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arXiv:1905.02345 · Physical Review A

Neural ensemble decoding for topological quantum error-correcting codes

Milap Sheth, Sara Zafar Jafarzadeh, Vlad Gheorghiu

Published Mar 23, 2020• 25 citations
arXivHTMLPDF

Abstract

Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We classify each error syndrome to the decoder which is more likely to decode it correctly using machine learning techniques. We apply our framework to an ensemble of minimum-weight perfect matching (MWPM) and hard-decision re-normalization-group decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudothreshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.

DOI
10.1103/physreva.101.032338

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Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We classify each error syndrome to the decoder which is more likely to decode it correctly using machine learning techniques. We apply our framework to an ensemble of minimum-weight perfect matching (MWPM) and hard-decision re-normalization-group decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudothreshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.
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.
Quantum error correction is essential to achieve fault-tolerant quantum computation in the presence of noisy qubits. Among the most promising approaches to quantum error correction is the surface code, thanks to a scalable two-dimensional architecture, only nearest-neighbor interactions, and a high error threshold. Decoding the surface code, i.e. finding the most likely error chain given a syndrome measurement outcome is a computationally complex task. Traditional decoders rely on classical algorithms, which, especially for larger systems, can be slow and may not always converge to the optimal solution. This thesis presents a novel approach to decoding the surface code using graph neural networks. By mapping the syndrome measurements to a graph and performing graph classification, we find that the graph neural networks can predict the most likely error configuration with high accuracy. Our results show that the GNN-based decoder outperforms the classic minimum weight perfect matching (MWPM) decoder in terms of accuracy. With a phenomenological noise model with depolarizing noise and perfect syndrome measurements, our networks beat MWPM up to code-size 15 across all relevant error rates. Furthermore, the GNN is capable of surpassing MWPM under circuit-level noise up to code size 7. We also show that training the network on repetition code data from a recent experiment [Google Quantum AI, Nature 614, 676 (2023)] produces per-step error rates comparable to those achieved with a matching decoder specifically adapted to the error rates of the physical qubits. This indicates that graph neural network decoders are capable of learning the underlying error distribution on the qubits. Our findings advance the field of quantum error correction and provide a promising new direction for the development of efficient decoding algorithms.
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.
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 (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.
Abstract To address qubits' high environmental sensitivity and reduce the significant error rates in current quantum devices, quantum error correction stands as one of the most dependable approaches. The topological surface code, renowned for its unique qubit lattice structure, is widely considered a pivotal tool for enabling fault-tolerant quantum computation. Through redundancy introduced across multiple qubits, the surface code safeguards quantum information and identifies errors via state changes captured by syndrome qubits. However, simultaneous errors in data and syndrome qubits substantially escalate decoding complexity. Quantum Generative Adversarial Networks (QGANs) have emerged as promising deep learning frameworks, effectively harnessing quantum advantages for practical tasks such as image processing and data optimization. Consequently, a topological code trainer for quantum-classical hybrid GANs is proposed as an auxiliary model to enhance error correction in machine learning-based decoders, demonstrating significantly improved training accuracy compared to the traditional Minimum Weight Perfect Matching (MWPM) algorithm, which achieves an accuracy of 65 % . Numerical experiments reveal that the decoder achieves a fidelity threshold of P=0.1978, substantially surpassing the traditional algorithm's threshold of P=0.1024. To enhance decoding efficiency, a Transformer decoder is integrated, incorporating syndrome error outputs trained via QGANs into its framework. By leveraging its self-attention mechanism, the Transformer effectively captures long-range qubit dependencies at a global scale, enabling high-fidelity error correction over larger dimensions. Numerical validation of the surface code error threshold demonstrates an 8.5% threshold with a correction success rate exceeding 94%, whereas the local MWPM decoder achieves only 55% and fails to support large-scale computation at a 4% threshold.