Hybrid-FQNN: Quantum-Classical Dense Architectures

• Hybrid-FQNN is a class of hybrid quantum-classical models that combine fully connected classical layers with parameterized quantum circuits for nonlinear feature transformations.
• It employs both serial and parallel coupling patterns, enabling applications in anomaly detection, medical imaging, regression, and quantum many-body simulations.
• The architecture leverages diverse encoding methods, entanglement strategies, and gradient-based optimization to balance expressivity with practical resource constraints.

Searching arXiv for recent and relevant papers on Hybrid-Fully Connected Quantum Neural Network (H-FQNN). Hybrid-Fully Connected Quantum Neural Network (H-FQNN) denotes a family of hybrid quantum-classical neural architectures that combine classical fully connected processing with parameterized quantum circuits, typically in an end-to-end trainable pipeline. In the cited literature, the term is used in two closely related senses: first, for models in which a variational quantum circuit replaces or augments a classical dense hidden layer inside an otherwise classical network; second, for hybrid models whose quantum component itself has increased or fully entangling connectivity, approaching an all-to-all or highly connected quantum layer. Representative instances include ADS-B anomaly detection, dementia detection from MRI, continuous-variable regression, binary classification with variational-circuit neurons, and fully entangling quantum-neural ansätze for many-body simulation (Naaman et al., 19 Sep 2025, Kim, 2023, Jain et al., 2022, Zhang et al., 21 Jan 2025).

1. Terminology and conceptual scope

A common H-FQNN pattern is a classical-quantum-classical stack in which classical features are first transformed by dense layers, then processed by a variational quantum circuit (VQC), and finally mapped by a classical output layer. In the ADS-B anomaly-detection formulation, the network contains an input layer, a classical dense hidden layer with ReLU, a quantum layer implemented as a VQC, and a 2-neuron output layer for binary classification; the quantum layer replaces a classical dense layer that would otherwise appear in a fully connected neural network (FNN) baseline (Naaman et al., 19 Sep 2025). In a regression setting on the Boston Housing dataset, the architecture is described as [Input Layer] → [Quantum Layer] → [Output Layer], with the quantum layer wrapped as a KerasLayer inside TensorFlow/Keras, again functioning as a non-classical hidden transformation within a classical network (Jain et al., 2022).

A second usage emphasizes the connectivity of the quantum subnetwork. In the quantum many-body literature, the hybrid ansatz couples a neural network with a hardware-efficient parameterized quantum circuit (PQC), and the fully connected variant is characterized by entanglement in which every qubit can be connected to every other qubit (Zhang et al., 21 Jan 2025). The architecture-search literature makes this interpretation explicit by treating entanglement topology, gate composition, measurement design, and classical-quantum coupling as search dimensions of a broader HQNN design space (Marchisio et al., 18 May 2026).

This suggests that H-FQNN is better understood as an architectural class than as a single canonical model. What remains stable across these formulations is the hybridization principle: a quantum layer supplies a trainable nonlinear map or expressive quantum feature transformation, while classical dense modules handle preprocessing, postprocessing, or both.

2. Architectural motifs and coupling patterns

The dominant coupling pattern is sequential. A general HQNN workflow is described as input → classical pre-processing → classical layers → quantum layer (PQC) → measurement → classical post-processing, with classical and quantum parameters trained end-to-end (Marchisio et al., 18 May 2026). The ADS-B H-FQNN instantiates this directly as Classical features → Dense1 → ReLU → Dense2 → ReLU → Quantum layer (VQC) → Final Dense (output) → Loss, and the dementia-detection QCCNN inserts a VQC bottleneck between a ResNet-18 feature extractor and a final fully connected classifier, summarized as

$$QCNN = L_{4 \rightarrow 2} \cdot Q_{512 \rightarrow 4} \cdot L_{512}.$$

In that model, the VQC reduces a 512-dimensional classical representation to 4 expectation values before classical classification (Naaman et al., 19 Sep 2025, Kim, 2023).

