Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around 7.1% when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method.
Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around 7.1% when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method.
With the advent of noisy intermediate-scale quantum (NISQ) devices, practical quantum computing has seemingly come into reach. However, to go beyond proof-of-principle calculations, the current processing architectures will need to scale up to larger quantum circuits which will require fast and scalable algorithms for quantum error correction. Here, we present a neural network based decoder that, for a family of stabilizer codes subject to depolarizing noise and syndrome measurement errors, is scalable to tens of thousands of qubits (in contrast to other recent machine learning inspired decoders) and exhibits faster decoding times than the state-of-the-art union find decoder for a wide range of error rates (down to 1%). The key innovation is to autodecode error syndromes on small scales by shifting a preprocessing window over the underlying code, akin to a convolutional neural network in pattern recognition approaches. We show that such a preprocessing step allows to effectively reduce the error rate by up to 2 orders of magnitude in practical applications and, by detecting correlation effects, shifts the actual error threshold up to fifteen percent higher than the threshold of conventional error correction algorithms such as union find or minimum weight perfect matching, even in the presence of measurement errors. An in situ implementation of such a machine learning-assisted quantum error correction will be a decisive step to push the entanglement frontier beyond the NISQ horizon.
Quantum error correction technology is an important solution to solve the noise interference generated during the operation of quantum computers. In order to find the best syndrome of the stabilizer code in quantum error correction, we need to find a fast and close to the optimal threshold decoder. In this work, we build a convolutional neural network (CNN) decoder to correct errors in the toric code based on the system research of machine learning. We analyze and optimize various conditions that affect CNN, and use the RestNet network architecture to reduce the running time. It is shortened by 30%–40%, and we finally design an optimized algorithm for CNN decoder. In this way, the threshold accuracy of the neural network decoder is made to reach 10.8%, which is closer to the optimal threshold of about 11%.The previous threshold of 8.9%–10.3% has been slightly improved, and there is no need to verify the basic noise.
Surface code error correction offers a highly promising pathway to achieve scalable fault-tolerant quantum computing. When operated as stabilizer codes, surface code computations consist of a syndrome decoding step where measured stabilizer operators are used to determine appropriate corrections for errors in physical qubits. Decoding algorithms have undergone substantial development, with recent work incorporating machine learning (ML) techniques. Despite promising initial results, the ML-based syndrome decoders are still limited to small scale demonstrations with low latency and are incapable of handling surface codes with boundary conditions and various shapes needed for lattice surgery and braiding. Here, we report the development of an artificial neural network (ANN) based scalable and fast syndrome decoder capable of decoding surface codes of arbitrary shape and size with data qubits suffering from the depolarizing error model. Based on rigorous training over 50 million random quantum error instances, our ANN decoder is shown to work with code distances exceeding 1000 (more than 4 million physical qubits), which is the largest ML-based decoder demonstration to-date. The established ANN decoder demonstrates an execution time in principle independent of code distance, implying that its implementation on dedicated hardware could potentially offer surface code decoding times of O( μ sec), commensurate with the experimentally realisable qubit coherence times. With the anticipated scale-up of quantum processors within the next decade, their augmentation with a fast and scalable syndrome decoder such as developed in our work is expected to play a decisive role towards experimental implementation of fault-tolerant quantum information processing.
Abstract There has been a rise in decoding quantum error correction codes with neural network–based decoders, due to the good decoding performance achieved and adaptability to any noise model. However, the main challenge is scalability to larger code distances due to an exponential increase of the error syndrome space. Note that successfully decoding the surface code under realistic noise assumptions will limit the size of the code to less than 100 qubits with current neural network–based decoders. Such a problem can be tackled by a distributed way of decoding, similar to the renormalization group (RG) decoders. In this paper, we introduce a decoding algorithm that combines the concept of RG decoding and neural network–based decoders. We tested the decoding performance under depolarizing noise with noiseless error syndrome measurements for the rotated surface code and compared against the blossom algorithm and a neural network–based decoder. We show that a similar level of decoding performance can be achieved between all tested decoders while providing a solution to the scalability issues of neural network–based decoders.
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.
Abstract The decoding of error syndromes of surface codes with classical algorithms may slow down quantum computation. To overcome this problem it is possible to implement decoding algorithms based on artificial neural networks. This work reports a study of decoders based on convolutional neural networks, tested on different code distances and noise models. The results show that decoders based on convolutional neural networks have good performance and can adapt to different noise models. Moreover, explainable machine learning techniques have been applied to the neural network of the decoder to better understand the behaviour and errors of the algorithm, in order to produce a more robust and performing algorithm.