Variational quantum circuits are increasingly studied as continuous-function approximators, but quantum regression remains difficult to train when global losses, finite-shot stochasticity, and circuit-depth growth combine to produce weak or ill-conditioned gradient signals. We study this trainability problem in a controlled hybrid quantum--classical regression design. The central ingredient is a capacity-controlled classical embedding that acts as a learnable geometric preconditioner: it reshapes the input distribution seen by a data-reuploading variational circuit while preserving a low-dimensional quantum bottleneck. We pair this representation design with a curriculum protocol that grows circuit depth progressively and switches from SPSA-based stochastic exploration to Adam-based analytic-gradient fine-tuning. We formalize the mechanism through a local quantum-tangent contraction statement: in the linearized quantum-parameter dynamics, the embedding changes the empirical Gram matrix that controls residual contraction and one-step loss decrease. Across finite-size statevector audits on PDE-informed regression benchmarks and small-data tabular tasks, the Hybrid QNN lowers error relative to Pure QNN baselines under matched quantum-model budgets. Strong classical references remain competitive, and in several cases are better in absolute error; the evidence therefore supports a trainability claim for the hybrid QNN design rather than a claim of classical or hardware quantum advantage.
Variational quantum circuits are increasingly studied as continuous-function approximators, but quantum regression remains difficult to train when global losses, finite-shot stochasticity, and circuit-depth growth combine to produce weak or ill-conditioned gradient signals. We study this trainability problem in a controlled hybrid quantum--classical regression design. The central ingredient is a capacity-controlled classical embedding that acts as a learnable geometric preconditioner: it reshapes the input distribution seen by a data-reuploading variational circuit while preserving a low-dimensional quantum bottleneck. We pair this representation design with a curriculum protocol that grows circuit depth progressively and switches from SPSA-based stochastic exploration to Adam-based analytic-gradient fine-tuning. We formalize the mechanism through a local quantum-tangent contraction statement: in the linearized quantum-parameter dynamics, the embedding changes the empirical Gram matrix that controls residual contraction and one-step loss decrease. Across finite-size statevector audits on PDE-informed regression benchmarks and small-data tabular tasks, the Hybrid QNN lowers error relative to Pure QNN baselines under matched quantum-model budgets. Strong classical references remain competitive, and in several cases are better in absolute error; the evidence therefore supports a trainability claim for the hybrid QNN design rather than a claim of classical or hardware quantum advantage.
Abstract Hybrid quantum–classical learning has emerged as a promising paradigm for leveraging near-term quantum devices in practical machine learning tasks. However, many existing hybrid architectures rely on tightly coupled end-to-end training schemes that are difficult to optimize, sensitive to quantum noise, and strongly dependent on specific encoder designs. In this work, we introduce Quantum LEGO Learning, a modular framework for hybrid artificial intelligence that decomposes learning into reusable classical and quantum blocks with explicitly separated roles. Within this framework, a pre-trained classical neural network acts as a frozen feature extractor, while a variational quantum circuit (VQC) serves as the only trainable, adaptive module that operates on structured latent representations. This design enables stable optimization, reduces parameter coupling between classical and quantum components, and supports architecture-agnostic integration across different classical backbones. We develop a block-wise theoretical analysis that decomposes the learning behavior of the hybrid model into components of approximation, estimation, optimization, and noise-induced error. The analysis shows that once expressive classical representations are fixed, the overall model's approximation capability depends primarily on the classical feature block, while the trainable quantum module governs the complexity of downstream adaptation. We further derive optimization and noise-resilience bounds demonstrating how freezing the classical block suppresses the propagation of noisy gradients during training. To validate the proposed framework, we conduct systematic experiments on quantum dot classification and genome transcription factor binding site prediction using multiple frozen feature extractors, including ResNet and tensor-train network backbones, combined with both quantum and classical adaptive heads. Experimental results from noiseless simulations, realistic quantum-noise models, and IBM Heron quantum hardware demonstrate stable optimization, robustness to noise, and improved performance of VQC-based adaptive heads under matched parameter budgets. These results suggest that modular hybrid architectures provide a practical and scalable pathway for quantum-enhanced learning in the NISQ regime.