A parallel coupling pattern also appears. The Parallel Hybrid Network (PHN) passes the same input simultaneously to a classical multilayer perceptron (MLP) and a VQC, then linearly combines their outputs:

$o_m = s^c_m c_m + s^q_m q_m.$

The stated motivation is that the quantum branch learns a smooth sinusoidal foundation, while the classical branch fills non-harmonic gaps in the landscape (Kordzanganeh et al., 2023). Although PHN is not labeled H-FQNN in the source paper, it formalizes an important variant of hybrid fully connected design: dense classical and quantum pathways can be combined without forcing a strictly serial bottleneck.

Related photonic hybrids extend the same logic to continuous-variable (CV) quantum hardware. Austin et al. replace classical hidden layers in a photonic neural network with a trainable CV quantum neural network, leaving classical fully connected input and output layers in place (Austin et al., 2024). Across these cases, the architectural question is not whether classical and quantum modules should be combined, but where the quantum layer should sit, how strongly it should be coupled to dense classical components, and whether serial or parallel composition is preferable.

3. Quantum layer construction, encoding, and measurement

Feature encoding is one of the principal determinants of H-FQNN behavior. The architecture-search literature treats angle encoding and amplitude encoding as explicit search-space choices (Marchisio et al., 18 May 2026). In the ADS-B H-FQNN, the quantum layer uses angle embedding via $R_X$ rotations, followed by a strongly entangling variational circuit in PennyLane; in the Boston Housing model, classical inputs are mapped to qubits with AngleEmbedding, typically using $R_y(x_i)$ rotations (Naaman et al., 19 Sep 2025, Jain et al., 2022). A different compression strategy is used in QuClassi, where two classical features are encoded per qubit by sequential $RY$ and $RZ$ rotations, reducing qubit requirements for classification (Stein et al., 2021).

Ansatz design varies substantially across H-FQNN-related systems. The ADS-B model uses strongly entangling circuits with learnable rotations around the $Z$, $Y$, and $Z$ axes together with CNOT entanglement (Naaman et al., 19 Sep 2025). The binary-classification HNN based on Qiskit uses the RealAmplitudes ansatz with one repetition and full entanglement, with each hidden-layer “neuron” realized by an independent VQC (Arthur et al., 2022). In the many-body setting, the PQC component is a hardware-efficient ansatz,

$$U(\theta) = \prod_{l=1}^{N_l} \left[ \prod_{(m,n)} \text{CNOT}_{m,n} \prod_{j=1}^{N_q} R_Z(\theta^Z_{l,j}) R_X(\theta^X_{l,j}) \right] \prod_{k=1}^{N_q} H_k,$$

and the fully connected case permits every qubit pair to be entangled (Zhang et al., 21 Jan 2025). In CV photonic hybrids, the quantum layer is constructed from displacement, squeezing, non-Gaussian Kerr gates, and linear interferometers, with homodyne readout (Austin et al., 2024).

Readout mechanisms are equally heterogeneous. A broad template for hybrid models is

$o_m = s^c_m c_m + s^q_m q_m.$0

with $o_m = s^c_m c_m + s^q_m q_m.$1 commonly taken to be a Pauli observable (Jain et al., 2022). The ADS-B H-FQNN measures Pauli-$o_m = s^c_m c_m + s^q_m q_m.$2 expectation values; the dementia-detection QCCNN measures qubits in the $o_m = s^c_m c_m + s^q_m q_m.$3-basis to obtain a 4-dimensional classical vector; QuClassi instead uses a SWAP test to compute quantum state fidelity as the similarity signal for classification (Naaman et al., 19 Sep 2025, Kim, 2023, Stein et al., 2021).

The phrase “fully connected” therefore admits two concrete technical readings in this literature: dense classical coupling around a quantum layer, and dense or all-to-all entangling connectivity within the quantum layer itself. The architecture-search and genetic-search papers both treat this as a design variable rather than a fixed property (Marchisio et al., 18 May 2026, Ahmed et al., 16 Apr 2026).

