Organizations: IRIF, CNRS and Université Paris Cité and QC Ware, France · QC Ware, France · IonQ · IRIF, CNRS and Université Paris Cité and Quantum Signals · Quantum Signals
Abstract
Training quantum neural networks (QNNs) on quantum hardware is currently bottlenecked by the cost of gradient estimation: standard parameter-shift methods require a number of circuit evaluations that grows quadratically with the number of trainable parameters, making hardware-based optimisation impractical beyond small system sizes. In this work, we introduce a training framework that reduces this cost to logarithmic in the number of qubits, making gradient-based QNN optimisation feasible on near-term hardware at increasing scales. Our framework combines three co-designed ingredients: (i) a structured, subspace-preserving Butterfly circuit architecture with O(nlogn) parameters and logarithmic depth; (ii) a layer-wise training strategy that confines on-hardware optimisation to one small, well-structured layer at a time; and (iii) a parallelised parameter-shift rule that exploits the commuting structure within each Butterfly layer to extract all gradients in a constant number of circuit executions. Together these reduce the number of distinct circuit evaluations per optimisation step from O(n2) to O(logn). We validate the framework on clinical data imputation using the MIMIC-III electronic health record dataset, a demanding benchmark sensitive to optimisation instability and model variance. Hybrid classical-quantum models are trained directly on IonQ Forte Enterprise trapped-ion hardware at 16 qubits without performance degradation relative to ideal or noisy simulation and via tensor-network simulation at 32 qubits, with 32-qubit inference executed on hardware. The resulting models match or exceed strong classical neural baselines in downstream patient survival prediction while exhibiting reduced variance across runs, demonstrating that the proposed framework enables practical, scalable QNN training under realistic hardware constraints.
Hybrid quantum-classical neural networks (HQNNs) are emerging as a practical approach for quantum machine learning in the noisy intermediate-scale quantum (NISQ) era, as they combine classical learning components with parameterized quantum circuits in an end-to-end trainable framework. However, their performance and efficiency depend strongly on architectural choices such as data encoding, circuit structure, measurement design, and the coupling between classical and quantum modules. This makes manual design increasingly difficult, especially when hardware limitations and resource constraints must also be taken into account. In this paper, we study the foundations of HQNNs and neural architecture search (NAS), discuss how NAS extends to quantum and hybrid settings, and demonstrate FLOPs-aware search (where FLOPs serve as a proxy for computational complexity), as an important hardware-aware direction for building HQNNs that are not only accurate but also computationally efficient and practically deployable.
Alberto Marchisio, Muhammad Kashif, Nouhaila Innan +1
Quantum neural networks (QNNs) have attracted growing interest for scientific machine learning, yet in regression settings they often suffer from limited trainability under noisy gradients and ill-conditioned optimization. We propose a hybrid quantum--classical regression framework designed to mitigate these bottlenecks. Our model prepends a lightweight classical embedding that acts as a learnable geometric preconditioner, reshaping the input representation to better condition a downstream variational quantum circuit. Building on this architecture, we introduce a curriculum optimization protocol that progressively increases circuit depth and transitions from SPSA-based stochastic exploration to Adam-based gradient fine-tuning. We evaluate the approach on PDE-informed regression benchmarks and standard regression datasets under a fixed training budget in a simulator setting. Empirically, the proposed framework consistently improves over pure QNN baselines and yields more stable convergence in data-limited regimes. We further observe reduced structured errors that are visually correlated with oscillatory components on several scientific benchmarks, suggesting that geometric preconditioning combined with curriculum training is a practical approach for stabilizing quantum regression.
Hybrid Quantum Neural Network (QNN) classifiers produce logits as expectation values of quantum measurement operators. For standard Pauli measurements, these outputs are intrinsically bounded to the interval [-1,1]. When such bounded logits are used directly with the cross-entropy loss applied to softmax-normalized logits for multi-class classification, the loss function operates in a regime of weak sensitivity to logit differences. As a consequence, parameter gradients are suppressed, leading to unstable optimization in variational quantum classifiers (VQCs). In this work, we identify this effect as measurement-induced logit contraction, a previously uncharacterized source of trainability degradation in hybrid QNNs. To address this limitation, we introduce a learnable scaling parameter, termed Quantum Measurement Temperature (QMT), which rescales quantum measurement outputs prior to the loss. Unlike post-hoc calibration, QMT acts during training and compensates for the physically imposed bounds on quantum measurement outputs. This rescaling increases gradient magnitude and variance, thereby improving loss sensitivity. The proposed mechanism is architecture-agnostic and does not modify the quantum ansatz, circuit depth, or measurement operators. Experiments on fluorescence microscopy images and a six-class variant of Fashion MNIST demonstrate that QMT consistently enhances logit separation, strengthens gradients, stabilizes training across random initializations, and improves classification accuracy, relative to unscaled measurement readouts. These results demonstrate that QMT enables stable and reliable training of hybrid QNNs for practical applications.