Hybrid Quantum Neural Networks (HQNN)

• Hybrid Quantum Neural Networks (HQNN) are hybrid quantum-classical models that embed parameterized quantum circuits within classical neural pipelines.
• They employ quantum feature extractors, convolutional modules, and parallel architectures to tackle tasks from image recognition to molecular property prediction under NISQ constraints.
• HQNNs are trained through classical optimization with quantum gradients computed via parameter-shift rules, achieving parameter efficiency and performance gains across diverse empirical domains.

Hybrid Quantum Neural Networks (HQNNs) are hybrid quantum-classical models in which a parameterized quantum circuit (PQC) or variational quantum circuit (VQC) is embedded within a classical neural architecture and trained end-to-end in the Noisy Intermediate-Scale Quantum (NISQ) regime. Across the literature represented here, HQNNs are used as quantum feature extractors, quantum classifier or regressor heads, quantum convolutional or quanvolutional modules, physics-informed encoders, and search objects in neural architecture optimization. Their common premise is that classical modules remain responsible for preprocessing, optimization, and postprocessing, while the quantum layer supplies a trainable nonlinear transformation through state preparation, entangling evolution, and measurement (Mittal et al., 2024, Ahmed et al., 24 Jan 2025, Kashif et al., 25 May 2026).

1. Definition and operating principle

A concise formalization of the HQNN paradigm is the hybrid composition

$$y = g_{\boldsymbol{\phi}}\big(f_{\boldsymbol{\theta}}(h_{\boldsymbol{\phi}}(x))\big),$$

where $h_{\boldsymbol{\phi}}$ denotes classical preprocessing, $f_{\boldsymbol{\theta}}$ the PQC, and $g_{\boldsymbol{\phi}}$ classical postprocessing (Kashif et al., 25 May 2026). This formulation captures the recurrent design pattern in which classical data are first reshaped into a qubit-compatible representation, then encoded into a quantum state, transformed by trainable gates, measured, and finally mapped back into task outputs.

The quantum subroutine is typically shallow and explicitly NISQ-compatible. The literature surveyed here uses angle embedding, amplitude embedding, data re-uploading, and fixed feature maps such as Qiskit’s ZZFeatureMap; trainable layers include RealAmplitudes, Basic Entangling Layers, Strongly Entangling Layers, and task-specific controlled-rotation ansatzes (Mittal et al., 2024, Kashif et al., 13 Nov 2025, Jeong et al., 14 Sep 2025, Rai et al., 19 May 2026). Measurement usually returns expectation values of Pauli observables, often one value per qubit, which become the interface to the classical head (Ahmed et al., 16 Apr 2026, Ratto et al., 17 Jan 2026).

The practical appeal of HQNNs in these studies is not fault-tolerant quantum advantage, but near-term deployability under qubit-count, depth, decoherence, and simulation-cost constraints. Several works explicitly frame HQNNs as a pragmatic NISQ-era compromise: the quantum component is kept small enough to be executable or simulatable, while the classical component absorbs the remainder of the modeling burden (Ahmed et al., 24 Jan 2025, Marchisio et al., 18 May 2026). A recurrent implication is that “hybrid” is not merely an implementation convenience; it changes both the representation and the optimization landscape (Kashif et al., 25 May 2026).

2. Architectural families

The most common HQNN structure is a sequential classical–quantum–classical pipeline. In indoor localization from received signal strength indicator fingerprints, the input is a 3D RSSI vector $\mathbf{x}=[x_1,x_2,x_3]$, encoded by ZZFeatureMap, processed by a 3-qubit RealAmplitudes circuit, and passed to two classical dense layers, with a hidden layer of 32 neurons and ReLU activation, to predict $\hat{\mathbf{z}}=[\hat{x},\hat{y}]$ (Mittal et al., 2024). In protein–ligand binding affinity prediction, HQDeepDTAF preserves the classical sequence-embedding and convolutional front end of a DeepDTAF-style model and replaces the fully connected tail with an HQNN comprising a classical embedding network, a quantum data re-uploading circuit, and a classical regression network (Jeong et al., 14 Sep 2025). In quantitative structure–property relationship modeling for CO$_2$-capturing amines, a variational quantum regressor is attached to either MLP or GNN backbones, with training variants HQSc, HQFi, and HQFr reflecting scratch training, fine-tuning, and frozen-backbone transfer (Cho et al., 1 Mar 2025).

