Hybrid Quantum-Classical Neural Networks

• Hybrid Quantum-Classical Neural Networks (HQCNNs) are models that embed trainable quantum circuits within classical pipelines to enhance feature extraction and learning.
• They employ recurring architectures—such as serial, quantum front-end, and parallel designs—that seamlessly fuse classical and quantum processing.
• These models have been applied across disciplines including image analysis, radar detection, and graph data, often improving performance under low-data or noisy conditions.

Hybrid Quantum-Classical Neural Network (HQCNN) denotes a family of models that embed a trainable quantum component inside a larger classical learning pipeline. In the literature, the acronym is used both broadly for hybrid quantum-classical neural networks and more narrowly for hybrid quantum-classical convolutional neural networks; in both usages, the defining pattern is a classical–quantum–classical workflow in which classical preprocessing or feature extraction is followed by quantum encoding, a parameterized quantum circuit or quantum convolutional block, quantum measurement, and classical postprocessing for prediction (Zaman et al., 2024, Marchisio et al., 18 May 2026).

1. Terminology and model scope

The term is not fully standardized. In the MNIST hyperparameter study, HQNNs are defined as QNNs embedded inside a larger classical neural-network workflow, with four standard steps: classical input preprocessing, classical-to-quantum feature mapping, training the PQC and measuring qubits, and classically postprocessing the measurement results (Zaman et al., 2024). In medical imaging and computer-vision papers, HQCNN often specifically means Hybrid Quantum-Classical Convolutional Neural Network, where a quantum convolutional or quanvolutional stage is coupled to a classical CNN tail (Houssein et al., 2021, Ng et al., 2024). Other works use closely related labels such as hybrid quantum neural network, hybrid classical–quantum classifier, or hybrid quantum-classical photonic neural network for architectures that are not convolutional in the classical sense but still follow the same hybrid design principle (Austin et al., 2024, Chen et al., 2021).

This scope includes several distinct but compatible constructions. Some HQCNNs use a single variational quantum circuit as a middle layer between classical dense stacks, as in connected-vehicle incident detection with a 4-qubit quantum layer inserted between classical layers (Khan et al., 2021). Others use quantum preprocessing on local image patches, producing feature maps that are then consumed by a classical CNN, as in COVID-19 chest X-ray classification and multiple-color-space image classification (Houssein et al., 2021, Ng et al., 2024). Still others are parallel hybrids rather than serial pipelines, feeding the same input to a classical branch and a quantum branch and combining their outputs only at the end (Kordzanganeh et al., 2023, Xu et al., 2024). Continuous-variable photonic variants replace a classical hidden layer with a trainable continuous-variable quantum circuit, retaining classical input and output layers (Austin et al., 2024).

A unifying description is therefore architectural rather than domain-specific: an HQCNN is a hybrid model in which the quantum block is not standalone, but is optimized jointly or functionally coupled with classical components. This suggests that the acronym names a design pattern rather than a single canonical network topology.

2. Recurrent architectural patterns

A common serial pattern is a classical-to-quantum-to-classical pipeline. In the MNIST study on quantum-specific hyperparameters, the model consists of classical preprocessing on MNIST digit images, angle-based quantum feature encoding, a parameterized quantum layer whose architecture is varied, and a single classical output layer producing 4-class predictions (Zaman et al., 2024). The connected-vehicle incident detector uses a feed-forward stack with six input features, classical dense layers $6 \rightarrow 48 \rightarrow 32 \rightarrow 4$, a 4-qubit quantum layer, a post-quantum dense layer $4 \rightarrow 4$, and a sigmoid output for binary classification (Khan et al., 2021). A related end-to-end classifier couples a tensor-network feature extractor to a variational quantum circuit, with both parts optimized simultaneously and the classical–quantum boundary treated as adjustable according to resource availability (Chen et al., 2021).

