Mitigating Measurement-Induced Training Instability in Hybrid Quantum Neural Networks for Protein Classification

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.

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.

Artificial intelligence has been transformed by deep neural networks, yet the search for new learning architectures continues. Quantum machine learning offers one such direction, and hybrid quantum neural networks, which combine classical neural-network components with quantum information processing units, have emerged as a practical framework for near-term quantum technologies. However, the rapid development of the field across diverse architectures, benchmarks and hardware assumptions makes it difficult to assess the utility of various proposals, identify where genuine advantages may arise, and determine how practitioners can use these models. While recent benchmarks caution that such gains have not yet been demonstrated at scale, theoretical work has identified tasks on which quantum models hold provable advantages, and hybrid approaches have delivered promising results on practical problems using deliberately compact quantum components and substantially fewer trainable parameters. Here, we review hybrid quantum neural networks for the machine-learning and quantum-machine-learning communities. We summarize their main theoretical and methodological foundations, survey some of the most promising architectures developed so far, and examine their implementation challenges and reported performance. By consolidating these perspectives, this review provides a structured view of the state of the field and helps identify promising paths for future research and application-driven development.