A second family inserts quantum processing into image models. The comparative image-classification literature distinguishes Quantum Convolutional Neural Networks (QCNNs), Quanvolutional Neural Networks (QuanNNs), and Quantum Transfer Learning (QTL) (Ahmed et al., 24 Jan 2025). QCNNs use quantum convolution and pooling, QuanNNs apply quantum filters to local patches, and QTL appends a VQC to a pretrained classical feature extractor. Related CNN-QNN hybrids use a classical CNN for spatial feature extraction followed by a quantum layer, sometimes with amplitude encoding and an additional classical output layer (Shi et al., 2023). Medical-image HQNNs also follow this pattern: a pre-trained ResNet-50 backbone can be followed by a 10-dimensional bottleneck and then either a 10-qubit VQC, a capacity-matched classical nonlinear block, or no intermediate transform, allowing isolation of the quantum contribution (Cruz et al., 22 May 2026).

A third family is explicitly parallel rather than sequential. The Parallel Hybrid Network (PHN) sends the same input simultaneously to a classical multilayer perceptron and a VQC, then linearly combines the outputs with trainable weights. The stated rationale is that the angle-embedded VQC supplies a smooth harmonic or Fourier-like base, while the MLP fills non-harmonic residual structure (Kordzanganeh et al., 2023). This is a distinct interpretation of HQNNs: not a replacement of a classical layer by a quantum layer, but a decomposition of function space into complementary inductive biases.

A fourth family treats architecture itself as a discrete search object. GAT-QNN defines a macroCircuit search space and samples microCircuits, training them with reintegration into shared weights and later performing a backend-aware inference-stage genetic search without retraining (Ahmed et al., 16 Apr 2026). FAQNAS and related neural architecture search work define HQNN design over qubit count, encoding, gate set, topology, and depth, often with FLOPs-aware multi-objective optimization via NSGA-II (Kashif et al., 13 Nov 2025, Marchisio et al., 18 May 2026).

3. Training objectives, gradients, and evaluation

HQNN training in the surveyed literature is almost always classical optimization over hybrid computational graphs. Regression-oriented HQNNs typically use mean squared error. The indoor localization model minimizes

$$L(\hat{\mathbf{z}}, \mathbf{z}) = \frac{1}{n}\sum_{i=1}^{n}\left[(\hat{x}_i-x_i)^2+(\hat{y}_i-y_i)^2\right],$$

with test performance reported as RMSE (Mittal et al., 2024). Binding-affinity HQNNs and variational quantum regressors for QSPR also use MSE, with AdamW or Adam as the classical optimizer (Jeong et al., 14 Sep 2025, Cho et al., 1 Mar 2025). Classification-oriented HQNNs use cross-entropy, negative log-likelihood, or focal loss, depending on the task (Kashif et al., 13 Nov 2025, Cruz et al., 22 May 2026, Syah et al., 18 Apr 2026).

Quantum gradients are commonly computed by parameter-shift rules. A standard form used in this literature is

$$\frac{\partial C}{\partial \theta_k} = \frac{1}{2}\left[C\!\left(\boldsymbol{\theta}+\frac{\pi}{2}\mathbf{e}_k\right)-C\!\left(\boldsymbol{\theta}-\frac{\pi}{2}\mathbf{e}_k\right)\right],$$

which is then combined with gradient-based updates on the classical parameters (Kashif et al., 25 May 2026). Indoor localization uses parameter-shift on the quantum side and Adam on the classical side; seismic inversion additionally compares adjoint differentiation, finite difference, and SPSA, reporting adjoint as exact and efficient in simulation while SPSA is faster but stochastic (Mittal et al., 2024, Vashisth et al., 6 Mar 2025).