A second recurring pattern is quantum front-end feature extraction followed by a classical convolutional or dense classifier. The COVID-19 HQCNN contains 1 quantum convolution layer, 3 classical Conv2D layers, 2 max-pooling layers, 2 fully connected layers, and 1 softmax output layer, for 9 layers total and 120,394 parameters; the quantum layer is used mainly as a preprocessing or feature-transformation layer before the classical CNN (Houssein et al., 2021). The multiple-color-space HQCNN similarly separates preprocessing and color-space conversion, quantum feature extraction with parameterized quantum circuits, and classical classification with BatchNorm1D, FC1, LeakyReLU, FC2, Softmax, and cross-entropy loss (Ng et al., 2024). In radar-based drone detection, the hybrid model keeps classical convolutional feature extraction on STFT spectrograms, inserts parallel variational quantum circuits after classical fully connected layers, concatenates their outputs, and applies a final classical classifier (Malarvanan, 2024).

A third pattern is non-serial coupling. The parallel hybrid network for periodic regression sends the same input simultaneously to a classical MLP branch and a variational quantum circuit branch, then combines the outputs linearly so that the quantum branch supplies a smooth sinusoidal foundation and the classical branch supplies non-harmonic corrections (Kordzanganeh et al., 2023). The Parallel Proportional Fusion of Quantum and Spiking Neural Networks processes MNIST images through a spiking branch and a variational quantum branch in parallel, then fuses the classical and quantum feature vectors according to $Q_h=\zeta Q_q+(1-\zeta)Q_c$ (Xu et al., 2024). In graph-structured quantum learning, residual skip structure is inserted between quantum layers, yielding a Res-HQCNN in which residual addition modifies the effective input to the next layer (Liang et al., 2024).

Pattern	Representative structure	Example
Serial hybrid	classical preprocessing $\rightarrow$ PQC $\rightarrow$ classical head	MNIST HQNN (Zaman et al., 2024)
Quanvolutional front end	quantum patch processing $\rightarrow$ classical CNN	COVID-19 HQCNN (Houssein et al., 2021)
Parallel hybrid	same input to classical and quantum branches, late fusion	PHN / PPF-SQNN (Kordzanganeh et al., 2023)

These patterns differ in where the quantum resource is placed, but they share the same operational role: the quantum block acts as a learned feature transformation whose outputs are made usable by classical layers through measurement and postprocessing.

3. Quantum components, encodings, and hyperparameters

The quantum block is typically built from three elements: data encoding, a trainable ansatz, and measurement. Angle encoding is especially common. The MNIST hyperparameter study uses RY angle encoding with the feature map

$S_{x}_{j} = \bigotimes_{i=1}^{N} U_i \hspace{0.2cm} where \hspace{0.2cm}U_i := \begin{bmatrix} \cos(x_j^{(i)}) & -\sin(x_j^{(i)})\ \sin(x_j^{(i)}) &\cos(x_j^{(i)}) \end{bmatrix},$

followed by one of three circuit families: Basic Entangling, Strongly Entangling, or Random Circuit (Zaman et al., 2024). The COVID-19 HQCNN also compares RX and RY angle encoding and reports that RY performs better, while decoding is done with Pauli-$Z$ measurement (Houssein et al., 2021). In radar, classical features are encoded by $R_x$-axis rotations, entangled by a closed chain of CNOT gates, and measured through Pauli-basis expectation values involving $Y_j$ (Malarvanan, 2024). In photonic hybrids, classical hidden activations are scaled into amplitudes and phases suitable for continuous-variable gates, and the qumodes are measured by homodyne detection in the position basis (Austin et al., 2024).

