On the Complementarity of Classical Convolution and Quantum Neural Networks in Image Classification

1. Introduction

This work investigates a hybrid quantum-classical model that combines a compact Convolutional Neural Network (CNN) [1,2,3] with a two-qubit Quantum Neural Network (QNN) [4,5,6] for binary image classification [7,8,9]. The motivation is preliminary testing of such hybrid architectures in a controlled environment where model stability, interpretability, and complementarity between branches can be observed clearly. To enable this type of controlled study, we deliberately use an artificially generated dataset consisting of small $16\times 16$ grayscale images containing one or more geometric shapes embedded in structured noise. This design choice allows us to isolate core visual features such as contrast, edge strength, and spatial compactness, without additional complications arising from real satellite imagery such as compression artifacts, atmospheric distortion, or variable illumination.

The objective of this work is not to outperform large classical architectures on high-resolution, real-world benchmarks. Instead, the aim is to answer whether a minimal quantum component can be trained stably alongside a classical convolutional backbone, and whether the quantum branch contributes a useful and complementary signal to the final decision. The use case should therefore be understood as a proof-of-concept filter for distinguishing prominent structure from background noise in constrained computational settings such as early onboard preprocessing or embedded lightweight detection pipelines.

We demonstrate that the hybrid model can be trained end-to-end without collapse, that its predicted probabilities are smooth and well-calibrated, and that late fusion of the CNN and QNN outputs provides strong performance in this controlled setting. The results indicate that hybrid models of this scale are viable and warrant further exploration on progressively more realistic datasets.

To situate this work within the broader context we point to the fact that in recent years, the field of quantum machine learning and quantum computing in general has broadened. For example the examined benchmarking and feature selection in quantum neural networks [10,11,12,13,14,15] has been examined and optimization in noisy variational settings [16,17,18,19,20] has been studied, and it has been shown, that the choice of the optimization strategies is vital, with multiple viable choices [21,22,23,24] available. Hybrid quantum classical approaches [25,26,27] have been explored in various settings, alongside with profound benchmarking of various phenomena [28,29,30,31]. Related advances in variational eigensolver theory and ansatz construction [32,33,34,35,36,37,38] further motivate this study. In this context, we evaluate whether a minimal two qubit circuit can provide complementary signal when fused with a lightweight convolutional branch. The results indicate that hybrid models of this scale are viable and warrant further exploration on progressively more realistic datasets.

2. Methodology

In this section, the dataset, its generation and the model are described, to precisely define the setup used to obtain the results, that are presented in sec:results.

Data Generation

The dataset is synthetically generated to allow precise control over shape structure and noise characteristics. Its negative samples, labeled by $y=0$, consist solely of random noise uniformly distributed in a limited intensity range, while the positive samples, labeled as $y=1$, contain either one or two randomly positioned shapes selected from {square, rectangle, circle, ellipse, right triangle}.

When a shape is selected, and a placement is calculated, it brightens the pixels and also applies a slight halo effect, which simulates diffuse scattering. All images are $16\times 16$ pixels and normalized to $[0,1]$, and an selected 16 are shown in Figure 1.

Model Architecture

The architecture of the employed model can be be described as follows, the input, which is a single-channel grayscale image,

$$X\in {\mathbb{R}}^{1\times 16\times 16}.$$

(1)

is processed by two parallel branches, a classical CNN branch that extracts spatial features directly from the image, and a QNN branch that operates on a reduced set of handcrafted global features. The outputs of the two branches are fused at the logit level to form the final classification.

CNN Branch

The classical branch learns local spatial patterns using convolutional filters. The first convolution increases the number of channels from one to eight and applies a nonlinear activation,

$$H_1=\sigma (W_1*X+b_1),W_1\in {\mathbb{R}}^{8\times 1\times 3\times 3},H_1\in {\mathbb{R}}^{8\times 16\times 16}.$$

(2)

Here, $\sigma (·)$ denotes the ReLU activation, which introduces non-linearity and helps the network learn more expressive features.

To reduce spatial resolution and improve invariance to small spatial shifts, a $2\times 2$ max-pooling operation is applied,

$$H_1^{\mathrm{pool}}={MaxPool}_{2\times 2}\left(H_1\right)\in {\mathbb{R}}^{8\times 8\times 8}.$$

(3)

A second convolutional layer further enriches feature complexity by mapping eight channels to sixteen,

$$H_2=\sigma (W_2*H_1^{\mathrm{pool}}+b_2),W_2\in {\mathbb{R}}^{16\times 8\times 3\times 3},H_2\in {\mathbb{R}}^{16\times 8\times 8}.$$

(4)

This is again followed by max-pooling to compress spatial dimensions,

$$H_2^{\mathrm{pool}}={MaxPool}_{2\times 2}\left(H_2\right)\in {\mathbb{R}}^{16\times 4\times 4}.$$

(5)

The resulting tensor is flattened into a feature vector,

$$h_{\mathrm{cnn}}=vec\left(H_2^{\mathrm{pool}}\right)\in {\mathbb{R}}^{256},$$

(6)

which is mapped to a single scalar logit by a fully connected layer,

$$z_{\mathrm{cnn}}=w_{\mathrm{cnn}}^{\top }{h}_{\mathrm{cnn}}+b_{\mathrm{cnn}},w_{\mathrm{cnn}}\in {\mathbb{R}}^{256},z_{\mathrm{cnn}}\in \mathbb{R}.$$

(7)

This logit represents the classical model’s confidence before sigmoid activation.

