Abstract Understanding how depth-dependent quantum evolution reshapes the geometric organization of data representations in quantum neural networks (QNNs) remains a central question in quantum machine learning. We study a hardware-efficient variational QNN—single-qubit R Y / R Z rotations per layer followed by a nearest-neighbor CNOT entangling ring, with circuit depth n ranging from 3 to 12 layers—and ask how its data representation geometry reorganizes as n increases. We report two findings across MNIST and BloodMNIST (15 independent random seeds for amplitude encoding: 5 MNIST, 10 BloodMNIST), with a BloodMNIST angle-encoding control (10 further seed-runs, 25 in total) used to check the second finding’s recurrence under a different encoding strategy. First, class separability shows power-law growth within the tested depth window, S ( n ) ∝ n b , where b = δ − ε captures the balance between inter-class expansion and intra-class diffusion; this description holds throughout the tested range for MNIST, whereas for BloodMNIST the growth is better described by a model approaching a depth-independent ceiling within the same range, with b retained as a common cross-dataset summary exponent. Second, we measure an auxiliary parameter-perturbation sensitivity statistic and find that its depth-trend exponent γ shows a dataset-level association with b via b ≈ k data ( 1 − | γ | ) . We explicitly tested whether this statistic tracks the textbook state-based quantum Fisher information by computing the latter rigorously for the same trained circuits, and found that it does not; we also find that the b – γ association is weak and not statistically significant within either dataset alone. We therefore report this coupling descriptively, as an open dataset-level pattern rather than an established physical law. To our knowledge, this is the first report of this depth-dependent separability-growth pattern in QNNs together with a practical, predictive heuristic for depth selection derived from it, using each dataset’s own functional form. In the tested depth regime ( n ∈ [ 3 , 12 ]
Abstract Understanding how depth-dependent quantum evolution reshapes the geometric organization of data representations in quantum neural networks (QNNs) remains a central question in quantum machine learning. We study a hardware-efficient variational QNN—single-qubit R Y / R Z rotations per layer followed by a nearest-neighbor CNOT entangling ring, with circuit depth n ranging from 3 to 12 layers—and ask how its data representation geometry reorganizes as n increases. We report two findings across MNIST and BloodMNIST (15 independent random seeds for amplitude encoding: 5 MNIST, 10 BloodMNIST), with a BloodMNIST angle-encoding control (10 further seed-runs, 25 in total) used to check the second finding’s recurrence under a different encoding strategy. First, class separability shows power-law growth within the tested depth window, S ( n ) ∝ n b , where b = δ − ε captures the balance between inter-class expansion and intra-class diffusion; this description holds throughout the tested range for MNIST, whereas for BloodMNIST the growth is better described by a model approaching a depth-independent ceiling within the same range, with b retained as a common cross-dataset summary exponent. Second, we measure an auxiliary parameter-perturbation sensitivity statistic and find that its depth-trend exponent γ shows a dataset-level association with b via b ≈ k data ( 1 − | γ | ) . We explicitly tested whether this statistic tracks the textbook state-based quantum Fisher information by computing the latter rigorously for the same trained circuits, and found that it does not; we also find that the b – γ association is weak and not statistically significant within either dataset alone. We therefore report this coupling descriptively, as an open dataset-level pattern rather than an established physical law. To our knowledge, this is the first report of this depth-dependent separability-growth pattern in QNNs together with a practical, predictive heuristic for depth selection derived from it, using each dataset’s own functional form. In the tested depth regime ( n ∈ [ 3 , 12 ]
Hybrid quantum neural networks (QNNs) pair a classical feature extractor with a variational quantum circuit, and the effective dimension of the circuit's quantum Fisher information matrix (QFIM) is widely used to assess such models. This scalar is often interpreted as an indicator of trainability or accuracy, although these readings are rarely tested under controlled conditions. We therefore investigate what it actually measures. We define the effective dimension as the participation ratio of the QFIM diagonal, which lies between one and the parameter count, and we evaluate it across two image datasets, four frozen backbones including a random-projection control, four circuit depths on eight qubits, and three seeds. During training, the effective dimension remains well below the parameter count, and its correlation with test accuracy ranges from -0.72 to +0.87, changing sign across seeds in half of the configurations. Increasing depth activates a larger share of the circuit directions, from 0.38 to 0.91 of the parameter count, while the initialization gradient variance remains between 1e-5 and 1e-4, so no barren plateau (BP) appears at eight qubits. At a fixed circuit, the four backbones share an initial effective dimension of 6 to 9, even though CIFAR-10 accuracy ranges from 23.14% to 86.44%. These results show that the effective dimension characterizes circuit geometry rather than model quality. We conclude with a practical protocol that assigns each diagnostic to the question it can answer.
