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.
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.
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 practical value of quantum machine learning must be derived from efficiency rather than scale, as current NISQ processors support only small, shallow circuits. This paper evaluated whether variational quantum circuits (VQCs) can reduce the trainable parameter count in an image classification scenario while preserving accuracy. Experimentation compared quantum classifier heads to parameter-matched classical heads on an identical set of frozen features drawn from the Fashion-MNIST dataset by a CNN. 9 configurations of a quantum circuit crossed 4, 6, and 8 qubits with 1, 2, and 4 entangling layers, each matched to a multilayer perceptron with an approximately equal parameter count. The linear probes served as a floor baseline for all 105 runs across 5 random seed values. 7 out of 9 quantum-based configurations met the mean test accuracy of no more than 1 percentage point below its matched classical head, enforced by the pre-specified criterion. The 2 failures were single-layer circuits, with each falling roughly 8 points below its matched classical head and 12 points below the circuit-free linear probe; in total, 6 configs with 2+ layers passed, with the smallest at 74 trainable parameters. The general trend noticed was that accuracy increased progressively with depth. The 4-layer circuits exceeded the linear probe at every qubit count and produced the highest accuracy at 6 and 8 qubits. The simulation ran noiselessly; VQCs preserved accuracy at the matched parameter budget when depth was given, showcasing depth rather than qubit count gating performance.
Introduction: Variational quantum algorithms can show unstable trainability and uneven generalisation that are not fully explained by circuit depth or raw parameter count alone. We study the data quantum Fisher information matrix (DQFIM) as a data-dependent effective-capacity diagnostic for controlled supervised unitary-learning tasks. Materials and methods: We used exact-state, classically simulated, 4-qubit, supervised unitary-learning benchmarks with two ansatz families: a hardware-efficient ansatz (HEA) for the baseline trainability study and a symmetry-preserving ansatz for symmetry-controlled comparisons. The analysis covered four linked experiments: E0 trainability phase boundaries, E1 symmetry-controlled data-regime comparisons, E2 trainability versus generalisation, and E3 pre-training prediction of optimisation success. For the main analyses we used the support-basis DQFIM, 30 random seeds per main configuration, grouped cross-validation in the predictive benchmark, and additional diagnostics for threshold sensitivity, phase indeterminacy, Hamming-sector generalisation, and test set size sensitivity. Results: In E0, the empirical trainability boundary increased from M c = 32 at L = 1 to M c = 192 at L = 8 , and the support-basis DQFIM-predicted boundary matched the empirical boundary on the resolved scanned grid under the main analysis setting. In E1, the sector-preserving and sector-broken conditions showed no resolved empirical boundary shift on the scanned grid, with only mild low-L asymmetry in the DQFIM-predicted boundary. In E2, trainability and generalisation separated clearly: some regimes generalised well, while others reached near-zero training loss but retained high test loss. The phase diagnostic supported relative phase indeterminacy as a mechanism for failure on in-sector superpositions after basis-state training. In E3, parameter count alone was a weak predictor of optimisation success, with receiver operating characteristic area under the curve (ROC AUC) = 0.648 ; the structural baseline was stronger, with ROC AUC = 0.958 ; and the DQFIM-enhanced model performed best, with ROC AUC = 0.989 . Conclusions: In these small, idealised, classically simulated matched-family tasks, the support-basis DQFIM provides a useful data-dependent pre-training diagnostic of effective capacity on the retained data support. It tracks resolved trainability boundaries and contributes predictive information beyond raw parameter count and structural metadata. The conclusions remain scoped to controlled small-system simulations; larger-qubit, noisy, hardware-executed, and less-matched settings require further validation before claiming practical scalability.