The circuit ansatz varies substantially across papers. The Basic Entangling family uses one-parameter single-qubit rotation gates on each qubit followed by a ring of CNOTs; the Strongly Entangling family uses three-parameter single-qubit rotation gates followed by entanglers; the Random Circuit family is generated from a random subset of Pauli gates, single-parameter and three-parameter rotation gates, and entangling gates such as CX or CZ (Zaman et al., 2024). The six-qubit HQCNN for color-space image classification combines encoding rotations $4 \rightarrow 4$0, $4 \rightarrow 4$1, and $4 \rightarrow 4$2 with IsingYY and IsingZZ operators, two quantum pooling layers, and an additional optimized unitary $4 \rightarrow 4$3 built from controlled rotations $4 \rightarrow 4$4, $4 \rightarrow 4$5, or $4 \rightarrow 4$6 (Ng et al., 2024). The medical-image HQCNN with a 4-qubit VQC adds cyclic $4 \rightarrow 4$7 encoding, superpositional entanglement through Hadamard, $4 \rightarrow 4$8, and cyclic CNOT gates, and a Fourier-inspired Quantum Attention-Fourier layer using Toffoli, $4 \rightarrow 4$9, controlled phase gates with phase schedule $Q_h=\zeta Q_q+(1-\zeta)Q_c$0, and Hadamard projection (Shahjalal et al., 16 Sep 2025). TunnElQNN, by contrast, keeps the quantum circuit small—2 qubits with AngleEmbedding, BasicEntanglerLayer(s), and Pauli-$Q_h=\zeta Q_q+(1-\zeta)Q_c$1 measurement—but modifies the classical nonlinearity through a tunnel-diode activation function (Abbas, 2 May 2025).

Quantum-specific hyperparameters materially affect both accuracy and compute. The MNIST study varies circuit depth from 2 to 6 layers, qubit counts of 4, 9, and 16, entanglement type or circuit family, measurement observables Pauli $Q_h=\zeta Q_q+(1-\zeta)Q_c$2, $Q_h=\zeta Q_q+(1-\zeta)Q_c$3, and $Q_h=\zeta Q_q+(1-\zeta)Q_c$4, and shot counts of 100 and 1024 (Zaman et al., 2024). Its reported findings are unusually explicit: depth generally increases training time; 3–4 layers often give the best accuracy overall; Pauli $Q_h=\zeta Q_q+(1-\zeta)Q_c$5 is generally the best observable overall; Random Circuit performs best overall; training time rises substantially with qubit count; the jump from 9 to 16 qubits is much more expensive than from 4 to 9 qubits; and 1024 shots help primarily for Basic and Strongly Entangling circuits rather than universally (Zaman et al., 2024). The paper interprets declining accuracy in large Basic and Strongly Entangling circuits as a possible sign of over-expressibility and barren plateaus. A plausible implication is that HQCNN design cannot be reduced to “more qubits, more depth, more entanglement”; the relevant object is the interaction among encoding, ansatz, measurement, and optimization budget.

4. Application domains and reported empirical behavior

Image classification is the most common application area. In COVID-19 chest X-ray analysis, the HQCNN using random quantum circuits as a quantum convolutional preprocessing layer reports 98.4\% accuracy and 99.3\% sensitivity on the binary COVID-19 vs Normal dataset, 99\% accuracy and 99.7\% sensitivity on the binary COVID-19 vs Pneumonia dataset, and 88.6\% accuracy with 88.7\% sensitivity on the three-class COVID-19 vs Normal vs Pneumonia dataset (Houssein et al., 2021). In general medical imaging, the 5-layer-CNN-plus-4-qubit-VQC HQCNN reports, among other results, 99.91\% accuracy and 100.00\% AUC on PathMNIST binary and 99.95\% accuracy on OrganAMNIST binary, while also reporting 87.18\% accuracy and 90.04\% AUC on the smaller and noisier BreastMNIST benchmark (Shahjalal et al., 16 Sep 2025). For color-space image classification, the six-qubit HQCNN achieves a best MNIST accuracy of 94.3\% in Lab, while the CNN baseline achieves 92.8\% in RGB, and the paper reports that the HQCNN outperforms the classical CNN across all tested color spaces for the ten-class MNIST task (Ng et al., 2024).

