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IEEE Pune Section International Conference (PuneCon)

Machine Learning Based Decoders for Near-Term Quantum Surface Codes: Performance and Pseudothreshold Analysis

Veeresh Kuruba, E. S. Shivaleela, Srinivas T

Published Dec 12, 2025• 0 citations
Publisher

Abstract

Quantum error correction (QEC) is crucial for achieving fault-tolerant quantum computing by addressing errors in noisy qubits. The surface code, a leading QEC framework, traditionally employs the Minimum Weight Perfect Matching (MWPM) decoder. We present a step-by-step methodology that formulates surface code decoding as a supervised machine learning classification problem within a two-level framework, where low-level modules address physical qubit errors and highlevel modules correct logical errors. We implement Feedforward Neural Networks (FFNN) and Graph Neural Networks (GNN), which are individually employed to perform low-level physical qubit error correction and high-level logical error correction for rotated surface codes of distances$d=3, d=5$, and$d=7$. Under depolarizing noise, both FFNN and GNN achieve lower logical error rates than MWPM, with GNN yielding up to 28% improvement at$d=7$. GNN achieves pseudothresholds of 0.1084, 0.1311, and 0.1441 for$d=3, d=5$, and$d=7$, surpassing MWPM's 0.0914, 0.1024, and 0.1126. Crossover points further indicate GNN's robustness at higher noise levels (e.g., 0.1632 vs. 0.1291 for MWPM). By learning error distributions in a two-level setting, both FFNN and GNN outperform MWPM, demonstrating their potential as scalable, high-performance decoders for near-term quantum devices.

DOI
10.1109/punecon67554.2025.11378723

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Quantum error correction (QEC) is crucial for achieving fault-tolerant quantum computing by addressing errors in noisy qubits. The surface code, a leading QEC framework, traditionally employs the Minimum Weight Perfect Matching (MWPM) decoder. We present a step-by-step methodology that formulates surface code decoding as a supervised machine learning classification problem within a two-level framework, where low-level modules address physical qubit errors and highlevel modules correct logical errors. We implement Feedforward Neural Networks (FFNN) and Graph Neural Networks (GNN), which are individually employed to perform low-level physical qubit error correction and high-level logical error correction for rotated surface codes of distancesd=3,d=5d=3, d=5d=3,d=5, andd=7d=7d=7. Under depolarizing noise, both FFNN and GNN achieve lower logical error rates than MWPM, with GNN yielding up to 28% improvement atd=7d=7d=7. GNN achieves pseudothresholds of 0.1084, 0.1311, and 0.1441 ford=3,d=5d=3, d=5d=3,d=5, andd=7d=7d=7, surpassing MWPM's 0.0914, 0.1024, and 0.1126. Crossover points further indicate GNN's robustness at higher noise levels (e.g., 0.1632 vs. 0.1291 for MWPM). By learning error distributions in a two-level setting, both FFNN and GNN outperform MWPM, demonstrating their potential as scalable, high-performance decoders for near-term quantum devices.
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Quantum computers can solve selected problems more efficiently than classical computers, but current devices are limited by high physical error rates. Quantum Error Correction (QEC) mitigates this by encoding many physical qubits into a logical qubit, with the surface code among the most widely studied approaches. Since syndrome measurements are produced continuously, the decoder must process them fast enough to avoid becoming a system bottleneck, making real-time decoding essential for fault-tolerant quantum computing. While most state-of-the-art real-time decoders rely on Minimum-Weight Perfect Matching (MWPM), we instead use a high-accuracy Graph Neural Network (GNN) that trades higher computational cost for lower logical error rates. To make this GNN practical for real-time decoding, we apply algorithm-hardware co-design. We reduce complexity through hardware-guided pruning and retraining, producing two hardware-friendly models that reduce parameter count by 3.1×3.1\times3.1× and 6.5×6.5\times6.5×. These target, respectively, an average decoding latency of one syndrome cycle and a worst-case latency within one syndrome cycle. We further reduce hardware cost using input-graph filtering and post-training quantization. Based on these optimized models, we propose an FPGA-based architecture for low-latency inference and real-time decoding. Evaluated on surface codes up to distance 7 under circuit-level noise at physical error rate p=10−3p=10^{-3}p=10−3, our decoder outperforms MWPM in decoding accuracy for both average-latency and max-latency settings. It reduces logical error rate by 40% at 1μs1μs1μs average latency, and by 13% under a strict 1μs1μs1μs deadline.
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.
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
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.
Errors in surface code have typically been decoded by Minimum Weight Perfect Matching (MWPM) based method. Recently, neural-network-based Machine Learning (ML) techniques have been employed for this purpose. Here we propose a two-level (low and high) ML-based decoding scheme, where the first level corrects errors on physical qubits and the second one corrects any existing logical errors, for different noise models. Our results show that our proposed decoding method achieves ∼10×\sim10 \times∼10× and ∼2×\sim2 \times∼2× higher values of pseudo-threshold and threshold respectively, than for MWPM. We show that usage of more sophisticated ML models with higher training/testing time, do not provide significant improvement in the decoder performance. Finally, data generation for training the ML decoder requires significant overhead hence lower volume of training data is desirable. We have shown that our decoder maintains a good performance with the train-test-ratio as low as 40:6040:6040:60.