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arXiv:2609.12500 · National Science Review

Quantum neural network equipped with backpropagation on a qudit processor

Yibo Yuan, Zhuoyue Xu, Zhenyue Du, Xingyu Zhao, Xu Cheng, Yue Li, Waner Hou, Yuqi Zhou, Zhaokai Li, Yiheng Lin

quant-phSubmitted Sep 11, 2026• Announced Sep 14, 2026• 0 citations
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Abstract

Quantum neural networks (QNNs), one of the fundamental algorithms in quantum machine learning, have been widely used in classification and identification tasks. However, the capabilities of QNNs are constrained by their size, which is determined by the dimension of the Hilbert space of the underlying quantum processor. Multi-level quantum digits (qudits) offer access to a higher-dimensional Hilbert space compared to two-level qubits, enabling the construction of more expressive QNNs. In this work, we report an experimental demonstration of qudit-based QNN using a trapped $\rm ^{40}Ca^+$ ion. We train the QNN using a hybrid quantum-classical implementation of backpropagation and achieve an experimental classification accuracy of $95.7\%$ on a test image set. This demonstration highlights the potential of qudit-based processors to QNN architectures and provides a framework for implementing qudit-based QNNs across various quantum devices.

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Quantum computers are the latest dream of humans in solving nondeterministic polynomial problems. These problems include machine learning and object classification. However, commercial quantum computers are still not available. In the meantime, scientists developed hybrid models by combining Convolutional Neural Networks (CNN) and Quantum Neural Networks (QNN) to apply quantum computing to artificial intelligence (AI) applications. The hybrid model accelerates the processing of an AI machine by combining the advantage of CNN in feature extraction with the parallel computing capability of QNN. In this study, we optimized the existing hybrid models with a fewer number of qubits to improve the performance of QNN on digit recognition. The study also presents the implementation of the model using Qiskit, describes the training process, and analyzes the impact of the number of qubits and quantum gates on the classification results. Experiments show that Hybrid Quantum-Classical Neural Networks (H-QNN) have the potential to outperform pure CNN. Finally, the study suggests future improvements, including the optimization of quantum algorithms and extending experiments to more complex data sets.

