Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.27682

Logical Neural Belief Propagation for Linear-Complexity Decoding of Surface Codes

Hee-Youl Kwak, Seong-Joon Park, Dae-Young Yun, Eliya Nachmani, Jae-Won Kim

quant-phcs.ITSubmitted Aug 27, 2026• Announced Aug 31, 2026• 0 citations
v1 · Aug 27, 2026v2 · Sep 13, 2026
arXivHTMLPDF

Abstract

Quantum error correction (QEC) requires decoders that achieve high logical accuracy while scaling efficiently with the code length. Belief propagation (BP) is attractive for its linear decoding complexity, but conventional BP decoders often fail to reach sufficient 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 decoding to logical-level decoding. L-NBP first runs a neural BP (NBP) module that produces posterior beliefs, and a logical classifier then transforms these beliefs into a continuous-valued soft syndrome and predicts the logical operator. Because all components in L-NBP are trainable by backpropagation, L-NBP is trained end-to-end, so that the NBP module learns to extract soft syndromes that are favorable for logical classification. On surface codes, L-NBP matches or outperforms the 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 $17.5\%$ under depolarizing noise. Moreover, under circuit-level noise, L-NBP matches the accuracy of BP-OSD on the distance-$9$ surface code while requiring only $0.2\%$ of its complexity. These results show that combining BP, neural weights, and logical-level decoding enables scalable and high-accuracy quantum decoding.

Comments
12 pages, 8 figures

Related papers

Achieving Thresholds via Standalone Belief Propagation on Surface Codes

Pedro Hack, Luca Menti, Francisco Lazaro +1 more

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.

quant-pharXiv:2603.05381•Mar 6, 2026•0 citations

Neural Decoders Reduce Logical Error Rates in Surface Code Quantum Systems

Lukas Schneider, Haruki Tanaka

Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical fault-tolerant quantum computing.

Frontiers in Applied Physics and Mathematics•Mar 25, 2026•0 citations

Developing a Practical Neural-Network Surface Code Decoder under Circuit-Level Noise

Aarush Pancholi

Abstract. Suppressing qubit errors is one of the main challenges in building fault-tolerant quantum computers, requiring quantum error correction to protect fragile quantum information during computation cycles. The surface code is a leading approach because of its high error tolerance and compatibility with realistic hardware, but it requires a decoder that can accurately and quickly use the syndrome measurements to discern whether a logical error has occurred. This study investigates whether a feedforward neural-network can serve as a practical decoder for the surface code under circuit-level noise. Using the Stim software package, this study generates surface code circuits at multiple code distances and physical error rates and compiles them into detector samplers, then trains a PyTorch neural-network to predict whether a logical observable flip occurred from the syndrome data. The neural-network decoder is compared against Minimum-Weight Perfect Matching (MWPM), focusing on metrics including logical error rate and inference speed. The results show that the neural-network decoder performs competitively with MWPM for code distances 3 and 5 and provides inference time advantages at higher physical error rates as its inference time remains relatively constant over increasing physical error rates. However, MWPM outperforms the neural-network at distance 7, suggesting that scalability remains a major limitation for neural-network decoders. These results show that neural-networks are a promising approach for tackling quantum error correction, although future work is still needed to improve these models for real systems at larger code distances.

•Aug 2, 2026•0 citations

Graph Dilution for Decoding Surface Codes

Boqing Zhang, Siyuan Niu

We present a low-complexity belief propagation (BP) decoding strategy for surface codes by dynamically diluting the decoding graphs based on both BP soft messages and the underlying topological product structure. Analytically, we show that the structural dilution breaks dominant short cycles of the decoding graphs while preserving the original distance of surface codes. Numerical results demonstrate that standard BP on these dynamically diluting graphs achieves a 9 % threshold under Pauli-X noise, closely matching the 9.1 % threshold of BP-OSD. Notably, the proposed soft decoder with almost linear complexity effectively solves the BP non-convergence issue under the code capacity noise model without introducing any postprocessing or heuristic adaptation, making it a highly promising candidate for efficient hardware implementation in real-time quantum decoding under circuit-level noise.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Aug 30, 2025•0 citations

Neural Belief-Matching Decoding for Topological Quantum Error Correction Codes

Luca Menti, Francisco Lázaro

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.

quant-pharXiv:2603.21730•Mar 24, 2026•0 citations

BP-LCGCD: A Gaussian-Elimination-Free and High-Performance Decoder for Surface Codes

