A promising strategy to protect quantum information from noise-induced errors is to encode it into the low-energy states of a topological quantum memory device. However, readout errors from such memory under realistic settings is less understood. We study the problem of decoding quantum information encoded in the groundspaces of topological stabilizer Hamiltonians in the presence of generic perturbations, such as quenched disorder. We first prove that the standard stabilizer-based error correction and decoding schemes work adequately well in such perturbed quantum codes by showing that the decoding error diminishes exponentially in the distance of the underlying unperturbed code. We then prove that Quantum Neural Network (QNN) decoders provide an almost quadratic improvement on the readout error. Thus, we demonstrate provable advantage of using QNNs for decoding realistic quantum error-correcting codes, and our result enables the exploration of a wider range of non-stabilizer codes in the near-term laboratory settings.
A promising strategy to protect quantum information from noise-induced errors is to encode it into the low-energy states of a topological quantum memory device. However, readout errors from such memory under realistic settings is less understood. We study the problem of decoding quantum information encoded in the groundspaces of topological stabilizer Hamiltonians in the presence of generic perturbations, such as quenched disorder. We first prove that the standard stabilizer-based error correction and decoding schemes work adequately well in such perturbed quantum codes by showing that the decoding error diminishes exponentially in the distance of the underlying unperturbed code. We then prove that Quantum Neural Network (QNN) decoders provide an almost quadratic improvement on the readout error. Thus, we demonstrate provable advantage of using QNNs for decoding realistic quantum error-correcting codes, and our result enables the exploration of a wider range of non-stabilizer codes in the near-term laboratory settings.
Finding optimal correction of errors in generic stabilizer codes is a computationally hard problem, even for simple noise models. While this task can be simplified for codes with some structure, such as topological stabilizer codes, developing good and efficient decoders still remains a challenge. In our paper, we systematically study a very versatile class of decoders based on feedforward neural networks. To demonstrate adaptability, we apply neural decoders to the triangular color and toric codes under various noise models with realistic features, such as spatially correlated errors. We report that neural decoders provide a significant improvement over leading efficient decoders in terms of the error-correction threshold. In particular, the neural decoder threshold for the two-dimensional color code is very close to the toric code threshold. Using neural networks simplifies the design of decoders and does not require prior knowledge of the underlying noise.
This paper proposes a novel architecture for quantum neural networks (QNNs) that leverages the benefits of topological protection and quantum error correction (QEC). The core idea is to construct a QNN based on a network of non-Abelian anyons, exploiting their inherent robustness against local perturbations. We introduce a framework where topological protection significantly reduces the impact of noise on the network, while LDPC (Low-Density Parity-Check) codes are employed to actively correct errors in the quantum information. The resulting system demonstrates a high tolerance to noise, a critical requirement for scalable QNN implementations. This approach represents a significant advancement in QNN design, merging the strengths of topological quantum computing and established QEC techniques. The theoretical framework outlines the key components and provides a pathway toward realizing fault-tolerant QNNs. The proposed architecture offers a promising route toward building practical and reliable quantum computing systems.
Decoding is an essential component of quantum error correction (QEC), translating stabilizer measurement outcomes into corrective actions that suppress logical errors and preserve logical quantum information. Building fault-tolerant architectures requires increasing the code distance, which in turn places growing demands on decoding accuracy, scalability, and practical deployability. While a wide range of decoding algorithms have been proposed and demonstrated, achieving reliable, scalable, and real-time decoding remains a significant challenge. Machine-learning (ML) approaches are particularly well suited to this setting, as quantum error decoding is fundamentally a problem of processing large volumes of classical data with complex spatiotemporal correlations. This chapter surveys ML-based methods for quantum error decoding, with a focus on topological codes and an emphasis on architectural principles, practical performance, and real-time considerations. We first frame decoding as a learning problem and outline key paradigms, including discriminative, generative, and reinforcement-learning formulations. We then introduce the neural network building blocks that underpin most contemporary neural decoders and discuss how these components can be integrated to balance expressivity, scalability, and latency. Building on this architectural perspective, we review recent progress and benchmarks in neural decoding for memory experiments, and discuss real-time decoding, open challenges, and future directions toward scalable fault-tolerant quantum computing.
This paper explores the potential of employing topological qubits as a robust memory solution for neural networks. The primary challenge in utilizing quantum computing for complex tasks, particularly within neural networks, lies in the decoherence of quantum states. Traditional qubits are highly susceptible to environmental noise, leading to rapid state decay and hindering long-term computations. Topological qubits, due to their inherent properties arising from non-Abelian braiding, offer a promising solution by providing a fundamentally different approach to quantum information storage. This work details a proposed architecture where a neural network is constructed utilizing these topologically protected qubits. We investigate the theoretical framework for encoding and processing information using these qubits, focusing on minimizing decoherence effects. The core claim presented here is that topological qubits can significantly reduce decoherence issues, enabling more complex and longer-term computations within neural networks. We outline a potential computational model and discuss key considerations for implementing such a system, including qubit connectivity, error correction strategies, and the scaling of the network. The research introduces a new approach to quantum memory, directly addressing a critical bottleneck in the advancement of quantum neural networks.
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.
A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment—without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate L of the encoded logical qubit to values much below the error rate phys of the physical qubits—fitting the expected power law scaling , with d the code distance. The neural network incorporates the information from 'flag qubits' to avoid reduction in the effective code distance caused by the circuit. As a test, we apply the neural network decoder to a density-matrix based simulation of a superconducting quantum computer, demonstrating that the logical qubit has a longer life-time than the constituting physical qubits with near-term experimental parameters.