We describe an empirical approach to identify low-weight combinations of columns of the decoding matrices of a quantum circuit-level noise model, for which belief-propagation (BP) algorithms converge possibly very slowly. Focusing on the logical-idle syndrome cycle of the low-density parity check gross code, we identify criteria providing a characterization of the Tanner subgraph of such low-weight error syndromes. We analyze the dynamics of iterations when BP is used to decode weight-four and weight-five errors, finding statistics akin to exponential activation in the presence of noise or escape from chaotic phase-space domains. We study how BP convergence improves when adding to the decoding matrix relevant combinations of fault columns, and show that the suggested decoder amendment can result in the reduction of both logical errors and decoding time.
We describe an empirical approach to identify low-weight combinations of columns of the decoding matrices of a quantum circuit-level noise model, for which belief-propagation (BP) algorithms converge possibly very slowly. Focusing on the logical-idle syndrome cycle of the low-density parity check gross code, we identify criteria providing a characterization of the Tanner subgraph of such low-weight error syndromes. We analyze the dynamics of iterations when BP is used to decode weight-four and weight-five errors, finding statistics akin to exponential activation in the presence of noise or escape from chaotic phase-space domains. We study how BP convergence improves when adding to the decoding matrix relevant combinations of fault columns, and show that the suggested decoder amendment can result in the reduction of both logical errors and decoding time.
Quantum low-density parity check (QLDPC) codes are promising in realization of scalable, fault tolerant quantum memory for computation. Many of the QLDPC codes constructions suffer from unavoidable short cycles in their Tanner graph which degrade the decoding performance of the belief propagation (BP) algorithm. In this paper, we propose a syndrome based generalized belief propagation (GBP) algorithm for decoding of quantum LDPC codes and analyze how the proposed algorithm escapes from short cycle trapping sets effectively compared to the BP algorithm. Simulation results show improved decoding performance of the GBP algorithm over BP for the dual containing Calderbank, Shor and Steane (CSS) codes when cycles of length 4 are considered in the region based approach.
Decoding Bivariate Bicycle (BB) quantum error correction codes typically requires Belief Propagation (BP) followed by Ordered Statistics Decoding (OSD) post-processing when BP fails to converge. Whether BP will converge on a given syndrome is currently determined only after running BP to completion. We show that convergence can be predicted in advance by a single modulo operation: if the syndrome defect count is divisible by the code's column weight w, BP converges with high probability (100% at p <= 0.001, degrading to 87% at p = 0.01); otherwise, BP fails with probability >= 90%. The mechanism is structural: each physical data error activates exactly w stabilizers, so a defect count not divisible by w implies the presence of measurement errors outside BP's model space. Validated on five BB codes with column weights w = 2, 3, and 4, mod-w achieves AUC = 0.995 as a convergence classifier at p = 0.001 under phenomenological noise, dominating all other syndrome features (next best: AUC = 0.52). The false positive rate scales empirically as O(p^2.05) (R^2 = 0.98), confirming the analytical bound from Proposition 2. Among BP failures on mod-w = 0 syndromes, 82% contain weight-2 data error clusters, directly confirming the dominant failure mechanism. The prediction is invariant under BP scheduling strategy and decoder variant, including Relay-BP - the strongest known BP enhancement for quantum LDPC codes. These results apply directly to IBM's Gross code [[144, 12, 12]] and Two-Gross code [[288, 12, 18]], targeted for deployment in 2026-2028.
Accurate and efficient decoding is a crucial component for achieving fault-tolerant quantum computing. Realistic circuit-level noise introduces temporal correlations and degeneracy, making optimal (maximum-likelihood) decoding computationally intractable in general. As a result, practical decoders rely on heuristic approximations, and it is generally difficult to quantify how suboptimal they are, as this strongly depends on the code and noise model considered. In this work, we study the accuracy of practical decoding algorithms under circuit-level noise by comparing them against a maximum likelihood decoding benchmark. Our approach propagates the density matrix through the full memory experiment and computes the optimal decoding decision for each syndrome history. We introduce pruning techniques with rigorous bounds, allowing us to access larger numbers of syndrome-extraction rounds. We apply this framework to small instances of the repetition code and a cellular automaton code, and benchmark minimum-weight perfect matching (MWPM), belief propagation with ordered statistics decoding (BP+OSD), Tesseract, and Planar decoders against optimal decoding. While standard decoders remain close to optimal for the repetition code, we find significant deviations for the cellular automaton code, with BP+OSD deteriorating already in experimentally relevant noise regimes. Moreover, the pruning method developed here highlights that, at low physical error rates, only a narrow fraction of syndrome histories contributes significantly to the logical error rate.
Abstract Fault-tolerant quantum computers must be designed in conjunction with classical co-processors that decode quantum error correction measurement information in real-time. In this work, we introduce the belief propagation plus ordered Tanner forest (BP + OTF) algorithm as an almost-linear time decoder for quantum low-density parity-check codes. The OTF post-processing stage removes qubits from the decoding graph until it has a tree-like structure. Provided that the resultant loop-free OTF graph supports a subset of qubits that can generate the syndrome, BP decoding is then guaranteed to converge. To enhance performance under circuit-level noise, we introduce a technique for sparsifying detector error models. This method uses a transfer matrix to map soft information from the full detector graph to the sparsified graph, preserving critical error propagation information from the syndrome extraction circuit. Our BP+OTF implementation first applies standard BP to the full detector graph, followed by BP + OTF post-processing on the sparsified graph. Numerical simulations show that the BP+OTF decoder achieves similar logical error suppression compared to state-of-the-art inversion-based and matching decoders for bivariate bicycle and surface codes, respectively, while maintaining almost-linear runtime complexity across all stages.
Belief Propagation combined with Ordered Statistics Decoding (BP OSD) is an open-source decoding tool used widely within the community due to its versatility and ease of application to Quantum Low Density Parity Check (QLDPC) codes. However, the degeneracy of quantum codes has been observed to degrade the performance of BP OSD, particularly leading to error floors for low physical error rates. Here, we investigate the low-weight error configurations responsible for such failures in Bivariate Bicycle codes. We show that, at code capacity, BP OSD fails on errors whose weight is significantly smaller than d/2, arising from global symmetries of the decoding graph that prevent the convergence of parallel BP. Furthermore, our analysis reveals new pitfalls that can arise when analysing the performance of BP OSD in the rare events regime, such as loss of ergodicity in splitting methods and numerical instabilities in the decoder. We show how to mitigate these failure mechanisms by modifying the BP update rules and the decoder implementation, leading to substantial improvement in logical error rate.
A weighted belief propagation decoding algorithm for low-density parity-check decoding is proposed to accelerate the convergence of decoding and reduce the number of decoding iterations. We propose an early stopping criterion which is based on the average weighted reliability measure of the hard output of decoder to detect undecodable blocks in an early stage of the decoding process. In addition, an adaptive weighted iteration scheme is proposed to increase the convergence speed of belief propagation (BP) decoding and therefore improve the error performance. The evolution of the adaptive weighting factor with the number of decoding iterations is based on the relative entropy between two consecutive ”a posteriori” probabilities distribution at decoder output. Simulation results show that the proposed algorithm can achieve about 0.2-0.4 dB coding gain improvement on average while reducing up to about 50%-85% iterations for decoding at low to medium E(subscript b)/ N(subscript 0) , compared with the conventional BP based algorithms.