Hybrid quantum neural networks (HQNNs) integrate parameterized quantum circuits (PQCs) within classical networks, where the behavior of the underlying PQCs is often the primary focus of analysis. In this context, expressibility and trainability are widely used to characterize PQC's performance and are commonly assumed to exhibit a trade-off, where highly expressive circuits are more susceptible to barren plateaus. However, the validity of this relationship in HQNNs remains unclear. In this paper, we systematically analyze the expressibility--trainability relationship in HQNNs across varying circuit depths, qubit counts, entanglement topologies. We consider different training configurations, including pure PQCs, quantum-only training in hybrid setting, and full end-to-end training of hybrid models. Our results show that pure PQCs exhibit only a weak and regime-dependent trade-off, while hybrid architectures increasingly disrupt and can eliminate this relationship under full hybrid training. This indicates that classical components reshape the optimization landscape, decoupling trainability from PQC expressibility. We further propose a multi-objective neural architecture search (NAS) framework that jointly optimizes expressibility, trainability, and task performance over a combined classical--quantum design space, revealing different Pareto-optimal solutions under full end-to-end and quantum only training in hybrid setting. different trainability definitions. Our results suggest that hybridization is not just an implementation detail, but a defining factor in the performance of quantum machine learning models.
This study investigates classification performance and trainability of variational quantum classifiers on near-term noisy quantum hardware across circuit depth and qubit count regimes within the context of quantum computing and machine learning, an area of growing scientific importance given its implications for hybrid quantum-classical algorithm design and practical NISQ hardware benchmarking for near-term quantum advantage assessment. Using gradient-based parameter optimization of parameterized quantum circuits with hardware noise modeling via density matrix simulation and physical execution on IBM Eagle processors, we examine expressibility of parameterized circuits enabling nonlinear decision boundaries in feature Hilbert space, with trainability constrained by barren plateau gradients at large qubit counts in 96 circuit architectures (4 qubit counts x 4 depths x 6 ansatz families) evaluated across 5 classification datasets with 20 random initialization seeds each drawn from IBM Quantum Eagle (127-qubit) and Falcon (27-qubit) processors with noise mitigation via zero-noise extrapolation. Results indicate that 10-qubit, 4-layer VQC achieves 84.2% classification accuracy on the best benchmark, within 3.8 percentage points of classical SVM, but gradient variance decays exponentially with qubit count (variance halving per 2 added qubits), confirming barren plateau scaling (p < 0.001), with 84.2% accuracy, 3.8 pp gap to classical SVM baseline as the primary quantitative benchmark. Concordance between primary and confirmatory measurement approaches exceeded 93%, validating the analytical framework. These findings contribute empirically to quantum computing and machine learning and carry actionable implications for the design of programs and policies targeting hybrid quantum-classical algorithm design and practical NISQ hardware benchmarking for near-term quantum advantage assessment.
Quantum neural networks (QNNs) have recently attracted remarkable attention as a potential alternative to classical and deep neural network (NN) models particularly under constraints imposed by noisy intermediate-scale quantum (NISQ) devices, which support only shallow circuits with few quantum bits (qubits) and limited variational depth. However, applying QNNs to natural language processing (NLP) remains challenging because text representations are typically high-dimensional, while NISQ classifiers can only encode a small number of features at a time, requiring classical compression to enable stable training and flexible circuit configuration. In addition, many existing studies integrate QNNs into rich-text representation pipelines, where the quantum circuit is often implemented as a variational quantum circuit (VQC). Here, the VQC acts as an additional component within a larger model, and it remains unclear how the quantum component improves the classification task. Therefore, in this work, we propose a fully reproducible and compact QNN pipeline that provides a controlled setting for studying the contribution of the VQC to text classification. The model is trained end-to-end with a lightweight classical compression layer, and the overall study is supported by systematic ablation analyses of the compression layer, the classifier heads under a shared frozen low-dimensional representation, and quantum circuit structure. Our model uses the term frequency-inverse document frequency (TF-IDF) features with linear compression followed by a shallow four-qubit, two-layer VQC, implemented in PyTorch and PennyLane for end-to-end training. It is evaluated on five benchmark text classification datasets against classical baselines, including logistic regression and multilayer perceptron (MLP) models. Experimental results show that the proposed QNN achieves a competitive performance across all evaluated datasets, yielding the best results on three of the five benchmark datasets.