QNN Branch

In parallel, the quantum branch operates not on the image directly, but on a global handcrafted feature representation. For each image, six statistical and structural descriptors are extracted,

$$f\in {\mathbb{R}}^6,$$

(8)

these are further described in Table 1. To reduce redundancy and improve conditioning, Principal Component Analsis (PCA) maps these features to a two-dimensional latent representation,

$$u=W_{\mathrm{PCA}}(f-\mu ),W_{\mathrm{PCA}}\in {\mathbb{R}}^{2\times 6},u\in {\mathbb{R}}^2.$$

(9)

This vector is then scaled into rotation angles suitable for quantum state encoding,

$$\theta =\alpha u+\beta ,\theta \in {[0,2\pi )}^2.$$

(10)

The quantum circuit begins in the ground state,

$$|{\psi }_0\rangle =|00\rangle .$$

(11)

A feature-encoding map embeds the classical angles into the quantum state,

$$|{\psi }_{\mathrm{enc}}\left(\theta \right)\rangle =U_{\mathrm{FM}}\left(\theta \right)|{\psi }_0\rangle ,$$

(12)

after which a trainable variational layer applies rotations parameterized by $\varphi$,

$$|{\psi }_{\mathrm{var}}(\theta ,\varphi )\rangle =U_{\mathrm{VA}}\left(\varphi \right)|{\psi }_{\mathrm{enc}}\left(\theta \right)\rangle ,\varphi \in {\mathbb{R}}^k.$$

(13)

The circuit output is obtained via the expectation value of the Pauli operator $Z\otimes Z$,

$$q(\theta ,\varphi )=〈{\psi }_{\mathrm{var}}(\theta ,\varphi )\left|(Z\otimes Z)\right|{\psi }_{\mathrm{var}}(\theta ,\varphi )〉,q(\theta ,\varphi )\in [-1,1].$$

(14)

A linear projection converts this value to a logit matching the classical branch output scale,

$$z_{\mathrm{qnn}}=w_{q}q(\theta ,\varphi )+b_q,w_q\in \mathbb{R},z_{\mathrm{qnn}}\in \mathbb{R}.$$

(15)

Fusion and Output

Finally, the two logits are combined through a learnable linear fusion layer,

$$z_{\mathrm{final}}=w_{f,1}{z}_{\mathrm{cnn}}+w_{f,2}{z}_{\mathrm{qnn}}+b_f,z_{\mathrm{final}}\in \mathbb{R}.$$

(16)

The final classification probability is obtained via the sigmoid function,

$$p(y=1\mid X)={\displaystyle \frac{1}{1+e^{-z_{\mathrm{final}}}}}.$$

(17)

Training Procedure

The dataset is balanced and split into training and validation partitions. Optimization uses the Adam optimizer with distinct learning rates for the quantum and classical components. Across 12 epochs, we record training loss, validation probabilities, and confusion matrices to evaluate model behavior and threshold sensitivity.

Figure 2. Training loss over 12 epochs for the hybrid model.

Figure 2. Training loss over 12 epochs for the hybrid model.

3. Results

This section evaluates the performance of the hybrid quantum-classical model on the held-out validation set. The model was trained for 12 epochs, during which the training loss decreased smoothly without instability. This indicates that the joint optimization of the classical and quantum parameters was well-behaved, and that the learning-rate separation between the branches was effective.

Using the standard decision threshold of $0.5$, the model achieved strong classification performance on the validation set. The confusion matrix in Figure 3 shows that the hybrid architecture successfully distinguishes between images containing geometric shapes and pure noise backgrounds. The final values are shown in Table 2.

The model produced no false positives, meaning that any sample classified as containing a shape indeed contained one. The remaining classification errors were false negatives, where faint or partially obscured shapes were classified as background. This suggests that the model is conservative in its detection behavior, preferring to avoid false alarms.

The results indicate that the classical and quantum components contribute complementary information, the CNN branch captures local spatial structure and shape boundaries directly from pixel data, while the QNN branch responds to global scene properties such as contrast and edge energy derived from the handcrafted descriptors. In the end, the fusion layer effectively combines these representations to improve confidence and robustness. The complementarity of the two branches is supported quantitatively. the CNN and QNN logits on the validation set are only weakly correlated, $r=0.28$, indicating that they respond to different aspects of the input. Moreover, removing either branch reduces the F1-score compared to the hybrid value of $0.97$, confirming that both components contribute unique predictive information.

This behavior shows that even a quantum circuit with only two qubits can provide a meaningful signal when integrated in a structured hybrid architecture, particularly when operating on compact, semantically interpretable feature inputs. The experiment therefore demonstrates the feasibility of stable, end-to-end training of hybrid models in small-scale, controlled settings.

4. Conclusions

This work demonstrates that hybrid quantum-classical architectures can be trained stably and effectively, even at very small quantum scales. By combining a compact convolutional network with a two-qubit variational quantum circuit, the model successfully integrates local spatial information with global statistical descriptors derived from the input. The resulting hybrid classifier achieves high validation performance, with an F1-score of $0.97$, and shows clear evidence that the quantum and classical branches contribute complementary information rather than redundant representations.

The experiment therefore provides a controlled proof-of-concept that a quantum component can enhance a lightweight classical model when the quantum input is appropriately structured and semantically meaningful. While the setting is deliberately simplified, the core result is definitive, the quantum branch can act as a useful co-processor. This suggests that hybrid models merit further investigation in more realistic scenarios, including higher-resolution imagery and resource-constrained onboard analysis, where compact yet expressive representations are valuable.