4. Training and optimization strategies

Most H-FQNNs are trained with hybrid gradient-based optimization. The ADS-B model uses Adam with $o_m = s^c_m c_m + s^q_m q_m.$4, 150 epochs, and either BCEWithLogitsLoss or CrossEntropyLoss; gradients of the variational parameters are computed as part of the overall hybrid workflow (Naaman et al., 19 Sep 2025). The Boston Housing model uses mean squared error,

$o_m = s^c_m c_m + s^q_m q_m.$5

and trains end-to-end with classical optimization after embedding the quantum circuit as a KerasLayer (Jain et al., 2022). The binary-classification HNN with VQC neurons uses negative log-likelihood with gradients estimated by the parameter-shift rule and classical stochastic or batch gradient descent (Arthur et al., 2022).

Software frameworks make these training loops practical. PennyLane appears repeatedly as the interface for implementing VQCs and coupling them to PyTorch or TensorFlow/Keras (Naaman et al., 19 Sep 2025, Jain et al., 2022). For CV photonic hybrids, the cited implementation uses TensorFlow/Keras for classical layers and PennyLane/Strawberry Fields for quantum layers (Austin et al., 2024). VQNet formalizes the general pattern by treating a VQC as a quantum operator in a symbolic computational graph, with qop and qop_pmeasure providing differentiable quantum nodes inside hybrid models (Chen et al., 2019).

Beyond backpropagation, two additional optimization directions are prominent. First, the GAT-QNN framework introduces a two-stage genetic algorithm in which a macroCircuit defines the search space, microCircuits are sampled and trained with parameter sharing, and a second genetic stage performs backend-aware inference-time architecture selection without retraining (Ahmed et al., 16 Apr 2026). The paper explicitly states that this methodology is extensible to H-FQNNs by adding genes for entanglement or connectivity options such as all-to-all, nearest-neighbor, or arbitrary patterns. Second, FLOPs-aware neural architecture search formulates HQNN design as a multi-objective problem over encoding, variational structure, measurement, and postprocessing,

$o_m = s^c_m c_m + s^q_m q_m.$6

with Pareto selection over accuracy and total FLOPs (Marchisio et al., 18 May 2026). A separate quantized-network line frames training as QBO/QCBO and uses Quantum Conditional Gradient Descent (QCGD), with the paper presenting this as relevant to hybrid fully connected quantum neural networks because it permits arbitrary activation and loss functions through spline interpolation (Li et al., 23 Jun 2025).

5. Empirical performance and application domains

Reported H-FQNN performance is application-dependent and should be interpreted in the context of backend assumptions, network scale, and baseline choice. The published results span anomaly detection, medical imaging, regression, binary classification, photonic classification, and quantum many-body simulation.

Domain	Representative configuration	Reported result
ADS-B anomaly detection	Dense + VQC + dense output	Accuracy 90.17%–94.05%; best F1 93.99% (Naaman et al., 19 Sep 2025)
MRI dementia detection	ResNet-18 + VQC bottleneck + FC	97.5% testing and 95.1% validation accuracy (Kim, 2023)
Binary classification with VQC neurons	Hidden-layer VQCs + output VQC	Roughly 10% higher accuracy than an individual VQC in simulation (Arthur et al., 2022)
Photonic hybrid classification	FC input/output + CVQNN middle layer	0.85 ± 0.03 vs 0.79 ± 0.03 for equal-size classical baseline (Austin et al., 2024)

In ADS-B anomaly detection, the H-FQNN is reported to achieve accuracies ranging from 90.17% to 94.05%, comparable to a traditional FNN at 91.50% to 93.37%, with the best H-FQNN result reaching 94.05% accuracy and 93.99% F1 score at 10,000 attack samples, BCEWithLogitsLoss, and 6 qubits (Naaman et al., 19 Sep 2025). This is one of the clearest examples in which the H-FQNN is explicitly benchmarked against a classical FNN with matched preprocessing and task definition.

In MRI dementia detection, the hybrid QCCNN uses a ResNet-18 feature extractor, a 4-qubit variational bottleneck, and a final dense classifier. The paper reports 97.5% and 95.1% testing and validation accuracies, respectively, compared with 91.5% and 89.2% for the classical neural network baseline; on a testing set of 100 normal and 100 dementia MRI images, the hybrid model detected normal and demented images correctly 95% and 98% of the time, compared to 89% and 91% for the CNN (Kim, 2023).

