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arXiv:2407.11523 · IEEE Transactions on Quantum Engineering

Improved Belief Propagation Decoding Algorithms for Surface Codes

Jiahan Chen, Zhengzhong Yi, Zhipeng Liang, Xuan Wang

Published Jan 1, 2025• 10 citations
arXivHTMLPDF

Abstract

Quantum error correction is crucial for universal fault-tolerant quantum computing. Highly accurate and low-time-complexity decoding algorithms play an indispensable role in ensuring quantum error correction works effectively. Among existing decoding algorithms, belief propagation (BP) is notable for its nearly linear time complexity and general applicability to stabilizer codes. However, BP's decoding accuracy without post-processing is unsatisfactory in most situations. This article focuses on improving the decoding accuracy of BP over GF(4) for surface codes. Inspired by machine learning optimization techniques, we first propose Momentum-BP and AdaGrad-BP to reduce oscillations in message updating, breaking the trapping sets of surface codes. We further propose EWAInit-BP, which adaptively updates initial probabilities and provides a 1 to 3 orders of magnitude improvement over traditional BP for planar surface code, toric code, and XZZX surface code without any post-processing method, showing high decoding accuracy even under parallel scheduling. The theoretical$O(1)$time complexity under parallel implementation and high accuracy of EWAInit-BP make it a promising candidate for high-precision real-time decoders.

DOI
10.1109/tqe.2025.3577769

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Quantum error correction is crucial for universal fault-tolerant quantum computing. Highly accurate and low-time-complexity decoding algorithms play an indispensable role in ensuring quantum error correction works effectively. Among existing decoding algorithms, belief propagation (BP) is notable for its nearly linear time complexity and general applicability to stabilizer codes. However, BP's decoding accuracy without post-processing is unsatisfactory in most situations. This article focuses on improving the decoding accuracy of BP over GF(4) for surface codes. Inspired by machine learning optimization techniques, we first propose Momentum-BP and AdaGrad-BP to reduce oscillations in message updating, breaking the trapping sets of surface codes. We further propose EWAInit-BP, which adaptively updates initial probabilities and provides a 1 to 3 orders of magnitude improvement over traditional BP for planar surface code, toric code, and XZZX surface code without any post-processing method, showing high decoding accuracy even under parallel scheduling. The theoreticalO(1)O(1)O(1)time complexity under parallel implementation and high accuracy of EWAInit-BP make it a promising candidate for high-precision real-time decoders.
In the quest for fault-tolerant quantum computation, the delicate nature of qubits demands the need for robust quantum error correction. Quantum error correcting codes, the quantum counterpart to classical error correcting codes in information theory, have been developed to address the unique challenges quantum bits face against the noise of their environment in storing information. These codes include but are not limited to the Kitaev surface code, toric code, rotated surface code, generalized bicycle code, and bivariate bicycle code. Fast and reliable decoders for these quantum error correcting codes are needed to correct errors and achieve fault tolerance on these qubits. The belief propagation algorithm, which is a efficient and reliable heuristic for decoding classical low-density parity check codes, has been generalized to different types of quantum error correction codes. Paired with the ordered statistics decoding algorithm as a post-processing step from the soft outputs of the belief propagation decoder, belief propagation and ordered statistics decoding has been one of the most promising avenues of efficient quantum error correction. This thesis combines the state-of-the-art belief propagation and ordered statistics decoding algorithms with the exploration of the neural belief propagation algorithm to leverage classical deep learning machines to learn message passing patterns in decoding the newest quantum codes, the rotated surface code and bivariate bicycle code. The neural network exploits the local lattice structure of these codes to adjust the trainable weights and biases to more learn message update patterns more efficiently than the vanilla belief propagation algorithm. In this thesis, we prove a new threshold for the physical error rate on the rotated surface code using a neural belief propagation algorithm and ordered statistics decoding that consistently outperforms the threshold established by the vanilla belief propagation algorithm with ordered statistics decoding. We additionally show the results for similar efforts on the bivariate bicycle codes, and discuss the difficulties in achieving a threshold using the current neural belief propagation algorithms.
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.
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.
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.
Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.
Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a quaternary BP for (additive) codes over GF(4), which has a higher check-node complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. In this paper, we propose a refined BP algorithm for decoding quantum codes by passing scalar messages. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but with a check-node complexity the same as binary BP. As every message is a scalar, the message normalization can be naturally applied to improve the performance. Another observation is that the message-update schedule affects the BP decoding performance against short cycles. We show that running BP with message normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error-floor in computer simulation.