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.
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.
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.
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.
To perform reliable information processing in quantum computers, quantum error-correction (QEC) codes are essential for the detection and correction of errors in the qubits. Among QEC codes, topological QEC codes are designed to interact between the neighboring qubits, which is a promising property for easing the implementation requirements. In addition, the locality to the qubits provides unusual tolerance to local errors. Recently, various decoding algorithms based on machine learning have been proposed to improve the decoding performance and latency of QEC codes. In this work, we propose a new decoding algorithm for surface codes, i.e., a type of topological codes, by using convolutional neural networks (CNN) tailored for the topological lattice structure of the surface codes. In particular, the proposed algorithm takes advantage of the syndrome pattern which is represented as a part of a rectangular lattice given to the CNN as its input. The remaining part of the rectangular lattice is filled with a carefully selected incoherent value for better logical error-rate performance. In addition, we introduce how to optimize the hyper-parameters in the CNN, according to the lattice structure of a given surface code. This reduces the overall decoding complexity and makes the CNN-based decoder computationally more suitable for implementation. The numerical results show that the proposed decoding algorithm effectively improves the decoding performance in terms of logical error-rate as compared to the existing algorithms, on various quantum error models.
Abstract Artificial neural networks (ANNs) are a promising approach to the decoding problem of quantum error correction (QEC), but have observed consistent difficulty when generalising performance to larger QEC codes. Recent scalability-focused approaches have split the decoding workload by using local ANNs to perform initial syndrome processing and leaving final processing to a global residual decoder. We investigated ANN surface code decoding under a scheme exploiting the spatiotemporal structure of syndrome data. In particular, we present a vectorised method for surface code data simulation and benchmark decoding performance when such data defines a multi-label classification problem and generative modelling problem for rotated surface codes with circuit noise after each gate and idle timestep. Performance was found to generalise to rotated surface codes of sizes up to d = 97, with depolarisation parameter thresholds of up to 0.7% achieved, competitive with minimum weight perfect matching (MWPM). Improved timings, compared with MWPM alone, were found starting at code distances of d = 33 and d = 89 under noise models above and below threshold respectively. These results suggest promising prospects for ANN-based frameworks for surface code decoding with performance sufficient to support the demands expected from fault-tolerant resource estimates.
Matching algorithms can be used for identifying errors in quantum systems, being the most famous the Blossom algorithm. Recent works have shown that small distance quantum error correction codes can be efficiently decoded by employing machine learning techniques based on neural networks (NN). Various NN-based decoders have been proposed to enhance the decoding performance and the decoding time. Their implementation differs in how the decoding is performed, at logical or physical level, as well as in several neural network related parameters. In this work, we implement and compare two NN-based decoders, a low level decoder and a high level decoder, and study how different NN parameters affect their decoding performance and execution time. Crucial parameters such as the size of the training dataset, the structure and the type of the neural network, and the learning rate used during training are discussed. After performing this comparison, we conclude that the high level decoder based on a Recurrent NN shows a better balance between decoding performance and execution time and it is much easier to train. We then test its decoding performance for different code distances, probability datasets and under the depolarizing and circuit error models.
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.