decoder reduces average latency to approximately $70\%$ of BP-OSD. When post-processing is parallelized the average latency is reduced by $55\%$ compared to the single process implementation, with the maximum latency reaching as low as $18\%$. These advantages make it particularly well-suited for real-time and resource-constrained quantum error correction systems.

Figures (17)

• Figure 1: (a) Example of BP failing to converge due to oscillations. Red Xs denote bits identified as erroneous by BP, and yellow squares represent unsatisfied syndrome checks. All four bits are oscillating. (b) One oscillating bit (e.g., the rightmost one) is selected, and its neighboring syndromes are flipped. BP then converges since the two competing error patterns now have different weights. (c) After convergence, the selected bit is flipped back to restore consistency with the original input syndrome.
• Figure 2: Ratio of unsuccessful BP decoding ($1-$convergence rate) on the $\llbracket144,12,12\rrbracket$ code under the circuit-level noise model. The maximum number of decoder iterations is set to 1,000 and number of samples is 10,000.
• Figure 3: Precision and recall probabilities of candidate bit selection on the $\llbracket144,12,12\rrbracket$ code. We evaluate the correlation between candidate bits and actual error locations by identifying the top 50 most frequently flipped bits among approximately 8,000 error mechanisms. The decoder is run with a maximum of 50 iterations, and statistics are collected over 1,000 decoding failures. This analysis reveals how well bit-level oscillation can serve as a heuristic for error localization.
• Figure 4: A simplified flowchart of the proposed decoder. The full procedure is described in Algorithm \ref{['alg:mybp']}.
• Figure 5: Error rates of the $\llbracket 154, 6, 16\rrbracket$ coprime-BB code under the code capacity model.