Quantum Neural Networks (QNNs) represent a compelling convergence of quantum mechanics and machine learning, yet their practical deployment remains constrained by noise, limited qubit counts, and the cost of full quantum execution. Hybrid quantum-classical (HQC) training offers a pragmatic path forward: parameterized quantum circuits (PQCs) process information in exponentially large Hilbert spaces while classical optimizers update parameters via gradient methods accessible on today's hardware. We present a modular HQC training framework, formalising the parameter-shift rule for hardware-compatible gradient estimation, characterising the barren-plateau phenomenon through a variance-decay analysis, and introducing noise-aware training strategies for NISQ devices. Empirical results from state-vector and shot-based simulation confirm measurable quantum advantage on quantum-structured tasks, with the phase classification benchmark improving +12.1% over a classical MLP baseline. The accompanying open-source library provides a clean abstraction between quantum simulation and classical optimisation, serving as a foundation for near-term quantum machine learning research.
Hybrid quantum-classical learning models increasingly integrate neural networks with variational quantum circuits (VQCs) to exploit complementary inductive biases. However, many existing approaches rely on tightly coupled architectures or task-specific encoders, limiting conceptual clarity, generality, and transferability across learning settings. In this work, we introduce Quantum LEGO Learning, a modular and architecture-agnostic learning framework that treats classical and quantum components as reusable, composable learning blocks with well-defined roles. Within this framework, a pre-trained classical neural network serves as a frozen feature block, while a VQC acts as a trainable adaptive module that operates on structured representations rather than raw inputs. This separation enables efficient learning under constrained quantum resources and provides a principled abstraction for analyzing hybrid models. We develop a block-wise generalization theory that decomposes learning error into approximation and estimation components, explicitly characterizing how the complexity and training status of each block influence overall performance. Our analysis generalizes prior tensor-network-specific results and identifies conditions under which quantum modules provide representational advantages over comparably sized classical heads. Empirically, we validate the framework through systematic block-swap experiments across frozen feature extractors and both quantum and classical adaptive heads. Experiments on quantum dot classification demonstrate stable optimization, reduced sensitivity to qubit count, and robustness to realistic noise.
Hybrid quantum-classical computing offers a promising pathway to harness quantum advantages while mitigating the hardware limitations of noisy intermediate-scale quantum (NISQ) devices. This work proposes a modular hybrid architecture that combines classical preprocessing, quantum-enhanced feature mapping through parameterized quantum circuits (PQCs), and adaptive classical optimization. The framework is designed to improve robustness and generalization of machine learning models in noisy environments, addressing critical issues such as overfitting, noise sensitivity, and hardware-induced errors. Experimental evaluations were conducted on benchmark datasets, including Iris, reduced-dimension MNIST, synthetic noisy XOR, Wine Quality, and Breast Cancer Wisconsin. The proposed model was tested under multiple noise models, such as Gaussian, salt-and-pepper, Poisson, and quantum-specific errors (bit-flip, phase-flip, and depolarizing), using 5-fold cross-validation and repeated trials for statistical rigor. Key performance metrics included Classification Accuracy (CA), Robustness Index (RI), and Training Stability (TS), with results reported as mean ± standard deviation and supported by paired t-tests (p < 0.05). Results demonstrate that the hybrid model consistently outperforms classical support vector machines (SVM) and standalone variational quantum classifiers (VQC) in noisy conditions, achieving smaller accuracy degradation and higher stability. The combination of quantum- enhanced feature mapping, optimizer synergy, and modular design is shown to be central to its resilience.
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 Language modeling in natural language processing has predominantly relied on large-scale classical neural architectures, yet constructing expressive and resource-efficient models compatible with Noisy Intermediate-Scale Quantum (NISQ) devices remains a fundamental open challenge. In this work, two Ising-inspired variational quantum-classical architectures are proposed to enhance representational capacity by leveraging spin-coupling mechanisms and Ising-type Hamiltonian evolution within hybrid classical-quantum layers. Specifically, a quantum Ising Ansatz circuit parameterized by an Ising Hamiltonian is constructed, and a quantum approximate optimization algorithm (QAOA) is incorporated to reduce the number of trainable circuit parameters while enabling learnable quantum evolution of hidden states. To achieve stable and efficient hybrid training under NISQ constraints, a joint optimization strategy is introduced that combines the Adam optimizer with quantum simultaneous perturbation stochastic approximation (SPSA), yielding convergence acceleration of up to 64% relative to parameter-shift rule and Adam-only baselines. Empirical evaluations on the WikiText-2 benchmark demonstrate that the proposed architectures effectively capture long-range semantic dependencies, achieving a competitive perplexity of PPL=39.8 (lower values indicate stronger language modeling performance). This result surpasses existing quantum language models, including Quixer, Quantum LSTM, and Quantum GRU, and approaches the performance of the classical NanoGPT baseline (PPL=40.5). Collectively, these findings establish the viability of Ising-based hybrid quantum-classical networks as a promising direction for practical quantum natural language processing.