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Analyzing the Expressibility and Trainability of Quantum Convolutional Neural Networks

Asif Akhtab Ronggon, Tuhin Hossain, Tahani Jaser Alahmadi, Mohammad Ali Moni

Published May 25, 2026• 0 citations
Publisher

Abstract

Abstract Quantum Convolutional Neural Networks(QCNN) are a promising framework in QuantumMachine Learning(QML). The study analyzes QCNNs’ expressibility, trainability, and optimizers theoretically. This approach utilizes statistical learning and quantum information theory to assess QCNNs’ effectiveness in imitating quantum states, optimizing parameters, and executing intricate machine learning(ML) tasks. For this purpose, the Haar measure is used to evaluate the QCNN expressibility, yielding a score of 0.7720, acoverage entropy of 1.4023, and an effective state-space dimension of 166.59/200. This indicates that QCNNs can simulate massive quantum states and achieve a peak expressibility score of 0.8120 at layer 3, before decreasing due to overparameterization. Additionally, we examine gradient variance and find that the barren plateau indicator is 0.0818 and the mean gradient norm is 1.1837, both of which are below the theoretical threshold of 0.1, proving that the QCNN can avoid barren plateaus and train perfectly. Furthermore, the obtained Fisher Information Matrix(FIM) condition number of 2.05 ×10³ indicates a well-conditioned optimization landscape. We also utilized the Hessian matrix for optimization analysis, identifying a saddle point with 56 zero, 11 positive,and 5 negative eigenvalues. Where QCNNs required only 72 parameters, an 11.8-fold reduction from classical CNNs’ 848 parameters, but expressibility stays the same, assuming smooth convergence. Despite a gradient noise factor near 1, QCNNs have only a 43% expressibility decrease at 5% noise levels. Finally, the outcomes prove that QCNNs outperform classical CNNs in ML applications with respect to expressibility, trainability, and optimization.

