Degeneracy Cutting: A Local and Efficient Post-Processing for Belief Propagation Decoding of Quantum Low-Density Parity-Check Codes

TL;DR

This work addresses the degeneracy-limited performance of belief propagation decoding for quantum LDPC codes by introducing degeneracy cutting (DC), a local, linear-time post-processing that prunes a single qubit per stabilizer generator based on BP marginals and then re-runs BP. The DC method preserves the favorable linear scaling and parallelism of BP, and, when extended with a detector degeneracy matrix, remains effective under phenomenological and circuit-level noise models, achieving performance close to BP+OSD with substantially lower cost. Numerically, BP+DC approaches BP+OSD for surface codes and can even outperform BP+OSD for BB codes under code-capacity noise; this advantage extends to realistic noise models with the detector degeneracy framework. The approach offers a practical path to real-time, scalable quantum decoding by balancing accuracy, efficiency, and parallelizability, and it lays groundwork for further enhancements through refined degeneracy modeling and integration with advanced BP techniques.

Abstract

Quantum low-density parity-check (qLDPC) codes are promising for realizing scalable fault-tolerant quantum computation due to their potential for low-overhead protocols. A common approach to decoding qLDPC codes is to use the belief propagation (BP) decoder, followed by a post-processing step to enhance decoding accuracy. For real-time decoding, the post-processing algorithm is desirable to have a small computational cost and rely only on local operations on the Tanner graph to facilitate parallel implementation. To address this requirement, we propose degeneracy cutting (DC), an efficient post-processing technique for the BP decoder that operates on information restricted to the support of each stabilizer generator. DC selectively removes one variable node with the lowest error probability for each stabilizer generator, significantly improving decoding performance while retaining the favorable computational scaling and structure amenable to parallelization inherent to BP. We further extend our method to realistic noise models, including phenomenological and circuit-level noise models, by introducing the detector degeneracy matrix, which generalizes the notion of stabilizer-induced degeneracy to these settings. Numerical simulations demonstrate that BP+DC achieves decoding performance approaching that of BP followed by ordered statistics decoding (BP+OSD) in several settings, while requiring significantly less computational cost. Our results present BP+DC as a promising decoder for fault-tolerant quantum computing, offering a valuable balance of accuracy, efficiency, and suitability for parallel implementation.

Figures (5)

• Figure 1: An illustration of degeneracy in belief propagation decoding and degeneracy cutting (DC). Circles represent variable nodes (qubits) in the support of an $X$-type stabilizer generator, while squares represent check nodes (parity checks) connected to the variable nodes. (a) A physical error pattern affecting two qubits (red circles with a diagonal pattern). (b) BP cannot distinguish this pattern from an alternative two-qubit error (blue circles with vertical pattern in panel (a)) that produces the identical syndrome. This ambiguity arises from the degeneracy of the code. Consequently, the decoder assigns nearly equal estimated error probabilities (such as $\sim$0.5) in Eq. \ref{['eq_marginal_error_probability']} to all four involved qubits (purple circles with cross pattern), leading to a decoding failure. (c) The variable node with the lowest error probability is identified for each stabilizer generator and removed from the Tanner graph. If several variable nodes have the same lowest error probability, the nodes to be removed are selected at random. This process breaks the degeneracy introduced by the stabilizers, enabling an accurate error estimate to be obtained by rerunning BP.
• Figure 2: Performance of the BP decoder, BP+OSD decoder, BP+DC decoder, and BP+DC+OSD decoder for surface codes and BB codes under the code-capacity noise model. The x-axis indicates the physical error rate $p$, which represents the probability of a bit-flip on each physical qubit, while the y-axis shows the decoding failure probability, corresponding to the probability of decoding failure under different decoding strategies. The BP decoder is run up to $T_{\mathrm{iter}}=n$ iterations, where $n$ is the number of physical qubits, and we use the product-sum variant of BP for surface codes and the minimum-sum variant for BB codes. Error bars represent 95% confidence intervals based on binomial statistics, but are smaller than the marker size and thus may not be visible in the plots. While reducing the computational cost of decoding from $O(n^3)$ to $O(n)$, the BP+DC and BP+DC+OSD decoder achieves comparable performance to the BP+OSD decoder for surface codes. For BB codes, the BP+DC decoder achieves better performance than the BP+OSD decoder.
• Figure 3: Schematic illustration of degeneracy arising from measurement errors in the phenomenological noise model. Circles and squares represent error mechanisms and detectors, respectively. Circles on the planar layers represent bit-flip errors on data qubits in the support of an $X$-type stabilizer generator; the lower and upper planes correspond to errors in the previous and current measurement rounds. Circles connecting the two planes represent measurement errors. Data-qubit errors on two consecutive rounds (red circles with diagonal hatching) are indistinguishable from measurement errors on the associated syndrome measurements (blue circles with vertical hatching). Therefore, if an error occurs on the red qubits, the BP decoder assigns the same error probabilities to all four error mechanisms, resulting in a decoding failure.
• Figure 4: Performance of the BP decoder, BP+OSD decoder, BP+DC decoder, and BP+DC+OSD decoder for surface codes and BB codes under the phenomenological noise model. The x-axis indicates the physical error rate $p$, which represents the probability of an error on each physical qubit and syndrome measurements, while the y-axis shows the decoding failure probability, corresponding to the probability of decoding failure under different decoding strategies. The maximum number of BP iterations is set to $T_{\mathrm{iter}}=1000$ following the simulations performed in Ref. iolius2024almost, and we use the product-sum variant of BP for surface codes and the minimum-sum variant for BB codes. Error bars indicate the 95% confidence intervals obtained by assuming binomial statistics for the number of decoding failures, but are smaller than the marker size and thus may not be visible in some plots. The BP+DC decoder achieves logical error suppression within an order of magnitude of the BP+OSD decoder while reducing computational cost from $O(N^3)$ to $O(N)$.
• Figure 5: Performance of the BP decoder, BP+OSD decoder, BP+DC decoder, and BP+DC+OSD decoder for surface codes and BB codes under the circuit-level noise model. The x-axis indicates the physical error rate $p$, while the y-axis shows the decoding failure probability, corresponding to the probability of decoding failure under different decoding strategies. The maximum number of BP iterations is set to $T_{\mathrm{iter}}=1000$ following the simulations performed in Ref. iolius2024almost, and we use the product-sum variant of BP for surface codes and the minimum-sum variant for BB codes. Error bars indicate the 95% confidence intervals obtained by assuming binomial statistics for the number of decoding failures, but are smaller than the marker size and thus may not be visible in some plots. The BP+DC decoder achieves logical error suppression within an order of magnitude of the BP+OSD decoder while reducing computational cost from $O(N^3)$ to $O(N)$.