We propose Pathfinder-BP, a recursive decoder for topological quantum codes that combines belief propagation with controlled bit-flipping. An initial BP pass identifies and permanently fixes high-confidence errors. Upon convergence failure, the decoder recursively explores alternative error patterns by tentatively flipping low-confidence variables. On surface codes, Pathfinder-BP achieves a noise threshold and logical error rate comparable to minimum-weight perfect matching (MWPM). A crucial advantage of Pathfinder-BP is its highly parallelizable structure, which yields a worst-case decoding latency quasi-linear in the code distance, making it a suitable candidate for low-latency hardware implementation.
We propose Pathfinder-BP, a recursive decoder for topological quantum codes that combines belief propagation with controlled bit-flipping. An initial BP pass identifies and permanently fixes high-confidence errors. Upon convergence failure, the decoder recursively explores alternative error patterns by tentatively flipping low-confidence variables. On surface codes, Pathfinder-BP achieves a noise threshold and logical error rate comparable to minimum-weight perfect matching (MWPM). A crucial advantage of Pathfinder-BP is its highly parallelizable structure, which yields a worst-case decoding latency quasi-linear in the code distance, making it a suitable candidate for low-latency hardware implementation.
Surface codes have gained widespread popularity as quantum error-correction codes due to their unique structural feature that encoders and decoders can be implemented by leveraging interactions between physically neighboring qubits. The implementational advantage of surface codes has driven extensive research into decoding algorithms for surface codes, aimed at improving error-rate performance and/or reducing decoding complexity. In this paper, we propose a low-complexity postprocessing algorithm that effectively resolves failures of a syndrome-based belief-propagation (SB-BP) decoder for surface codes. In particular, we carefully analyze the topological structure of surface codes and present a method for selecting reliable qubits, enabling efficient recovery operator identification through a simple syndrome matching technique with a lookup table. Furthermore, we present an efficient way to construct lookup tables tailored to the proposed postprocessing algorithm. Numerical results demonstrate that the proposed postprocessing algorithm requires substantially reduced complexity as compared to that of a competing postprocessing algorithm, the ordered statistics decoding in the depolarizing error model, while providing the same logical error-rate (LER) performance. It is also demonstrated that the combination of SB-BP decoding and the proposed postprocessing algorithm outperforms minimum weight perfect matching, a more computationally expensive decoding algorithm, in terms of LER and threshold at drastically reduced complexity.
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.
The standard approach to Belief Propagation (BP) decoding passes messages on the Tanner graph of the quantum error-correcting (QEC) code, and is known to lack a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain thresholds (both for code capacity and circuit-level noise) via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.
Quantum error correction is a prerequisite for quantum computing; however, the performance critically depends on the accuracy of the decoding algorithm. To address these challenges, we propose a hybrid decoding architecture, BP + UF + BP. The protocol initiates with a truncated global BP stage to extract probabilistic gradients without requiring full convergence. This soft information guides a reliability-based Union-Find (UF) algorithm to prioritize high-likelihood error mechanisms. Finally, a local subgraph BP refinement maximizes correction accuracy. Numerical simulations on rotated surface codes under circuit-level depolarizing noise demonstrate a fault-tolerance threshold of approximately 0.72%. This significantly outperforms standard Minimum Weight Perfect Matching (MWPM) and Union-Find (UF) baselines. Notably, our method significantly reduces the logical error rate compared to the conventional decoders. With its empirically near-linear scaling under fixed iteration, the proposed architecture presents a scalable solution for real-time fault-tolerant quantum computing.
Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We classify each error syndrome to the decoder which is more likely to decode it correctly using machine learning techniques. We apply our framework to an ensemble of minimum-weight perfect matching (MWPM) and hard-decision re-normalization-group decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudothreshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.
Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.