Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2112.15002

Toward Trainability of Deep Quantum Neural Networks

Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao

Published Dec 30, 2021• 5 citations
arXivHTMLPDF

Abstract

Quantum Neural Networks (QNNs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep QNN will not be feasible. In this work, we provide the first viable solution to the vanishing gradient problem for deep QNNs with theoretical guarantees. Specifically, we prove that for circuits with controlled-layer architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behaviour on parameter space consisting of rotation angles, as employed in almost any QNNs, instead of relying on impractical 2-design assumptions. We explicitly construct examples where only our QNNs are trainable and converge, while others in comparison cannot.

DOI
10.48550/arxiv.2112.15002

Related papers

Toward Trainability of Quantum Neural Networks

Kaining Zhang, Min-Hsiu Hsieh, Liu Liu +1 more

Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing gradient with rate exponential to the input qubit number. The vanishing gradient could seriously influence the applications of large-size QNNs. In this work, we provide a viable solution with theoretical guarantees. Specifically, we prove that QNNs with tree tensor and step controlled architectures have gradients that vanish at most polynomially with the qubit number. We numerically demonstrate QNNs with tree tensor and step controlled structures for the application of binary classification. Simulations show faster convergent rates and better accuracy compared to QNNs with random structures.

arXiv:2011.06258•Nov 12, 2020•35 citations

Deep Variational Quantum Circuits with Barren-Plateau-Free Architectures

Kaining Zhang, Min-Hsiu Hsieh, Dacheng Tao

Variational quantum algorithms (VQAs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep circuit will not be feasible. In this work, we provide a viable solution to the vanishing gradient problem for deep VQAs with theoretical guarantees. Specifically, we prove that for quantum controlled-layer and quantum residual network (QResNet), architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behavior on parameter space consisting of rotation angles, as employed in almost all VQAs, instead of relying on impractical 2-design assumptions. We conduct several numerical experiments as verifications, where only our circuits are trainable and converge, while hardware-efficient and random circuits with similar number of parameters in comparison cannot converge.

Artificial Intelligence Science and Engineering•Mar 1, 2026•1 citation

Escaping from the Barren Plateau Via Gaussian Initializations in Deep Variational Quantum Circuits

Kaining Zhang, Liu Liu, Min-Hsiu Hsieh +1 more

Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.

arXiv:2203.09376•neural information processing systems•Jan 1, 2022•32 citations

Training Theory of Variational Quantum Machine Learning

Kaining Zhang

Recent advancements in machine learning have revolutionised research across various fields. Despite their success, conventional learning techniques are hindered by their significant computational resources and energy requirements. Prompted by recent experimental breakthroughs in quantum computing, variational quantum machine learning (QML) – machine learning integrated with variational quantum circuits (VQCs) – has emerged as a promising alternative. Nonetheless, the theoretical framework underpinning the advantages of variational QML is still rudimentary. Specifically, the training of VQCs faces several challenges, such as the barren plateau problem, where the gradient diminishes exponentially with an increasing qubits. A related issue arises in variational QML training, where the convergence rate is exponentially small. In this thesis, we present theoretically guaranteed solutions to these challenges. First, we construct innovative circuit architectures to address the vanishing gradient problem in deep VQCs. We propose quantum controlled-layer and quantum ResNet structures, demonstrating that the expected gradient norm's lower bound is unaffected by the increase in qubits and circuit depth. Next, we introduce an initialization strategy to mitigate the vanishing gradient issue in general deep quantum circuits. We prove that Gaussian-initialized parameters ensure the gradient norm's decay rate remains inversely polynomial despite the increase in qubit numbers and circuit depth. Finally, we propose a novel and effective theory for analysing the training of quantum neural networks with moderate depths. We prove that, under certain randomness conditions in the circuits and datasets, training converges linearly with a rate inversely proportional to the dataset size. Our approach surpasses previous results, achieving exponentially larger convergence rates with modest depth, or conversely, requiring exponentially less depth for equivalent rates.

•Jan 1, 2023•0 citations

Exponential Hardness of Optimization from the Locality in Quantum Neural Networks

Hao-Kai Zhang, Chengkai Zhu, Geng Liu +1 more

Quantum neural networks (QNNs) have become a leading paradigm for establishing near-term quantum applications in recent years. The trainability issue of QNNs has garnered extensive attention, spurring demand for a comprehensive analysis of QNNs in order to identify viable solutions. In this work, we propose a perspective that characterizes the trainability of QNNs based on their locality. We prove that the entire variation range of the loss function via adjusting any local quantum gate vanishes exponentially in the number of qubits with a high probability for a broad class of QNNs. This result reveals extra harsh constraints independent of gradients and unifies the restrictions on gradient-based and gradient-free optimizations naturally. We showcase the validity of our results with numerical simulations of representative models and examples. Our findings, as a fundamental property of random quantum circuits, deepen the understanding of the role of locality in QNNs and serve as a guideline for assessing the effectiveness of diverse training strategies for quantum neural networks.

