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Physical Review A

Topological blocking decoder for surface codes

Hyun-Woo Jung, Jeongseok Ha

Published Apr 18, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.111.042424

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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 error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.
To perform reliable information processing in quantum computers, quantum error-correction (QEC) codes are essential for the detection and correction of errors in the qubits. Among QEC codes, topological QEC codes are designed to interact between the neighboring qubits, which is a promising property for easing the implementation requirements. In addition, the locality to the qubits provides unusual tolerance to local errors. Recently, various decoding algorithms based on machine learning have been proposed to improve the decoding performance and latency of QEC codes. In this work, we propose a new decoding algorithm for surface codes, i.e., a type of topological codes, by using convolutional neural networks (CNN) tailored for the topological lattice structure of the surface codes. In particular, the proposed algorithm takes advantage of the syndrome pattern which is represented as a part of a rectangular lattice given to the CNN as its input. The remaining part of the rectangular lattice is filled with a carefully selected incoherent value for better logical error-rate performance. In addition, we introduce how to optimize the hyper-parameters in the CNN, according to the lattice structure of a given surface code. This reduces the overall decoding complexity and makes the CNN-based decoder computationally more suitable for implementation. The numerical results show that the proposed decoding algorithm effectively improves the decoding performance in terms of logical error-rate as compared to the existing algorithms, on various quantum error models.
Quantum error correction (QEC) requires accurate and efficient decoders, yet belief propagation (BP), despite its linear decoding complexity, often provides insufficient logical accuracy on surface codes. We propose Logical Neural Belief Propagation (L-NBP), a BP-based neural decoder that redirects the decoding objective from physical-level to logical-level decoding. L-NBP uses a neural BP (NBP) module to produce posterior beliefs, which a logical classifier transforms into a continuous-valued soft syndrome for logical-operator prediction. Trained end-to-end by backpropagation, the NBP module learns soft syndromes that are favorable for logical classification. On surface codes, L-NBP matches or outperforms BP with ordered-statistics decoding (BP-OSD) and minimum-weight perfect matching (MWPM) while retaining the linear complexity of BP, and achieves a threshold of 17.5%17.5\%17.5% under depolarizing noise. Under circuit-level noise, L-NBP matches the accuracy of BP-OSD on the distance-999 surface code while requiring only 0.2%0.2\%0.2% of its complexity.
Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02 % . This is larger than the threshold achieved by previous matching-based decoders ( 14.88 ± 0.02 % ), though still below the known upper bound of ∼ 18.9 % .
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.
Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of 17% under independent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6% ) and a threshold value of 14% under depolarizing data noise (compared to the ideal threshold of 18.9% ), which are on par with thresholds achieved by non-BP post-processing methods.