Evaluating Learned Decoders under Compound Noise Models in Surface Codes

The practical realization of fault-tolerant quantum computing critically hinges on the capacity of quantum error correction (QEC) schemes to robustly mitigate physical noise. Among the foremost candidates, the surface code is distinguished by its high threshold and local stabilizer checks. In recent years, neural network-based decoders, particularly those leveraging deep reinforcement learning (RL) and convolutional architectures, have demonstrated competitive performance against algorithmic staples such as Minimum Weight Perfect Matching (MWPM) and Union-Find. However, the vast majority of these learned decoders are trained and evaluated under idealized phenomenological or circuit-level depolarizing noise models. Real-world quantum processors exhibit complex, compound noise profiles encompassing crosstalk, leakage, non-Markovian temporal drift, and spatially correlated errors. In this work, we empirically evaluate the robustness of state-of-the-art neural decoders against highly structured compound noise models. We introduce a comprehensive stress-testing methodology tailored to quantify the degradation of the logical error rate when decoders are subjected to out-of-distribution noise characteristics. Our empirical results reveal that while neural decoders can overfit to their training noise profile, strategically designed training curricula and architecture regularizations enable significant resilience. We provide extensive numerical benchmarks validating that learned decoders can maintain near-optimal pseudothresholds across a broad spectrum of shifting noise conditions, highlighting their readiness for real-world superconducting qubit deployments.The practical realization of fault-tolerant quantum computing critically hinges on the capacity of quantum error correction (QEC) schemes to robustly mitigate physical noise. Among the foremost candidates, the surface code is distinguished by its high threshold and local stabilizer checks. In recent years, neural network-based decoders, particularly those leveraging deep reinforcement learning (RL) and convolutional architectures, have demonstrated competitive performance against algorithmic staples such as Minimum Weight Perfect Matching (MWPM) and Union-Find. However, the vast majority of these learned decoders are trained and evaluated under idealized phenomenological or circuit-level depolarizing noise models. Real-world quantum processors exhibit complex, compound noise profiles encompassing crosstalk, leakage, non-Markovian temporal drift, and spatially correlated errors. In this work, we empirically evaluate the robustness of state-of-the-art neural decoders against highly structured compound noise models. We introduce a comprehensive stress-testing methodology tailored to quantify the degradation of the logical error rate when decoders are subjected to out-of-distribution noise characteristics. Our empirical results reveal that while neural decoders can overfit to their training noise profile, strategically designed training curricula and architecture regularizations enable significant resilience. We provide extensive numerical benchmarks validating that learned decoders can maintain near-optimal pseudothresholds across a broad spectrum of shifting noise conditions, highlighting their readiness for real-world superconducting qubit deployments.

Jul 21, 20260 citations

Compound Error Modeling and Synthesis in Deep Reinforcement Learning for Quantum Error Correction

Hiroshi Tanaka, Ming Liu, Jian Wang

The practical realization of fault-tolerant quantum computing depends critically on the efficiency and adaptability of quantum error correction (QEC) decoders. Recent advancements have highlighted deep reinforcement learning (DRL) as a promising approach for decoding topological codes. However, training these agents typically assumes simplified, uniform noise models, which fail to capture the complex, compounding errors characteristic of near-term quantum hardware. In this paper, we rigorously formulate the synthesis of compound error models—encompassing thermal relaxation, spatial cross-talk, and non-Markovian memory effects—within the framework of deep reinforcement learning for surface code decoding. We propose a methodological framework for simulating these mismatch environments during training to foster robust policy generalization. Extensive simulations demonstrate that DRL agents trained on our synthesized compound error models achieve up to a $24\%$ reduction in logical error rates under realistic noise conditions compared to agents trained on standard depolarizing noise. This work provides a rigorous foundation for synthesizing accurate quantum noise environments for reinforcement learning applications.The practical realization of fault-tolerant quantum computing depends critically on the efficiency and adaptability of quantum error correction (QEC) decoders. Recent advancements have highlighted deep reinforcement learning (DRL) as a promising approach for decoding topological codes. However, training these agents typically assumes simplified, uniform noise models, which fail to capture the complex, compounding errors characteristic of near-term quantum hardware. In this paper, we rigorously formulate the synthesis of compound error models—encompassing thermal relaxation, spatial cross-talk, and non-Markovian memory effects—within the framework of deep reinforcement learning for surface code decoding. We propose a methodological framework for simulating these mismatch environments during training to foster robust policy generalization. Extensive simulations demonstrate that DRL agents trained on our synthesized compound error models achieve up to a $24\%$ reduction in logical error rates under realistic noise conditions compared to agents trained on standard depolarizing noise. This work provides a rigorous foundation for synthesizing accurate quantum noise environments for reinforcement learning applications.

Jul 21, 20260 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.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 MathematicsMar 25, 20260 citations

Machine Learning Optimal Quantum Error Correction Thresholds

Dominik Seip, Luis Colmenarez, Markus Schmitt +1 more

As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.

quant-pharXiv:2606.22194Jun 23, 20260 citations

Compound Error Modeling and Synthesis in Deep Reinforcement Learning for Quantum Error Correction

Liu Ming, Tanaka Hiroshi, Wang Jian

Jul 21, 20260 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.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, 20260 citations

24\%

reduction in logical error rates under realistic noise conditions compared to agents trained on standard depolarizing noise. This work provides a rigorous foundation for synthesizing accurate quantum noise environments for reinforcement learning applications.

24\%