Performance patterns outside standard vision are more conditional. In radar-based drone detection and classification, the HQNN does not dominate uniformly: when SNR is high, the comparable classical CNN outperforms the HQNN for detection, but in the low-SNR regime the HQNN becomes superior, with detector F1 rising relative to the CNN at $Q_h=\zeta Q_q+(1-\zeta)Q_c$6, $Q_h=\zeta Q_q+(1-\zeta)Q_c$7, and $Q_h=\zeta Q_q+(1-\zeta)Q_c$8 dB and classifier F1 remaining slightly higher across all tested SNRs from $Q_h=\zeta Q_q+(1-\zeta)Q_c$9 dB to $\rightarrow$0 dB (Malarvanan, 2024). In connected-vehicle incident detection, the 4-qubit hybrid model achieves recall of 98.9\%, 98.3\%, and 96.6\% on DS-1, DS-2, and DS-3 respectively, and the average improvement in F2-score over classical models is 1.9\% on DS-2 and 7.8\% on DS-3, with the advantage concentrated in insufficient-data settings rather than the abundant-data regime (Khan et al., 2021). TunnElQNN reports 97\% validation accuracy, versus 87\% for the ReLUQNN baseline, on a three-class interleaving half-circles problem, and the paper attributes the gain to the combination of TDAF and the quantum layer rather than to either component alone (Abbas, 2 May 2025).

Several papers extend the concept beyond conventional supervised image tasks. Res-HQCNN for graph-structured quantum data combines fidelity-based supervision with a graph-aware Hilbert–Schmidt term and reports that using graph information improves learning compared with $\rightarrow$1, while residual connections improve performance compared with non-residual QNNs, especially as depth increases (Liang et al., 2024). A hybrid quantum-classical neural network for continuous-variable prediction slightly outperforms a classical neural network on the Boston Housing regression task in its qubit-based form, while the photonic version remains constrained by simulation cost (Jain et al., 2022). A molecular HQCNN trained by energy minimization rather than standard supervised labels generates accurate ground-state potential-energy curves for $\rightarrow$2, LiH, and BeH$\rightarrow$3, and intermediate measurements reduce both training and testing error dramatically relative to the same circuit without intermediate measurements (Xia et al., 2019). For quantum phase recognition, a hardware-in-the-loop HQCNN combining a learned quantum disentangler with a classical classifier reaches accuracies above 85\% in single-shot measurements and above 99\% when averaging over ten measurements on surface-code states (Scarato et al., 26 Jun 2026).

The empirical record is therefore heterogeneous. Some papers report strong improvements on specific datasets, some report advantage only in low-data or low-SNR regimes, and some present proof-of-concept benefits that remain modest or simulator-bound. This suggests that HQCNNs are better understood as task- and regime-dependent models than as uniformly superior replacements for classical networks.

5. Learnability, trainability, and architecture search

A persistent theme is that HQCNN behavior is dominated by design choices that are absent in ordinary DNN tuning. The MNIST hyperparameter study makes this explicit by treating HQNN design as a quantum hyperparameter optimization problem rather than a classical one, and by showing that depth, qubit count, entanglement type, measurement observable, and shot count interact nontrivially in their effect on accuracy, convergence, and training time (Zaman et al., 2024). The paper’s best-performing configuration—16 qubits, 1024 shots, 3 layers, Random Circuit, Pauli $\rightarrow$4—reaches 95.3\% accuracy but requires 708.57 minutes of training time, directly illustrating the accuracy–cost tradeoff (Zaman et al., 2024).

Later work complicates the usual PQC-centric intuition. The expressibility–trainability study argues that the standard claim that “more expressive circuits are harder to train” is not reliable in HQNNs: pure PQCs show only a weak and regime-dependent trade-off, hybrid architectures weaken it further, and full end-to-end hybrid training can eliminate it entirely because classical components reshape the optimization landscape (Kashif et al., 25 May 2026). The paper therefore distinguishes quantum-only trainability from full-model trainability and proposes a multi-objective NAS framework that jointly optimizes expressibility, trainability, and task performance across a combined classical–quantum design space (Kashif et al., 25 May 2026). A companion NAS paper makes a related point from the efficiency side, treating HQNN architecture search as a structured search over encoding, variational circuit structure, measurement, and post-processing, and demonstrating FLOPs-aware NSGA-II search over 23,328 hybrid architectures per dataset with number of qubits $\rightarrow$5, encoding methods $\rightarrow$6, rotation gates $\rightarrow$7, entangling gates $\rightarrow$8, topology $\rightarrow$9, and depth $\rightarrow$0 (Marchisio et al., 18 May 2026).