Evaluation protocols are highly task-dependent. Reported metrics include RMSE, MAE, concordance index, Pearson correlation $R$, balanced accuracy, macro precision, macro recall, macro F1-score, ROC/AUC, and normalized RMSE (Jeong et al., 14 Sep 2025, Ratto et al., 17 Jan 2026, Cruz et al., 22 May 2026, Rai et al., 19 May 2026). Several studies explicitly separate architectural efficiency from predictive quality. One benchmark selects, for each task complexity, the first model sorted by FLOPs that satisfies training accuracy $h_{\boldsymbol{\phi}}$0 and validation accuracy $h_{\boldsymbol{\phi}}$1, thereby emphasizing minimal resource use at fixed performance rather than unconstrained best accuracy (Kashif et al., 2024). Another line of work systematically varies quantum-specific hyperparameters—depth, qubits, entanglement type, shots, and observables—and shows that their effects on both accuracy and training time are nontrivial and architecture-dependent (Zaman et al., 2024).

4. Empirical domains and reported performance

HQNNs have been applied to localization, radar sensing, molecular property prediction, semiconductor co-optimization, medical imaging, and physics-informed geoscience inversion. The reported empirical profile is mixed: some tasks show clear gains over selected classical or quantum baselines, while others show parity, domain dependence, or persistent deficits relative to strong classical models.

Domain	Representative HQNN configuration	Representative reported result
Indoor RSSI localization	3 qubits, ZZFeatureMap, RealAmplitudes, 32-neuron head	Zigbee Sc-3: Quantum Fingerprint 2.616, HQNN simulator 1.298, HQNN hardware 1.471 (Mittal et al., 2024)
Complexity-scaling benchmark	Hybrid model with trainable quantum layer	FLOPs increase: 53.1% for HQNNs vs 88.1% for classical models; parameter growth: 81.4% vs 88.5% (Kashif et al., 2024)
Protein–ligand affinity	HQDeepDTAF-NN-Angle with 9 qubits	MAE $h_{\boldsymbol{\phi}}$2, RMSE $h_{\boldsymbol{\phi}}$3, vs DeepDTAF MAE $h_{\boldsymbol{\phi}}$4, RMSE $h_{\boldsymbol{\phi}}$5 (Jeong et al., 14 Sep 2025)
Blood-cell classification	ResNet-50 + 10-D bottleneck + 10-qubit VQC	Blood Cell Images macro F1 0.9153 vs 0.8862 and 0.8784; PBC macro F1 0.9869 vs 0.9854 and 0.9851 (Cruz et al., 22 May 2026)
Radar occupancy classification	2-qubit HQNN with compact CNN frontend	99.7% synthetic, 97.0% real, with 0.066M parameters and up to 170x fewer parameters (Ratto et al., 17 Jan 2026)
MIS-HEMT co-optimization	Circuit (13, 5) at $h_{\boldsymbol{\phi}}$6, 4 qubits	Overall normalized RMSE reduced by 24.4% relative to ANN (Rai et al., 19 May 2026)

In indoor localization, the contrast between trainable HQNNs and fixed quantum fingerprinting is especially sharp. The reported interpretation is that quantum fingerprinting uses a fixed similarity circuit with no trainable parameters and therefore adapts poorly in sparse, high-interference environments, whereas HQNNs learn both quantum and classical parameters. The hardware results on real IBM devices are usually slightly worse than simulation, but the gap is not large (Mittal et al., 2024).

