Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.
Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.
Due to code degeneracy and the graph structure of quantum low-density parity-check (QLDPC) codes, the performance of conventional belief propagation (BP) decoding can be poor. Recently, belief propagation guided decimation (BPGD) has shown promise to combat these challenges. In this paper, we investigate various decimation approaches to improve the error correcting performance and convergence speed of BPGD. We first consider soft decimation, where the BP equations are modified via several tuneable parameters. This approach exhibits linear complexity relative to the length of the block code and is shown to outperform hard decimation approaches for careful selection of the algorithm parameters. We then combine the approaches in a "soft-hard" BPGD variant, where hard decisions are periodically made and those symbols are permanently fixed throughout the remainder of the decoding process. Simulation results show that further performance improvement can be observed in this case at the cost of increasing the algorithmic complexity.
Quantum low-density parity-check (QLDPC) codes are a leading approach to quantum error correction, yet conventional belief propagation (BP) decoders often perform poorly, primarily due to non-convergence exacerbated by stabilizer constraints, which induce short cycles and degeneracy. We propose two scheduling variants, sequential check node scheduling (SCNS) and sequential variable node scheduling (SVNS), that improve BP's error-correction ability by processing check nodes (CNs) or variable nodes (VNs), respectively, in a fixed order, stabilizing message updates and reducing stalls. We also employ this technique to an improved BP-variant called BP guided decimation (BPGD), where symbols are progressively fixed during decoding iterations. Here, we demonstrate that the sequential BPGD (SBPGD) decoder can further improve the convergence properties and performance of the decoder. On standard QLDPC benchmarks under a Pauli-X noise model, our sequential schedules are shown to lower the block error rate relative to conventional BP, and SBPGD outperforms BPGD while using significantly fewer decimation rounds, translating to lower computational cost. These results demonstrate that changing the update schedule, without altering the code, can improve both the reliability and efficiency of BP-based decoding for QLDPC codes. For the [[1922,50,16]] C2 hypergraph-product code with independent X errors, SVNS-BP surpasses BP-OSD-0 in error correction at roughly the same complexity as standard BP.
Quantum low density parity-check (QLDPC) codes represent an attractive candidate for achieving fault-tolerant quantum computation. Erasure decoding of QLDPC codes has gained traction recently as many physical systems such as neutral-atom systems, photonic systems, trapped-ion systems etc. suffer from qubit erasures. Techniques for erasure decoding of QLDPC codes via belief propagation (BP) have been proposed in the literature, such as BP with guided decimation (BP-GD), where decimation refers to sequentially fixing hard values for the variable nodes. We provide an alternative decimation-based BP algorithm, which we term the BP with degree-based decimation (BP-DD) algorithm, that intelligently chooses a check node based on its degree and subsequently decimates a variable node in its neighborhood. We apply the BP-DD algorithm to two families of QLDPC codes, namely hypergraph product codes and lifted product codes, and show improved logical error performance over the BP-GD algorithm, without incurring additional complexity. Our simulations show that the BP-DD algorithm provides a versatile solution for erasure decoding of QLDPC codes, in contrast to some techniques in the literature that are applicable to only specific classes of QLDPC codes.
In this paper, we propose a belief-propagation (BP)-based decoder, termed the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for quantum low-density parity-check (QLDPC) codes. The key idea is to generate \emph{structured decoding diversity} by constructing multiple redundant parity-check representations via cycle-free subtree decompositions of the Tanner graph, and running BP decoding in parallel across these representations. This extends the classical Multiple-Bases Belief-Propagation (MBBP) framework to the quantum setting while preserving the linear-time complexity and efficiency of standard BP decoding, and avoids the need for super-linear post-processing. Simulation results demonstrate that MBBP-LD improves upon existing BP-based decoders, including BP with ordered statistics decoding (BP-OSD) and belief propagation with guided decimation (BPGD) across several QLDPC codes, while requiring substantially fewer total BP iterations. For bivariate bicycle codes [[144,12,12]] and [[288,12,18]], MBBP-LD achieves up to 20% reduction in error rate compared to BPGD and up to 30% compared to BP-OSD in the low- and moderate-error regimes. For the larger B1 code [[882,24,18≤d≤24]], MBBP-LD attains comparable or improved performance relative to BPGD while maintaining BP-like decoding latency under parallel implementation.
Quantum low-density parity-check (QLDPC) codes are promising candidates for low-overhead fault-tolerant quantum computation, but their practical use requires fast, low-complexity, and reliable decoders. Belief propagation (BP) is attractive because of its local message-passing structure, yet standard flooding BP often suffers from convergence failures on QLDPC codes due to short cycles, degeneracy, and symmetric decoding trajectories. Learned sequential BP improves convergence by using a reinforcement-learning policy to choose variable-node update orders, but a single learned trajectory can still be sensitive to unfavorable local Pauli decisions. We propose a learned parallel bit-flipping sequential BP decoder with a new quaternary score combining syndrome gain, a quantized Pauli log-likelihood penalty, and Q-table lookahead at hypothetical post-flip states. Our decoder first runs learned sequential BP for a fixed number of iterations. If the syndrome is not satisfied, it constructs quaternary bit-flipping candidates, each corresponding to changing the current Pauli decision of one qubit. The selected candidates initialize independent learned sequential BP continuations from the same decoder state. Since these continuations are independent, they can be executed in parallel, so testing several candidates mainly increases parallel hardware resources rather than sequential decoding latency. Simulations on representative QLDPC codes over the depolarizing channel show that the proposed decoder improves the reliability of learned sequential BP while having a parallel low-latency structure.
Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.