For shallow feedforward HNNs whose neurons are themselves VQCs, simulation results show that the hybrid neural network achieves roughly 10% higher classification accuracy and 20% better minimization of cost than an individual VQC, but real-hardware performance remains strong only when qubit and gate counts are sufficiently small (Arthur et al., 2022). In CV photonic hybrids, small hybrid networks achieved 0.85 ± 0.03 accuracy versus 0.79 ± 0.03 for a matched classical network, and the classical network required approximately twice as many parameters to match hybrid performance (Austin et al., 2024).

The many-body literature evaluates an H-FQNN-like fully entangling hybrid ansatz under a different objective: relative energy rather than classification accuracy. There, the hybrid method achieves notably lower relative energy than standalone neural quantum states, and for LiH the stated error drops from $o_m = s^c_m c_m + s^q_m q_m.$7 Ha for a modest standalone NQS to $o_m = s^c_m c_m + s^q_m q_m.$8 Ha for the hybrid method with a 4-layer PQC (Zhang et al., 21 Jan 2025). This widens the scope of H-FQNN from conventional supervised learning to variational scientific computing.

6. Limitations, misconceptions, and current directions

A recurrent misconception is that any insertion of a quantum layer into a dense network yields a decisive accuracy advantage. The ADS-B anomaly-detection study instead reports competitive performance: the H-FQNN is comparable to the classical FNN, sometimes slightly better and sometimes marginally worse depending on the experimental condition (Naaman et al., 19 Sep 2025). The binary-classification HNN paper likewise shows that simulated improvements do not automatically transfer to present-day hardware; on quantum hardware, both the hybrid network and the individual VQC only perform well when the qubit and gate count is sufficiently small (Arthur et al., 2022).

Another limitation concerns inductive bias. The PHN paper argues that angle-embedded quantum neural networks tend to approximate functions with a truncated Fourier-series character and may therefore struggle with non-harmonic features; its solution is to place the VQC in parallel with an MLP rather than in strict series (Kordzanganeh et al., 2023). This suggests that some H-FQNN failures may reflect a mismatch between task structure and the spectral bias of the chosen quantum encoding or ansatz, rather than an inherent weakness of hybridization itself.

Noise, compilation variability, and backend heterogeneity remain central concerns. GAT-QNN addresses this by separating parameter optimization from deployment-time architecture selection and reports consistent 22–23% test accuracy gains for GA-driven inference across multiple backends in MNIST classification, while also allowing smaller deployed microCircuits with lower gate count (Ahmed et al., 16 Apr 2026). FLOPs-aware NAS treats computational efficiency as a first-class objective, using Pareto search over accuracy and cost to avoid over-complex quantum circuits (Marchisio et al., 18 May 2026). These studies do not define a single H-FQNN architecture, but they supply concrete mechanisms for making H-FQNN design backend-aware and resource-aware.

Terminology is also unsettled. “HQFNN” in the fuzzy-neural literature may denote a Hierarchical Fused Quantum Fuzzy Neural Network (Wu et al., 2024) or a Highly Quantized Fuzzy Neural Network (Yao et al., 11 Jun 2025), not a Hybrid-Fully Connected Quantum Neural Network. At the same time, both fuzzy models share the broader hybrid design principle of coupling quantum subcircuits to classical feature extractors or fusion layers. A plausible implication is that future H-FQNN taxonomies will need to distinguish more carefully between dense classical-quantum coupling, fully entangling quantum connectivity, and task-specific hybrids such as fuzzy or photonic variants.

Current research directions therefore converge on three problems: enlarging the architecture space without losing trainability, selecting architectures under explicit hardware constraints, and determining when quantum layers contribute genuine task-relevant expressivity rather than merely additional parameters. Within that agenda, H-FQNN remains less a finalized model than a technically active template for hybrid dense-quantum learning.