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International Conference on Quantum Communications, Networking, and Computing (QCNC)

An Equivariant Machine Learning Decoder for 3D Toric Codes

Oliver Weißl, Evgenii Egorov

Published Mar 31, 2025• 0 citations
Publisher

Abstract

Research on mitigating errors in computing and communication systems has grown with their widespread use. In quantum computing, error correction is crucial as errors can quickly propagate, undermining results and the theoretical speedup over classical systems. Quantum error-correcting codes, particularly topological codes, are a key focus. These codes map parity check matrices to graphs on d-dimensional surfaces, with our research centered on the 3D toric code. Effective decoders must be robust to noise, with performance improving as code size increases, but their complexity should scale polynomially with lattice size for practicality. We propose a neural network with inductive bias, leveraging equivariance to generalize efficiently from a smaller input subset. Additionally, we explore transformer networks for error correction and compare these methods to existing techniques for decoding errors in the 3D toric code, highlighting their strengths and limitations.

DOI
10.1109/qcnc64685.2025.00112

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Research on mitigating errors in computing and communication systems has grown with their widespread use. In quantum computing, error correction is crucial as errors can quickly propagate, undermining results and the theoretical speedup over classical systems. Quantum error-correcting codes, particularly topological codes, are a key focus. These codes map parity check matrices to graphs on d-dimensional surfaces, with our research centered on the 3D toric code. Effective decoders must be robust to noise, with performance improving as code size increases, but their complexity should scale polynomially with lattice size for practicality. We propose a neural network with inductive bias, leveraging equivariance to generalize efficiently from a smaller input subset. Additionally, we explore transformer networks for error correction and compare these methods to existing techniques for decoding errors in the 3D toric code, highlighting their strengths and limitations.
Mitigating errors in computing and communication systems has seen a great deal of research since the beginning of the widespread use of these technologies. However, as we develop new methods to do computation or communication, we also need to reiterate the method used to deal with errors. Within the field of quantum computing, error correction is getting a lot of attention since errors can propagate fast and invalidate results, which makes the theoretical exponential speed increase in computation time, compared to traditional systems, obsolete. To correct errors in quantum systems, error-correcting codes are used. A subgroup of codes, topological codes, is currently the focus of many research papers. Topological codes represent parity check matrices corresponding to graphs embedded on a ddd-dimensional surface. For our research, the focus lies on the toric code with a 3D square lattice. The goal of any decoder is robustness to noise, which can increase with code size. However, a reasonable decoder performance scales polynomially with lattice size. As error correction is a time-sensitive operation, we propose a neural network using an inductive bias: equivariance. This allows the network to learn from a rather small subset of the exponentially growing training space of possible inputs. In addition, we investigate how transformer networks can help in correction. These methods will be compared with various configurations and previously published methods of decoding errors in the 3D toric code.
Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.
Quantum error correction is a critical component for scaling up quantum computing. Given a quantum code, an optimal decoder maps the measured code violations to the most likely error that occurred, but its cost scales exponentially with the system size. Neural network decoders are an appealing solution since they can learn from data an efficient approximation to such a mapping and can automatically adapt to the noise distribution. In this work, we introduce a data efficient neural decoder that exploits the symmetries of the problem. We characterize the symmetries of the optimal decoder for the toric code and propose a novel equivariant architecture that achieves state of the art accuracy compared to previous neural 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.
In the quest for fault-tolerant quantum computation, the delicate nature of qubits demands the need for robust quantum error correction. Quantum error correcting codes, the quantum counterpart to classical error correcting codes in information theory, have been developed to address the unique challenges quantum bits face against the noise of their environment in storing information. These codes include but are not limited to the Kitaev surface code, toric code, rotated surface code, generalized bicycle code, and bivariate bicycle code. Fast and reliable decoders for these quantum error correcting codes are needed to correct errors and achieve fault tolerance on these qubits. The belief propagation algorithm, which is a efficient and reliable heuristic for decoding classical low-density parity check codes, has been generalized to different types of quantum error correction codes. Paired with the ordered statistics decoding algorithm as a post-processing step from the soft outputs of the belief propagation decoder, belief propagation and ordered statistics decoding has been one of the most promising avenues of efficient quantum error correction. This thesis combines the state-of-the-art belief propagation and ordered statistics decoding algorithms with the exploration of the neural belief propagation algorithm to leverage classical deep learning machines to learn message passing patterns in decoding the newest quantum codes, the rotated surface code and bivariate bicycle code. The neural network exploits the local lattice structure of these codes to adjust the trainable weights and biases to more learn message update patterns more efficiently than the vanilla belief propagation algorithm. In this thesis, we prove a new threshold for the physical error rate on the rotated surface code using a neural belief propagation algorithm and ordered statistics decoding that consistently outperforms the threshold established by the vanilla belief propagation algorithm with ordered statistics decoding. We additionally show the results for similar efforts on the bivariate bicycle codes, and discuss the difficulties in achieving a threshold using the current neural belief propagation algorithms.
Abstract Building a large-scale quantum computer requires effective strategies to correct errors that inevitably arise in physical quantum systems 1 . Quantum error-correction codes 2 present a way to reach this goal by encoding logical information redundantly into many physical qubits. A key challenge in implementing such codes is accurately decoding noisy syndrome information extracted from redundancy checks to obtain the correct encoded logical information. Here we develop a recurrent, transformer-based neural network that learns to decode the surface code, the leading quantum error-correction code 3 . Our decoder outperforms other state-of-the-art decoders on real-world data from Google’s Sycamore quantum processor for distance-3 and distance-5 surface codes 4 . On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk and leakage, utilizing soft readouts and leakage information. After training on approximate synthetic data, the decoder adapts to the more complex, but unknown, underlying error distribution by training on a limited budget of experimental samples. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.