In radar sensing, the picture is domain-specific. For drone detection and classification on micro-Doppler spectrograms, CNNs are competitive or slightly better at high SNR in detection, but the HQNN becomes superior at lower SNR. For detection, F1 at $h_{\boldsymbol{\phi}}$7 dB is $h_{\boldsymbol{\phi}}$8 for HQNN and $h_{\boldsymbol{\phi}}$9 for CNN; for five-class classification at $f_{\boldsymbol{\theta}}$0 dB, F1 is $f_{\boldsymbol{\theta}}$1 for HQNN and $f_{\boldsymbol{\theta}}$2 for CNN (Malarvanan, 2024). By contrast, in indoor occupancy classification from range–Doppler maps, the compact HQNN is highly parameter-efficient and competitive in clean settings, but CNNs are more sample-efficient on real data and recover more sharply at positive SNR in the real domain (Ratto et al., 17 Jan 2026).

In molecular and materials applications, HQNNs are often presented as parameter-efficient replacements for classical fully connected tails rather than full-stack replacements. The protein–ligand study reports that an angle-based 9-qubit hybrid variant slightly outperforms DeepDTAF while using 100,016 classical parameters plus 540 quantum parameters, compared with 154,114 classical parameters for DeepDTAF (Jeong et al., 14 Sep 2025). The CO$f_{\boldsymbol{\theta}}$3-capturing amine study finds the strongest results for pre-trained 9-qubit hybrids, especially HQFi/9Q and HQFr/9Q, while also noting that improvements are property-dependent and not universal (Cho et al., 1 Mar 2025). Semiconductor-device co-optimization reports that the selected HQNN improves five of six targets relative to ANN and remains competitive on the sixth (Rai et al., 19 May 2026).

Medical-image results are likewise heterogeneous. Blood-cell classification reports a macro F1 gain of about 3.0%–3.7% over classical comparators on the Blood Cell Images dataset and a smaller but consistent advantage on the near-saturated PBC dataset (Cruz et al., 22 May 2026). Breast-cancer thermographic classification reports 98.11% accuracy, 98.13% F1, and AUC 0.991 for a 4-qubit hybrid model on a selected dataset subset, outperforming EfficientNet-B0 and ResNet-50 in that study; the same work also emphasizes that all quantum results are obtained by classical simulation and the sample size is small (Syah et al., 18 Apr 2026).

Several studies directly caution against reading these results as a universal superiority claim. A controlled image multi-classification comparison concludes that PCA-based angle-encoded QNNs break down in multi-class settings, CNN-QNN hybrids are more practical, but even an optimized HQNN remains below strong classical CNNs in accuracy (Shi et al., 2023). The computational-scaling benchmark also states that the evidence for advantage depends on the chosen task, complexity measure, and simulation context (Kashif et al., 2024).

5. Noise, hardware realization, and robustness

Noise analysis is central to HQNN research because most reported systems are designed for NISQ conditions. Comparative studies of QCNN, QuanNN, and QTL on reduced MNIST subsets report that QuanNN is the most robust architecture overall across Phase Flip, Bit Flip, Phase Damping, Amplitude Damping, and Depolarizing Channel, while QTL performs poorly and is excluded from the noise study due to underperformance (Ahmed et al., 24 Jan 2025). A complementary noise analysis sweeping probabilities from 0.1 to 1.0 on MNIST and Fashion-MNIST similarly concludes that QuanNN is generally more robust than QCNN, especially for phase-related noise, whereas amplitude damping and depolarizing noise are the most destructive channels (Ahmed et al., 6 May 2025).

Real or hardware-informed deployment is less common but no longer absent. The indoor localization HQNN is trained on an IBM cloud simulator and evaluated on real IBM hardware, specifically ibm_kyiv and ibm_nazca; the reported conclusion is that hardware RMSE is usually slightly worse than simulator RMSE, but not dramatically worse, suggesting robustness for a 3-qubit circuit under present constraints (Mittal et al., 2024). The blood-cell study executes the HQNN on IBM hardware through the ibm_fez backend on a reduced balanced subset, with accuracy decreasing from 0.9250 on simulator to 0.9000 on hardware and macro F1 from 0.9246 to 0.8963, described as a modest degradation (Cruz et al., 22 May 2026). The QSPR study does not run on physical hardware, but uses IBM-device calibration data to simulate depolarizing noise, amplitude damping, phase damping, and readout error, concluding that performance changes are often small and statistically negligible for several tasks (Cho et al., 1 Mar 2025).

