Scalable On-Hardware Training of Quantum Neural Networks and Application to Clinical Data Imputation

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 $O(n \log n)$ 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.

• The paper presents a novel Butterfly QNN architecture that reduces gradient estimation cost from O(n^2) to O(log n) for efficient training.
• It employs a hierarchical layer-wise training strategy and a parallel parameter-shift rule to enable robust on-hardware optimization on trapped-ion quantum processors.
• Empirical results on the MIMIC-III dataset demonstrate competitive AUC performance and reduced variance compared to conventional deep learning models.

Scalable On-Hardware Training of Quantum Neural Networks for Clinical Data Imputation

Introduction

This paper introduces a significant advancement in the scalable, hardware-efficient training of Quantum Neural Networks (QNNs) with direct application to clinical data imputation tasks. The framework fundamentally reduces the bottleneck in gradient estimation—commonly a prohibitive $O(n^2)$ requirement in parameterised quantum circuits—down to $O(\log n)$ via a synergistic combination of algorithmic and architectural innovations. The approach facilitates practical, end-to-end gradient-based learning of QNNs on contemporary trapped-ion quantum processors and demonstrates competitive results relative to classical deep learning baselines in a stringent real-world benchmark. Empirical validation is provided on the MIMIC-III clinical dataset, a recognized standard for imputation protocol assessment.

Architecture and Training Pipeline

The core methodological contributions are threefold: the introduction of a structured subspace-preserving Butterfly QNN architecture, the adoption of a hierarchical layer-wise training regime, and the exploitation of gate commutativity for fully parallelized parameter-shift gradient calculation. The quantum component consists of a modular stack: (i) Entangled, non-Gaussian block-encoded initial states, (ii) feature angle encoding on each qubit, and (iii) a log-depth Butterfly circuit using reconfigurable beam splitter gates.

Figure 1: The quantum component of the hybrid imputation pipeline, which includes (i) Non-Gaussian state initialization, (ii) angle embedding for classical features, and (iii) a logarithmic-depth, subspace-preserving Butterfly circuit.

This design critically allows control of model expressivity, favourable barren plateau scaling, and straightforward hardware implementation on platforms with all-to-all qubit connectivity (e.g., trapped ions). The Butterfly architecture uses $O(n \log n)$ parameters with only $O(\log n)$ circuit layers.

For training, a progressive, layer-wise approach is employed: small, independent Butterfly subcircuits are first trained—potentially using classical simulation—and then assembled with a coupling layer, which is optimised on quantum hardware. This reduces direct hardware optimisation to a tractable regime while leveraging classical pre-training where feasible.

Parallel Parameter-Shift Rule

The expressive capacity of quantum models is frequently limited by the cost of gradient estimation. Unlike generic QNNs that require a unique shifted-circuit evaluation per parameter, the commuting structure within each Butterfly architecture layer admits the simultaneous application of parameter shifts. All gradients in a layer can be precisely extracted from only a constant (four) number of circuit executions, and only $O(\log n)$ such layers exist per QNN.

This enables hardware-compatible gradient-based optimisation at qubit counts well beyond the regime of naive, fully-dense QNN architectures. The overall circuit evaluation budget per step is thus $O(\log n)$—a foundational improvement for practical QML scaling.

Hybrid Quantum-Classical Imputation Pipeline

The experimental pipeline involves hybrid networks where a QNN replaces the central hidden layers of a classical MLP, resulting in architectures of the form $$d \rightarrow 128 \rightarrow n \rightarrow n \rightarrow 128 \rightarrow 1$$ ($n =$ number of qubits). The QNN is fully specified by the aforementioned circuit stack and trained directly on quantum hardware for 16-qubit models; for larger (e.g., 32-qubit) models, scalable tensor network (MPS) simulation is utilised for training, with the resulting circuits executed on quantum devices for inference.

An overview of the imputation pipeline is presented as follows:

Figure 2: Schematic of the hybrid classical-quantum imputation protocol, where observed clinical features seed the quantum module for prediction of missing entries.

Imputation quality is measured by downstream survival prediction AUC, leveraging the principle that improved imputation yields tangible clinical decision-support gains.

Empirical Results

Hardware-Based Training at Scale

A central result is that direct, gradient-based training of 16-qubit QNNs on IonQ's Forte Enterprise is feasible, robust, and does not incur performance degradation relative to classical or simulated comparison models of the same effective width.

Figure 3: Performance comparison: the hardware-trained 16-qubit hybrid DeepImputer achieves higher median AUC and lower variance than the fully classical 16-unit model, with all other architecture details matched.

Variance reduction in the quantum model is noteworthy, suggesting an inductive bias conferred by the Butterfly structure and layer-wise protocol.

Hardware/Simulation Parity

Extensive benchmarks reveal negligible loss between ideal, noisy, and hardware-based training, demonstrating that both noise resilience and gradient estimation efficiency are effectively addressed.

Figure 4: Parity in AUC performance across classical simulation, noisy quantum simulation, and real hardware execution at 16 qubits.

Inference at 32 Qubits

For 32-qubit circuits, the hybrid DeepImputer, with quantum layers trained via MPS methods and evaluated on hardware, matches the downstream AUC of a fully classical network of equivalent width. This substantiates both the feasibility of near-term quantum inference at moderate system sizes and the quality of the quantum-classical architecture's function class.

Figure 5: At 32 qubits, the hybrid model’s AUC matches its classical counterpart, supporting scalability in both training and hardware execution.

Network Capacity and Scaling Trends

Systematic analysis of hidden-layer width in classical models identifies $d \rightarrow 128$ as an optimal regime for this task, with models suffering from overfitting or poor convergence for wider networks. The scaling correspondence supports 128-qubit circuits as the relevant near-term target for quantum enhancement.

Figure 6: Classical MSE convergence saturates at 128 units, motivating the focus on quantum models of this scale.

Theoretical and Practical Implications

This work demonstrates that architectural and algorithmic co-design (structured ansätze, parallel parameter-shift, layer-wise training) enables practical, scalable, hardware-based QNN learning, eliminating the dominant gradient estimation bottleneck. The Butterfly circuit's expressivity, barren plateau resilience, and hardware-alignment mark it as a preferred ansatz for NISQ device deployments in real-world settings.

Pragmatically, quantum-enhanced imputers achieve parity or even slight superiority over strong classical deep learning benchmarks on the MIMIC-III survival prediction task. The observed variance reduction suggests that the QNN may offer improved robustness against stochastic training artifacts, a potentially valuable property for critical clinical applications.

Importantly, no statistically significant AUC drop is observed when moving from ideal simulation to real quantum hardware, demonstrating that hardware noise and realistic sampling rates are compatible with practical machine learning outcomes—given thoughtful architectural design.

Looking forward, as quantum hardware scales past 32 and toward 128+ qubits, the methodologies put forth provide a compelling route to full quantum enhancement in high-stakes, high-dimensional data imputation tasks. The modular training strategy further suggests compatibility with other QML problems, including generative modeling and quantum circuits for scientific applications.

Conclusion

The presented approach achieves a major milestone in practical quantum machine learning by reducing gradient evaluation cost from $O(n^2)$ to $O(\log n)$0 and validating, on hardware, QNNs that are competitive with and potentially more stable than their classical analogs. Empirical evidence on a clinically demanding imputation benchmark confirms that the method can serve as a robust, hardware-applicable component of hybrid machine learning workflows. These results lay groundwork for near-term, scalable, quantum-enhanced data analysis in medicine and beyond, with a well-defined path to leverage future hardware advances for high-dimensional, fully quantum models.

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