DOI
10.21203/rs.3.rs-8600748/v1

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Abstract Quantum Convolutional Neural Networks(QCNN) are a promising framework in QuantumMachine Learning(QML). The study analyzes QCNNs’ expressibility, trainability, and optimizers theoretically. This approach utilizes statistical learning and quantum information theory to assess QCNNs’ effectiveness in imitating quantum states, optimizing parameters, and executing intricate machine learning(ML) tasks. For this purpose, the Haar measure is used to evaluate the QCNN expressibility, yielding a score of 0.7720, acoverage entropy of 1.4023, and an effective state-space dimension of 166.59/200. This indicates that QCNNs can simulate massive quantum states and achieve a peak expressibility score of 0.8120 at layer 3, before decreasing due to overparameterization. Additionally, we examine gradient variance and find that the barren plateau indicator is 0.0818 and the mean gradient norm is 1.1837, both of which are below the theoretical threshold of 0.1, proving that the QCNN can avoid barren plateaus and train perfectly. Furthermore, the obtained Fisher Information Matrix(FIM) condition number of 2.05 ×10³ indicates a well-conditioned optimization landscape. We also utilized the Hessian matrix for optimization analysis, identifying a saddle point with 56 zero, 11 positive,and 5 negative eigenvalues. Where QCNNs required only 72 parameters, an 11.8-fold reduction from classical CNNs’ 848 parameters, but expressibility stays the same, assuming smooth convergence. Despite a gradient noise factor near 1, QCNNs have only a 43% expressibility decrease at 5% noise levels. Finally, the outcomes prove that QCNNs outperform classical CNNs in ML applications with respect to expressibility, trainability, and optimization.
Abstract Quantum convolutional neural networks (QCNNs) are structured variational quantum architectures for quantum machine learning(QML). This study examines their practical reliability through the balance between expressibility, trainability, optimization geometry, and noise resilience. We propose a multi-metric QCNN framework integrating Haar-based expressibility analysis, gradient sensitivity assessment, finite-size scaling, optimizer benchmarking, empirical Fisher information, Hessian spectral analysis, parameter efficiency evaluation, and depolarizing noise characterization. Our applied 6-qubit, 3-layer QCNN-inspired architecture, with 72 trainable parameters, achieved a Kolmogorov-Smirnov(KS) similarity of (0.8826±0.0144) and a Kullback-Leibler(KL) divergence of (0.05648±0.00578) relative to the Haar-random fidelity distribution. Increasing circuit depth improved expressibility, increasing KS similarity from (0.7400±0.0153) to (0.9471±0.0152) and reducing KL divergence from (0.1812±0.0363) to (0.01815±0.00702). However, 30 of 72 parameters exhibited near-zero sensitivity, and the gradient variance decreased from (0.13939±0.08380) to (0.00929±0.00429) across 2–8 qubits, indicating finite-size gradient suppression without establishing asymptotic barren-plateau(BP) behavior. Furthermore, Adam achieved lower mean squared error(MSE) than gradient descent (0.24388 vs. 0.44056), while Fisher and Hessian analyses revealed anisotropic optimization landscapes. The QCNN required 72 parameters compared with 906 for a dense neural network(DNN) baseline. Under depolarizing noise, MSE increased from 0.39506 to 0.63404 and purity decreased from 1.0 to 0.0893, demonstrating that expressibility alone does not ensure robust trainability.
Quantum neural networks (QNNs) have generated excitement around the possibility of efficiently analyzing quantum data.But this excitement has been tempered by the existence of exponentially vanishing gradients, known as barren plateau landscapes, for many QNN architectures.Recently, quantum convolutional neural networks (QCNNs) have been proposed, involving a sequence of convolutional and pooling layers that reduce the number of qubits while preserving information about relevant data features.In this work, we rigorously analyze the gradient scaling for the parameters in the QCNN architecture.We find that the variance of the gradient vanishes no faster than polynomially, implying that QCNNs do not exhibit barren plateaus.This result provides an analytical guarantee for the trainability of randomly initialized QCNNs, which highlights QCNNs as being trainable under random initialization unlike many other QNN architectures.To derive our results, we introduce a novel graph-based method to analyze expectation values over Haar-distributed unitaries, which will likely be useful in other contexts.Finally, we perform numerical simulations to verify our analytical results.
This study explores Quantum Convolutional Neural Network (QCNN) starting from foundational quantum operations, such as the Rx gate for encoding MNIST image data into quantum states. We implemented quantum convolutional and pooling layers using one_unitary and two_unitary circuits, enabling effective feature extraction and dimensionality reduction while preserving critical information. Expressibility analysis revealed varying capabilities across different one_unitary circuits, with Rx, Ry, and Rz combinations demonstrating promising results akin to Haar random states. The proposed QCNN model exhibited robust performance metrics (accuracy: 95.98%, precision: 94.44%, recall: 96.59%, F1-score: 0.9551, AUC: 0.9604) in classification tasks, supported by efficient convergence during optimization. Future directions include expanding QCNN applications to handle more complex datasets and optimizing architectures to enhance quantum machine learning capabilities, particularly in image processing. This study underscores the potential of QCNNs in advancing quantum computing applications in neural network architectures. Keywords: MNIST, classification, CNN, expressibility
Abstract Recently, interest in quantum computing has significantly increased, driven by its potential advantages over classical techniques. Quantum machine learning (QML) exemplifies one of the important quantum computing applications that are expected to surpass classical machine learning in a wide range of instances. This paper addresses the performance of QML in the context of high-energy physics (HEP). As an example, we focus on the top-quark tagging, for which classical convolutional neural networks (CNNs) have been effective but fall short in accuracy when dealing with highly energetic jet images. In this paper, we use a quantum convolutional neural network (QCNN) for this task and compare its performance with CNN using a classical noiseless simulator. We compare various setups for the QCNN, varying the convolutional circuit, type of encoding, loss function, and batch sizes. For every quantum setup, we design a similar setup to the corresponding classical model for a fair comparison. Our results indicate that, using a classical simulator, QCNN with proper setups tend to perform better than their CNN counterparts, especially when the convolution block has a lower number of parameters. For the higher parameter regime, the QCNN circuit was adjusted according to the dimensional expressivity analysis (DEA) to lower the parameter count while preserving its optimal structure. The DEA circuit demonstrated improved results over the comparable classical CNN model.
This study presents an advanced Quantum Convolutional Neural Network (QCNN) architecture designed to improve classification performance and training stability within the constraints of Noisy Intermediate-Scale Quantum (NISQ) devices. The proposed model incorporates a multi-dimensional quantum feature encoding stage, followed by entanglement-enriched convolutional layers and a refined quantum pooling mechanism. These design innovations increase the circuit’s expressive capacity while reducing susceptibility to barren plateau phenomena, thereby supporting more efficient optimization. Extensive numerical simulations indicate that the model achieves a classification accuracy of 95.2%, with a final loss of 0.24, a low output variance of 0.031, and improved expressibility characterized by a mean fidelity of 0.07. Collectively, these results demonstrate that the enhanced QCNN provides a robust and scalable framework for quantum machine learning tasks suitable for current and near-term quantum hardware.
This study presents an advanced Quantum Convolutional Neural Network (QCNN) architecture tailored for improved classification accuracy and enhanced training stability within the constraints of Noisy Intermediate-Scale Quantum (NISQ) devices. The proposed model incorporates a multi-dimensional quantum feature encoding stage, followed by entanglement-enriched convolutional layers and a refined quantum pooling mechanism. These design innovations substantially increase the circuit’s expressive capacity while reducing susceptibility to barren plateau phenomena, thereby supporting more efficient optimization. Extensive numerical simulations indicate that the model achieves a classification accuracy of 95.2%, alongside a final loss of 0.24, low output variance of 0.031, and improved expressibility characterized by a mean fidelity of 0.07. Collectively, these results demonstrate that the enhanced QCNN offers a robust and scalable framework for quantum machine learning tasks suitable for current and near-term quantum hardware. Also, modern applications from qunatum devices with the abovementioned result is met and leads to practical systems.