Proceedings of the AAAI Conference on Artificial Intelligence•Mar 24, 2024•5 citations

Trainability of Dissipative Perceptron-Based Quantum Neural Networks

Kunal Sharma, Marco Cerezo, Łukasz Cincio +1 more

Several architectures have been proposed for quantum neural networks (QNNs), with the goal of efficiently performing machine learning tasks on quantum data. Rigorous scaling results are urgently needed for specific QNN constructions to understand which, if any, will be trainable at a large scale. Here, we analyze the gradient scaling (and hence the trainability) for a recently proposed architecture that we call dissipative QNNs (DQNNs), where the input qubits of each layer are discarded at the layer's output. We find that DQNNs can exhibit barren plateaus, i.e., gradients that vanish exponentially in the number of qubits. Moreover, we provide quantitative bounds on the scaling of the gradient for DQNNs under different conditions, such as different cost functions and circuit depths, and show that trainability is not always guaranteed. Our work represents the first rigorous analysis of the scalability of a perceptron-based QNN.

arXiv:2005.12458•Physical Review Letters•May 6, 2022•164 citations
Quantum Neural Networks (QNNs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep QNN will not be feasible. In this work, we provide the first viable solution to the vanishing gradient problem for deep QNNs with theoretical guarantees. Specifically, we prove that for circuits with controlled-layer architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behaviour on parameter space consisting of rotation angles, as employed in almost any QNNs, instead of relying on impractical 2-design assumptions. We explicitly construct examples where only our QNNs are trainable and converge, while others in comparison cannot.
Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing gradient with rate exponential to the input qubit number. The vanishing gradient could seriously influence the applications of large-size QNNs. In this work, we provide a viable solution with theoretical guarantees. Specifically, we prove that QNNs with tree tensor and step controlled architectures have gradients that vanish at most polynomially with the qubit number. We numerically demonstrate QNNs with tree tensor and step controlled structures for the application of binary classification. Simulations show faster convergent rates and better accuracy compared to QNNs with random structures.
Variational quantum algorithms (VQAs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep circuit will not be feasible. In this work, we provide a viable solution to the vanishing gradient problem for deep VQAs with theoretical guarantees. Specifically, we prove that for quantum controlled-layer and quantum residual network (QResNet), architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behavior on parameter space consisting of rotation angles, as employed in almost all VQAs, instead of relying on impractical 2-design assumptions. We conduct several numerical experiments as verifications, where only our circuits are trainable and converge, while hardware-efficient and random circuits with similar number of parameters in comparison cannot converge.
Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.
Recent advancements in machine learning have revolutionised research across various fields. Despite their success, conventional learning techniques are hindered by their significant computational resources and energy requirements. Prompted by recent experimental breakthroughs in quantum computing, variational quantum machine learning (QML) – machine learning integrated with variational quantum circuits (VQCs) – has emerged as a promising alternative. Nonetheless, the theoretical framework underpinning the advantages of variational QML is still rudimentary. Specifically, the training of VQCs faces several challenges, such as the barren plateau problem, where the gradient diminishes exponentially with an increasing qubits. A related issue arises in variational QML training, where the convergence rate is exponentially small. In this thesis, we present theoretically guaranteed solutions to these challenges. First, we construct innovative circuit architectures to address the vanishing gradient problem in deep VQCs. We propose quantum controlled-layer and quantum ResNet structures, demonstrating that the expected gradient norm's lower bound is unaffected by the increase in qubits and circuit depth. Next, we introduce an initialization strategy to mitigate the vanishing gradient issue in general deep quantum circuits. We prove that Gaussian-initialized parameters ensure the gradient norm's decay rate remains inversely polynomial despite the increase in qubit numbers and circuit depth. Finally, we propose a novel and effective theory for analysing the training of quantum neural networks with moderate depths. We prove that, under certain randomness conditions in the circuits and datasets, training converges linearly with a rate inversely proportional to the dataset size. Our approach surpasses previous results, achieving exponentially larger convergence rates with modest depth, or conversely, requiring exponentially less depth for equivalent rates.
Quantum neural networks (QNNs) have become a leading paradigm for establishing near-term quantum applications in recent years. The trainability issue of QNNs has garnered extensive attention, spurring demand for a comprehensive analysis of QNNs in order to identify viable solutions. In this work, we propose a perspective that characterizes the trainability of QNNs based on their locality. We prove that the entire variation range of the loss function via adjusting any local quantum gate vanishes exponentially in the number of qubits with a high probability for a broad class of QNNs. This result reveals extra harsh constraints independent of gradients and unifies the restrictions on gradient-based and gradient-free optimizations naturally. We showcase the validity of our results with numerical simulations of representative models and examples. Our findings, as a fundamental property of random quantum circuits, deepen the understanding of the role of locality in QNNs and serve as a guideline for assessing the effectiveness of diverse training strategies for quantum neural networks.
Several architectures have been proposed for quantum neural networks (QNNs), with the goal of efficiently performing machine learning tasks on quantum data. Rigorous scaling results are urgently needed for specific QNN constructions to understand which, if any, will be trainable at a large scale. Here, we analyze the gradient scaling (and hence the trainability) for a recently proposed architecture that we call dissipative QNNs (DQNNs), where the input qubits of each layer are discarded at the layer's output. We find that DQNNs can exhibit barren plateaus, i.e., gradients that vanish exponentially in the number of qubits. Moreover, we provide quantitative bounds on the scaling of the gradient for DQNNs under different conditions, such as different cost functions and circuit depths, and show that trainability is not always guaranteed. Our work represents the first rigorous analysis of the scalability of a perceptron-based QNN.