International Conference on Integrated Circuits, Design, and Verification (ICDV)•Jun 16, 2025•0 citations
Quantum neural networks (QNNs), one of the fundamental algorithms in quantum machine learning, have been widely used in classification and identification tasks. However, the capabilities of QNNs are constrained by their size, which is determined by the dimension of the Hilbert space of the underlying quantum processor. Multi-level quantum digits (qudits) offer access to a higher-dimensional Hilbert space compared to two-level qubits, enabling the construction of more expressive QNNs. In this work, we report an experimental demonstration of qudit-based QNN using a trapped 40Ca+\rm ^{40}Ca^+40Ca+ ion. We train the QNN using a hybrid quantum-classical implementation of backpropagation and achieve an experimental classification accuracy of 95.7%95.7\%95.7% on a test image set. This demonstration highlights the potential of qudit-based processors to QNN architectures and provides a framework for implementing qudit-based QNNs across various quantum devices.
Deep learning and quantum computing have achieved dramatic progresses in recent years. The interplay between these two fast-growing fields gives rise to a new research frontier of quantum machine learning. In this work, we report an experimental demonstration of training deep quantum neural networks via the backpropagation algorithm with a six-qubit programmable superconducting processor. We experimentally perform the forward process of the backpropagation algorithm and classically simulate the backward process. In particular, we show that three-layer deep quantum neural networks can be trained efficiently to learn two-qubit quantum channels with a mean fidelity up to 96.0% and the ground state energy of molecular hydrogen with an accuracy up to 93.3% compared to the theoretical value. In addition, six-layer deep quantum neural networks can be trained in a similar fashion to achieve a mean fidelity up to 94.8% for learning single-qubit quantum channels. Our experimental results indicate that the number of coherent qubits required to maintain does not scale with the depth of the deep quantum neural network, thus providing a valuable guide for quantum machine learning applications with both near-term and future quantum devices.
Neural quantum kernels (NQKs) construct quantum kernels by pretraining a quantum neural network (QNN) and subsequently reusing the trained circuit as a task-adapted embedding. Extending this framework to qudits, with local unitaries in SU(d)\mathrm{SU}(d)SU(d), provides a natural route to richer data embeddings through the increased local degrees of freedom and a direct interface for multiclass classification via intrinsically multi-level quantum systems. In this work, focusing on qutrits (d=3d=3d=3), we extend NQKs to the qudit setting and perform a systematic study of key design choices, including the number of encoded features, the number of qutrits, the kernel construction (1-to-nnn and nnn-to-nnn), and the parameterization of SU(3)\mathrm{SU}(3)SU(3) unitaries. Across binary and three-class tasks on four benchmark datasets, qutrit NQKs improve over the corresponding QNN baselines in nearly all settings considered and can benefit from scaling both the feature budget and the system size, although the magnitude of these gains may saturate, is dataset-dependent, and depends on the chosen parameterization. In particular, an ablation over SU(3)\mathrm{SU}(3)SU(3) parameterizations shows that the unitary representation can substantially impact both optimization behaviour and classifier performance. These findings highlight the potential of qudit-based quantum models not only as a straightforward generalization of qubit-based architectures, but also as a promising means to better exploit complex data structures in quantum machine learning.
As quantum devices scale toward practical machine learning applications, the binary qubit paradigm faces expressivity and resource efficiency limitations. Multi-level quantum systems, or qudits, offer a promising alternative by harnessing a larger Hilbert space, enabling richer data embeddings, more compact variational circuits, and support for multi-valued problem structures. In this work, we review the role of qudits in quantum machine learning techniques, mainly variational quantum algorithms and quantum neural networks. Drawing on recent experimental demonstrations, including high-level superconducting transmons, qutrit-based combinatorial optimization, and single-qudit classifiers, we highlight how qudit architectures can reduce circuit depth and parameter counts while maintaining competitive fidelity. We further assess the evolving software ecosystem, from specialized simulators and differentiable-programming libraries to extensions of mainstream frameworks. We also identify key challenges in control complexity, noise management, and tooling maturity.
Image classification is a fundamental task in deep learning, and recent advances in quantum computing have generated significant interest in quantum neural networks. Traditionally, Convolutional Neural Networks (CNNs) are employed to extract image features, while Multilayer Perceptrons (MLPs) handle decision making. However, parameterized quantum circuits offer the potential to capture complex image features and define sophisticated decision boundaries. In this paper, we present a novel Hybrid Quantum–Classical Neural Network (H-QNN) for image classification, and demonstrate its effectiveness using the MNIST dataset. Our model combines quantum computing with classical supervised learning to enhance classification accuracy and computational efficiency. In this study, we detail the architecture of the H-QNN, emphasizing its capability in feature learning and image classification. Experimental results demonstrate that the proposed H-QNN model outperforms conventional deep learning methods in various training scenarios, showcasing its effectiveness in high-dimensional image classification tasks. Additionally, we explore the broader applicability of hybrid quantum–classical approaches in other domains. Our findings contribute to the growing body of work in quantum machine learning, and underscore the potential of quantum-enhanced models for image recognition and classification.
Quantum machine learning has established as an interdisciplinary field to overcome limitations of classical machine learning and neural networks. This is a field of research which can prove that quantum computers are able to solve problems with complex correlations between inputs that can be hard for classical computers. This suggests that learning models made on quantum computers may be more powerful for applications, potentially faster computation and better generalization on less data. The objective of this paper is to investigate how training of quantum neural network (QNNs) can be done using quantum optimization algorithms for improving the performance and time complexity of QNNs. A classical neural network can be partially quantized to create a hybrid quantum-classical neural network which is used mainly in classification and image recognition. In this paper, a QNN structure is made where a variational parameterized circuit is incorporated as an input layer named as Variational Quantum Neural Network (VQNNs). We encode the cost function of QNNs onto relative phases of a superposition state in the Hilbert space of the network parameters. The parameters are tuned with an iterative quantum approximate optimisation (QAOA) mixer and problem hamiltonians. VQNNs is experimented with MNIST digit recognition (less complex) and crack image classification datasets (more complex) which converges the computation in lesser time than QNN with decent training accuracy.
Quantum computers are the latest dream of humans in solving nondeterministic polynomial problems. These problems include machine learning and object classification. However, commercial quantum computers are still not available. In the meantime, scientists developed hybrid models by combining Convolutional Neural Networks (CNN) and Quantum Neural Networks (QNN) to apply quantum computing to artificial intelligence (AI) applications. The hybrid model accelerates the processing of an AI machine by combining the advantage of CNN in feature extraction with the parallel computing capability of QNN. In this study, we optimized the existing hybrid models with a fewer number of qubits to improve the performance of QNN on digit recognition. The study also presents the implementation of the model using Qiskit, describes the training process, and analyzes the impact of the number of qubits and quantum gates on the classification results. Experiments show that Hybrid Quantum-Classical Neural Networks (H-QNN) have the potential to outperform pure CNN. Finally, the study suggests future improvements, including the optimization of quantum algorithms and extending experiments to more complex data sets.