Modeling quantum neural network gradient with reinforcement learning

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 $O(n \log n)$ 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 $O(n^2)$ to $O(\log n)$. 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.

Variational quantum algorithms are often limited by barren plateaus: gradients vanish as circuit size and depth increase, making quantum neural networks difficult to train. We propose Q-Capsule, a localized capsule-based quantum neural architecture that mitigates this problem through register partitioning, local readout, sparse inter-capsule coupling, trainable data re-uploading, and Quantum Fisher Information Matrix (QFIM)-guided adaptive depth growth. By restricting the dominant support of each observable to a small capsule and controlling inter-capsule entanglement, Q-Capsule preserves useful gradient signals while retaining communication between local quantum representations. As the register width increases, Q-Capsule consistently maintains stable gradient variance, whereas globally entangling baselines exhibit exponential suppression with a log-gradient-variance slope near -ln 2 per qubit. Q-Capsule also produces more structured optimization landscapes, higher parameter efficiency, improved robustness to depolarizing noise, and lower measurement requirements. Its adaptive policy achieves 98.1% accuracy on binary classification and 97.7% on four-class classification, while using approximately 73% fewer two-qubit gates than the fixed-deep model on the multiclass task.

Quantum Machine Learning (QML) is fundamentally limited by two challenges: barren plateaus (exponentially vanishing gradients) and the fragility of parameterized quantum circuits under noise. Despite extensive empirical studies, a unified theoretical framework remains lacking. We introduce LieTrunc-QNN, an algebraic-geometric framework that characterizes trainability via Lie-generated dynamics. Parameterized quantum circuits are modeled as Lie subalgebras of u(2^n), whose action induces a Riemannian manifold of reachable quantum states. Expressivity is reinterpreted as intrinsic manifold dimension and geometry. We establish a geometric capacity-plateau principle: increasing effective dimension leads to exponential gradient suppression due to concentration of measure. By restricting to structured Lie subalgebras (LieTrunc), the manifold is contracted, preventing concentration and preserving non-degenerate gradients. We prove two main results: (1) a trainability lower bound for LieTrunc-QNN, and (2) that the Fubini-Study metric rank is bounded by the algebraic span of generators, showing expressivity is governed by structure rather than parameter count. Compact Lie subalgebras also provide inherent robustness to perturbations. Importantly, we establish a polynomial trainability regime where gradient variance decays polynomially instead of exponentially. Experiments (n=2-6) validate the theory: LieTrunc-QNN maintains stable gradients and high effective dimension, while random truncation leads to metric rank collapse. At n=6, full metric rank is preserved (rank=16). Results support a scaling law between gradient variance and effective dimension. This work provides a unified geometric framework for QNN design, linking Lie algebra, manifold geometry, and optimization.