We report an empirical study of how quantum neural networks (QNNs) train under global and local costfunctions, with layerwise training included as a third point of comparison. The main experiment is a set of 180 training runson binary MNIST (digits 3 and 6), using an 8-qubit hardware-efficient ansatz written in TensorFlow Quantum. A second,separate study measures gradient landscapes at a fixed input across 48 circuit configurations with 4 to 10 qubits. Wecompare three procedures: end-to-end training with a global product-Z readout (the baseline), a layerwise curriculum thatreuses trained parameters between stages, and training on a local averaged-Z cost. Test accuracy barely moves betweenmethods. The global baseline still reaches 87.2% at 8 layers, so we see no accuracy collapse in this regime, andmatched-seed tests at 8 layers find no significant difference (Friedman p = 0.654; every pairwise paired test isnon-significant, as are the tests at 4 and 6 layers). Gradient statistics tell the methods apart much more clearly. At depths 4,6 and 8, the local cost ends training with 1.5 to 2.4 times lower gradient variance than the global baseline (paired p <= 9.4x 10⁻³ after Holm correction; Cohen's d_z up to 2.3). It also costs far more to simulate at depth: 6.57 h per run at 8 layers,against 1.66 h for the baseline. In the landscape study, the local observable's gradient variance is 1.4 to 10 times lower thanthe global observable's over the whole configuration grid, and 4 to 10 times lower at depth 8. Layerwise training follows adifferent path, with gradient variance falling during optimization, yet it gives no
Hybrid quantum neural networks are increasingly explored for classification, yet it remains unclear how their performance and quantum behavior scale with circuit depth and qubit count. We present a controlled scaling study of hybrid quantum-classical classifiers along two axes: (1) increasing the number of quantum layers L at fixed qubits Q, and (2) increasing the number of qubits Q at fixed depth L. Across multiple datasets, we evaluate predictive performance using Accuracy, PR-AUC, Precision, Recall, and F1, and track quantum-specific metrics (QCE, EEE, QGN) to characterize how quantum properties evolve under scaling. Our results summarize scaling trends, saturation regimes, and dataset-dependent sensitivity, and further analyze how quantum metrics relate to predictive performance. This study provides practical guidance for selecting (Q,L) in hybrid QNN classifiers and establishes a consistent evaluation protocol for scaling analysis.
The performance of quantum neural network models depends strongly on architectural decisions, including circuit depth, placement of parametrized operations, and data-encoding strategies. Selecting an effective architecture is challenging and closely related to the classical difficulty of choosing suitable neural-network topologies, which is computationally hard. This work investigates automated quantum-circuit construction for regression tasks and introduces a genetic-algorithm framework that discovers Reduced Regressor QNN architectures. The approach explores depth, parametrized gate configurations, and flexible data re-uploading patterns, formulating the construction of quantum regressors as an optimization process. The discovered circuits are evaluated against seventeen classical regression models on twenty-two nonlinear benchmark functions and four analytical functions. Although classical methods often achieve comparable results, they typically require far more parameters, whereas the evolved quantum models remain compact while providing competitive performance. We further analyze dataset complexity using twelve structural descriptors and show, across five increasingly challenging meta-learning scenarios, that these measures can reliably predict which quantum architecture will perform best. The results demonstrate perfect or near-perfect predictive accuracy in several scenarios, indicating that complexity metrics offer powerful and compact representations of dataset structure and can effectively guide automated model selection. Overall, this study provides a principled basis for meta-learning-driven quantum architecture design and advances the understanding of how quantum models behave in regression settings--a topic that has received limited exploration in prior work. These findings pave the way for more systematic and theoretically grounded approaches to quantum regression.
Abstract Quantum machine learning, focusing on quantum neural networks (QNNs), remains a vastly uncharted field of study. Current QNN models primarily employ variational circuits on an ansatz or a quantum feature map, often requiring multiple entanglement layers. This methodology not only increases the computational cost of the circuit beyond what is practical on near-term quantum devices but also misleadingly labels these models as neural networks, given their divergence from the structure of a typical feed-forward neural network (FFNN). Moreover, the circuit depth and qubit needs of these models scale poorly with the number of data features, resulting in an efficiency challenge for real-world machine learning tasks. We introduce a bona fide QNN model, which seamlessly aligns with the versatility of a traditional FFNN in terms of its adaptable intermediate layers and nodes, absent from intermediate measurements such that our entire model is coherent. This model stands out with its reduced circuit depth and number of requisite CNOT gates, achieving a more than 50% reduction in both compared to prevailing QNN models. Furthermore, the qubit count in our model remains unaffected by the data’s feature quantity. We test our proposed model on various benchmarking datasets such as the breast cancer diagnostic (Wisconsin) and credit card fraud detection datasets. Our model achieved an accuracy of 91% on the breast cancer dataset and 85% on the credit card fraud detection dataset, outperforming existing QNN methods by 5–10% while requiring approximately 50% fewer quantum resources. These results showcase the advantageous efficacy of our approach, paving the way for the application of quantum neural networks to relevant real-world machine learning problems.
Quantum computing is expected to fundamentally change computer systems in the future. Recently, a new research topic of quantum computing is the hybrid quantum–classical approach for machine learning, in which a parameterized quantum circuit, also called quantum neural network (QNN), is optimized by a classical computer. This hybrid approach can have the benefits of both quantum computing and classical machine learning methods. In this early stage, it is of crucial importance to understand the new characteristics of quantum neural networks for different machine learning tasks. In this paper, we will study quantum neural networks for the task of classifying images, which are high-dimensional spatial data. In contrast to previous evaluations of low-dimensional or scalar data, we will investigate the impacts of practical encoding types, circuit depth, bias term, and readout on classification performance on the popular MNIST image dataset. Various interesting findings on learning behaviors of different QNNs are obtained through experimental results. To the best of our knowledge, this is the first work that considers various QNN aspects for image data.