A multi-metric characterization of quantum convolutional neural networks: expressibility, trainability, and noise sensitivity

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.