Quantum machine learning (QML) has become a promising area for real world applications of quantum computers, but near-term methods and their scalability are still important research topics. In this context, we analyze the trainability and controllability of specific Hamming weight preserving variational quantum circuits (VQCs). These circuits use qubit gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight k . In this work, we first design and prove the feasibility of new heuristic data loaders, performing quantum amplitude encoding of (nk) -dimensional vectors by training an n -qubit quantum circuit. These data loaders are obtained using controllability arguments, by checking the Quantum Fisher Information Matrix (QFIM)'s rank. Second, we provide a theoretical justification for the fact that the rank of the QFIM of any VQC state is almost-everywhere constant, which is of separate interest. Lastly, we analyze the trainability of Hamming weight preserving circuits, and show that the variance of the l2 cost function gradient is bounded according to the dimension (nk) of the subspace. This proves conditions of existence/lack of Barren Plateaus for these circuits, and highlights a setting where a recent conjecture on the link between controllability and trainability of variational quantum circuits does not apply.
Quantum machine learning (QML) has become a promising area for real world applications of quantum computers, but near-term methods and their scalability are still important research topics. In this context, we analyze the trainability and controllability of specific Hamming weight preserving variational quantum circuits (VQCs). These circuits use qubit gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight k . In this work, we first design and prove the feasibility of new heuristic data loaders, performing quantum amplitude encoding of (nk) -dimensional vectors by training an n -qubit quantum circuit. These data loaders are obtained using controllability arguments, by checking the Quantum Fisher Information Matrix (QFIM)'s rank. Second, we provide a theoretical justification for the fact that the rank of the QFIM of any VQC state is almost-everywhere constant, which is of separate interest. Lastly, we analyze the trainability of Hamming weight preserving circuits, and show that the variance of the l2 cost function gradient is bounded according to the dimension (nk) of the subspace. This proves conditions of existence/lack of Barren Plateaus for these circuits, and highlights a setting where a recent conjecture on the link between controllability and trainability of variational quantum circuits does not apply.
Abstract Barren plateaus are the main obstacle to training quantum machine-learning models. On a barren plateau, the cost-function gradient shrinks exponentially as the circuit grows, and training stalls. Today the only reliable test is to measure it. That means sampling many gradients for every candidate circuit, which is expensive. We ask a different question. Can a classical model predict trainability from the circuit architecture alone? The architecture here means qubit count, depth, ansatz type, entanglement pattern, entangler gate, and cost-observable locality. We test this directly. We use an exact statevector simulator to build a dataset of \Nrows{} random variational circuits. For each circuit we estimate the gradient variance Var[∂C/∂θ] from \Samples{} random parameter vectors, using the parameter-shift rule. This variance is the label. A gradient-boosted regressor then predicts log10Var with $R^2 = \RsqHold{}$ on held-out architectures. A classifier separates barren from trainable circuits with an area under the ROC curve of \AucHold{}. We also test transfer to larger circuits. We train only on small circuits and test on the two largest unseen sizes (n=11,12). The model still reaches $R^2 = \RsqExtrap{}$. Permutation importance then recovers established theory. Cost-observable locality and entanglement structure matter most. We make no quantum-advantage claim. This is classical machine learning about quantum circuits. It offers a cheap way to screen ans\"atze, and it shows which design choices cause untrainability.
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.
Recent advances in Machine Learning have transformed numerous industrial sectors, yet classical paradigms face fundamental limitations: rapidly growing data volumes, rising computational costs, significant energy consumption, and the physical scaling limits of conventional hardware architectures. Quantum computing has emerged as a promising computational paradigm to address these challenges, giving rise to the field of Quantum Machine Learning (QML). In this thesis, the theoretical foundations of QML are investigated, with a focus on near-term and future practical applications. Three central challenges are addressed: the trainability of variational quantum circuits, their expressivity, and their resistance to efficient classical simulation. The trainability of Hamming-weight preserving variational quantum circuits is first studied, and theoretical guarantees are established that resolve an open conjecture on the absence of barren plateaus for this circuit family. Subspace-preserving QML algorithms are then introduced, including photonic circuits and quantum convolutional neural networks, and are designed to mimic classical ML subroutines while offering polynomial quantum advantage. Finally, variational quantum circuits are analyzed as quantum Fourier models, and a framework is derived to jointly characterize expressivity and trainability, from which conditions are obtained under which quantum models provably separate from their classical counterparts. These contributions are intended to advance the theoretical roadmap for harnessing near-term and future quantum technologies in real-world applications.
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.
Variational quantum circuits (VQCs) suffer from barren plateaus, where gradients vanish exponentially with system size, limiting effective optimization. While prior mitigation strategies focus on circuit design and initialization, the effect of circuit compression – specifically, which gates are removed – on trainability remains comparatively unexplored. In this work we study quantum gate pruning from a trainability perspective in a controlled, noiseless 6-qubit simulation testbed, and report numerical evidence that gate-selection criterion plays a substantial role in preserving the optimization landscape of compressed circuits. We propose Q-AGASP, a saliency-based pruning framework that ranks parameterized quantum gates using a combination of gradient-weight sensitivity and quantum activation, and introduce a Quantum Fisher Information (QFI)-motivated saliency variant that targets parameters associated with low gradient variance. Both approaches are applied through an iterative pruning schedule with frozen weights. Experiments on a hybrid quantum-classical model (6-qubit, 144-parameter VQC) show that saliency-guided pruning improves the gradient variance of the surviving parameters relative to uninformed methods – by 6–9 $\times $ at 90% gate sparsity in our tested regime – while random or magnitude-based pruning can produce near-zero gradient variance in some seeds. We further find that post-fine-tuning accuracy converges across methods under our 15-epoch fine-tuning budget, which suggests that accuracy and pre-fine-tuning gradient variance are complementary rather than interchangeable diagnostics, with accuracy becoming less discriminative precisely when the fine-tuning budget is generous relative to circuit size. To analyze how gate rankings evolve during compression we introduce saliency drift, a Spearman-rank-correlation-based metric, and derive a distribution-free bound relating the observed correlation to the magnitude of ranking reshuffles; drift increases with sparsity in our experiments, consistent with iterative recomputation capturing structural changes not visible to one-shot pruning. We are explicit that these findings are numerical observations from a single ansatz, qubit count, and dataset rather than architecture-independent theorems, and we discuss how ansatz topology, entanglement structure, and circuit depth would be expected to modulate them, together with the changes needed to extend the framework to noisy hardware and to chemistry-motivated ansätze.
Abstract A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called quantum machine learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called variational quantum algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work, we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.