Architectural devices intended to improve optimization recur across domains. Residual connections in Res-HQCNN are reported to help deeper quantum neural networks learn graph-structured quantum data faster and better (Liang et al., 2024). Parallel hybrids are introduced partly to avoid information bottlenecks associated with serial classical-to-quantum or quantum-to-classical coupling, and their authors report that balanced contribution from both branches is important for best performance (Kordzanganeh et al., 2023). In continuous-variable photonic hybrids, trainability is improved sufficiently that hybrid networks achieve the same performance as fully classical networks that are twice the size, while noise analysis using effective number of bits identifies the quantum parameters most sensitive to reduced precision (Austin et al., 2024). These results do not establish a single trainability law, but they converge on a narrower point: in HQCNNs, optimization is a property of the full hybrid system rather than of the quantum circuit in isolation.

6. Limitations, robustness, and open problems

Many HQCNN results remain simulator-based or are explicitly constrained by NISQ-era limitations. The multiple-color-space HQCNN uses PennyLane and PyTorch in simulation and notes that backpropagation is feasible in simulation but parameter-shift would be required on actual hardware (Ng et al., 2024). The radar study is based on Martin–Mulgrew simulated returns rather than real radar recordings (Malarvanan, 2024). The continuous-variable regression paper reports that the photonic model becomes impractical beyond very small feature counts because of memory and runtime scaling (Jain et al., 2022). The graph-structured quantum-data study uses synthetic datasets with $\rightarrow$1 states and acknowledges that the residual addition can produce states that are not valid density matrices, so the resulting map is not completely positive (Liang et al., 2024). Even where hardware is used, results are mixed: an earlier binary-classification HQCNN built from VQC “neurons” improves accuracy by roughly 8–11\% over a single VQC on simulated hardware, but both models perform poorly on current quantum hardware except when qubit and gate count are sufficiently small (Arthur et al., 2022).

The evidence for practical advantage is therefore contingent rather than unconditional. The low-SNR radar paper states directly that the CNN outperforms the HQNN when SNR is high and that the advantage appears in the low-SNR regime of greatest practical interest (Malarvanan, 2024). The multiple-color-space image study reports that HQCNN does not always beat the strongest CNN result on CIFAR-10 and that performance is sensitive to color space and PQC design (Ng et al., 2024). The incident-detection paper finds only modest hybrid advantage when data are sufficient, with much clearer gains under data scarcity (Khan et al., 2021). This suggests that claims about HQCNN efficacy should be indexed to regime, dataset size, noise level, and architecture rather than treated as universal.

Security has also emerged as a distinct concern. The backdoor-attack study models HQNNs for image classification as $\rightarrow$2, with a classical CNN backbone followed by a quantum classifier, and reports that HQNNs are more robust than CNNs to standard patch and blend backdoor attacks, often requiring more significant image modifications for successful attacks (Guo et al., 2024). However, the same paper introduces the Qcolor backdoor, a color-shift trigger optimized with NSGA-II, and shows that HQNNs remain vulnerable to tailored attacks despite that increased robustness (Guo et al., 2024). The resulting picture is not that the quantum layer provides inherent security, but that hybrid architectures alter the attack surface.

Open directions in the surveyed literature are correspondingly concrete. Repeated recommendations include real-hardware validation beyond simulators, broader search over joint classical–quantum architectures, better handling of hardware noise and resource constraints, extension to larger and more realistic datasets, and more systematic analysis of when hybridization changes sample complexity or robustness in a practically useful way (Marchisio et al., 18 May 2026, Kashif et al., 25 May 2026). Taken together, these works position HQCNNs as a technically diverse research area whose central scientific questions concern not only representational expressivity, but also interface design, optimization geometry, hardware compatibility, and regime-specific utility.