Noise robustness does not imply generalized security. The first systematic backdoor study on HQNNs shows that standard patch and blend backdoors often require larger modifications to succeed against HQNNs than against comparable CNNs, and attributes part of this effect to the concentration of measure phenomenon in Hilbert space. However, the same study introduces the Qcolor backdoor, optimized with NSGA-II over color-channel scaling factors, and reports very high attack success rates together with high SSIM, thereby showing that HQNNs are not secure by default (Guo et al., 2024). A plausible implication is that robustness to hardware noise and robustness to adversarial manipulation are distinct properties.

6. Design automation, theory, and outstanding issues

A substantial recent shift in HQNN research is the movement from manually designed circuits to automated search and explicit efficiency criteria. FAQNAS formulates HQNN design as a Pareto problem that minimizes FLOPs and maximizes validation accuracy, measures quantum FLOPs as $f_{\boldsymbol{\theta}}$4, and explores 23,328 candidate quantum configurations across qubits, encoding, rotations, entanglers, topology, and depth (Kashif et al., 13 Nov 2025). Related work on hybrid quantum-classical neural architecture search argues that encoding, variational block design, measurement, and classical postprocessing should all be treated as search dimensions, with FLOPs serving as a practical proxy for simulation complexity in the NISQ era (Marchisio et al., 18 May 2026). GAT-QNN extends this logic further by separating macroCircuit training from backend-aware microCircuit selection, and reports consistent 22–23% test accuracy gains for GA-driven inference across PennyLane, AWS Braket simulator, and QASM simulator (Ahmed et al., 16 Apr 2026).

Theoretical interpretation remains unsettled. One benchmark argues that HQNNs show evidence of computational advantage because, from 10 to 110 features, FLOPs increase by 53.1% for HQNNs versus 88.1% for classical models, and parameter growth is slower as well; yet the same work explicitly states that the conclusion is benchmark-specific and does not constitute a universal proof or a realized hardware speedup (Kashif et al., 2024). Another study reconsiders the standard expressibility–trainability trade-off inherited from standalone PQCs and finds that pure PQCs show only a weak, regime-dependent trade-off, while hybrid architectures weaken or eliminate it under full end-to-end training. Its central claim is that classical components reshape the optimization landscape enough that trainability can become decoupled from PQC expressibility (Kashif et al., 25 May 2026).

Several recurring limitations cut across domains. Many reported gains rely on classical simulation rather than execution on physical quantum processors (Kashif et al., 2024, Syah et al., 18 Apr 2026, Vashisth et al., 6 Mar 2025). Scaling to larger datasets remains difficult, and some applications explicitly state that training directly on current hardware is too slow or costly (Mittal et al., 2024). Strong classical baselines can remain superior in high-SNR radar, in low-label real-data regimes, or in image classification benchmarks with mature CNN architectures (Malarvanan, 2024, Ratto et al., 17 Jan 2026, Shi et al., 2023). Security, noise mitigation, connectivity constraints, and hardware-aware compilation remain open concerns (Guo et al., 2024, Jeong et al., 14 Sep 2025).

The cumulative literature therefore supports a restrained but technically substantive characterization of HQNNs. They are neither interchangeable with standalone PQCs nor reducible to ordinary neural networks with an exotic layer. They are end-to-end hybrid systems whose behavior depends jointly on encoding, circuit family, measurement, classical coupling, optimization protocol, task structure, and deployment backend. This suggests that the most stable conclusions about HQNNs are local rather than universal: trainability, robustness, efficiency, and accuracy are architecture- and domain-dependent, and the decisive design question is usually not whether to use a quantum layer at all, but which hybrid decomposition is best matched to the task and the NISQ constraints under which it will be trained and deployed (Kashif et al., 25 May 2026, Marchisio et al., 18 May 2026).