Qianfan Wang, Jifan Liang, Lvzhou Li +2 more

Belief propagation (BP) combined with ordered statistics decoding (OSD) can achieve near-optimal logical error rates for surface codes. However, OSD requires high-latency and unstable-complexity Gaussian elimination (GE), limiting its practicality. In this paper, we propose BP-LCGCD, a GE-free and high-performance decoder that replaces the GE-based OSD with the GE-free LC-GCD. Moreover, in contrast to the original BP-OSD, which adopts a single fixed normalization factor α, we further propose a list-parameterized variant, BP-LCGCD+ α, which performs multiple BP decodings with different α to generate diverse posterior LLRs. We present complexity analysis, demonstrating that at low physical error rates, the average decoding complexity of the proposed algorithm approaches that of standard BP. Simulation results demonstrate that BP-LCGCD achieves logical error rates close to BP-OSD, while BP-LCGCD+α nearly matches the performance of the BP-OSD. They also show that both proposed decoders significantly outperform standard BP and minimum-weight perfect matching (MWPM) in terms of logical error rate and threshold.

IEEE Communications Letters•Dec 22, 2025•1 citation
Quantum error correction (QEC) requires decoders that achieve high logical accuracy while scaling efficiently with the code length. Belief propagation (BP) is attractive for its linear decoding complexity, but conventional BP decoders often fail to reach sufficient 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 decoding to logical-level decoding. L-NBP first runs a neural BP (NBP) module that produces posterior beliefs, and a logical classifier then transforms these beliefs into a continuous-valued soft syndrome and predicts the logical operator. Because all components in L-NBP are trainable by backpropagation, L-NBP is trained end-to-end, so that the NBP module learns to extract soft syndromes that are favorable for logical classification. On surface codes, L-NBP matches or outperforms the 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. Moreover, 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. These results show that combining BP, neural weights, and logical-level decoding enables scalable and high-accuracy quantum decoding.
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.
Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical fault-tolerant quantum computing.
Abstract. Suppressing qubit errors is one of the main challenges in building fault-tolerant quantum computers, requiring quantum error correction to protect fragile quantum information during computation cycles. The surface code is a leading approach because of its high error tolerance and compatibility with realistic hardware, but it requires a decoder that can accurately and quickly use the syndrome measurements to discern whether a logical error has occurred. This study investigates whether a feedforward neural-network can serve as a practical decoder for the surface code under circuit-level noise. Using the Stim software package, this study generates surface code circuits at multiple code distances and physical error rates and compiles them into detector samplers, then trains a PyTorch neural-network to predict whether a logical observable flip occurred from the syndrome data. The neural-network decoder is compared against Minimum-Weight Perfect Matching (MWPM), focusing on metrics including logical error rate and inference speed. The results show that the neural-network decoder performs competitively with MWPM for code distances 3 and 5 and provides inference time advantages at higher physical error rates as its inference time remains relatively constant over increasing physical error rates. However, MWPM outperforms the neural-network at distance 7, suggesting that scalability remains a major limitation for neural-network decoders. These results show that neural-networks are a promising approach for tackling quantum error correction, although future work is still needed to improve these models for real systems at larger code distances.
We present a low-complexity belief propagation (BP) decoding strategy for surface codes by dynamically diluting the decoding graphs based on both BP soft messages and the underlying topological product structure. Analytically, we show that the structural dilution breaks dominant short cycles of the decoding graphs while preserving the original distance of surface codes. Numerical results demonstrate that standard BP on these dynamically diluting graphs achieves a 9 % threshold under Pauli-X noise, closely matching the 9.1 % threshold of BP-OSD. Notably, the proposed soft decoder with almost linear complexity effectively solves the BP non-convergence issue under the code capacity noise model without introducing any postprocessing or heuristic adaptation, making it a highly promising candidate for efficient hardware implementation in real-time quantum decoding under circuit-level noise.
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.
Belief propagation (BP) combined with ordered statistics decoding (OSD) can achieve near-optimal logical error rates for surface codes. However, OSD requires high-latency and unstable-complexity Gaussian elimination (GE), limiting its practicality. In this paper, we propose BP-LCGCD, a GE-free and high-performance decoder that replaces the GE-based OSD with the GE-free LC-GCD. Moreover, in contrast to the original BP-OSD, which adopts a single fixed normalization factor α, we further propose a list-parameterized variant, BP-LCGCD+ α, which performs multiple BP decodings with different α to generate diverse posterior LLRs. We present complexity analysis, demonstrating that at low physical error rates, the average decoding complexity of the proposed algorithm approaches that of standard BP. Simulation results demonstrate that BP-LCGCD achieves logical error rates close to BP-OSD, while BP-LCGCD+α nearly matches the performance of the BP-OSD. They also show that both proposed decoders significantly outperform standard BP and minimum-weight perfect matching (MWPM) in terms of logical error rate and threshold.