QU-Net: Quantum-Enhanced U-Net for Self Supervised Embedding and Classification of Skin Cancer Images

Abstract

Background: Quantum Machine Learning (QML) has attracted significant attention in recent years. With quantum computing achievements in computationally costly domains, discovering its potential in improving the performance and efficiency of deep learning models in medical imaging has become a promising field of research. Methods: We investigate QML in healthcare by developing a novel quantum-enhanced U-Net (QU-Net). We experiment with six configurations of parameterized quantum circuits, varying the encoding technique (amplitude vs. angle), depth and entanglement. Using the ISIC-2017 skin cancer dataset, we compare QU-Net with classical U-Net on self-supervised image reconstruction and binary classification of benign and malignant skin cancer, where we combine bottleneck embeddings with patient metadata. Results: Our findings show that amplitude encoding stabilizes training, whereas angle encoding introduces fluctuations. The best performance is obtained with amplitude encoding and one layer. For reconstruction, QU-Net with entanglement converges faster (25 epochs vs. 44) with a lower Mean Squared Error per image (0.00015 vs. 0.00017) on unseen data. For classification, QU-Net with no entanglement embeddings reaches 79.03% F1-score compared with 74.14% for U-Net, despite compressing images to a smaller latent space (7 vs. 128). Conclusions: These results demonstrate that the quantum layer enhances U-Net’s expressive power with efficient data embedding.

1. Introduction

Quantum machine learning (QML) [1] is an emerging field that uses quantum computing’s capabilities to enhance traditional machine learning algorithms. With the quantum success in achieving an exponential speed-up in many classical tasks [2,3], one can wonder if we can achieve the same results to accelerate machine learning algorithms. Recently, the field has witnessed an increase in research to investigate the potential quantum advantage.

However, while many papers report that adding quantum components improves the results, it remains unclear whether such improvement can generalize to other tasks. Additionally, the quantum advantage may also be influenced by the commercialization of quantum technologies. Moreover, as shown by [4], recent research in the field seems to be biased by the notion of the eventual supremacy of quantum technology, often aiming to prove this assumption rather than conducting an objective investigation. This raises concerns about the rigor and impartiality of current studies, potentially biasing the understanding of Quantum Machine Learning’s true capabilities and limitations.

Skin cancer is among the most common types of cancer, where skin cells grow abnormally. It occurs most often on parts of the skin that are exposed to ultraviolet (UV) radiation, whether it comes from the sun or artificial tools such as tanning beds. Skin cancer is caused by DNA damage in the cell nucleus, leading to mutations that result in uncontrolled growth. There are three main types of skin cancer: basal cell carcinoma (BCC), squamous cell carcinoma (SCC), and melanoma, which is the least common but the most dangerous type of skin cancer as it spreads rapidly to other parts of the body’s skin, making it particularly lethal if not detected and treated early [5].

In this work, we propose a quantum-enhanced U-Net (QU-Net) that integrates a quantum layer as its bottleneck. QU-Net projects the image-data from the ISIC-2017 dataset [6] into a lower expressive space using a self-supervised approach [7,8] by training the model to reconstruct the images. The new embeddings are represented by the measurements on qubit states that are manipulated by a parameterized quantum circuit (PQC) in the quantum layer, and we experiment with several PQC configurations that vary in encoding strategy (amplitude vs. angle), circuit depth, and entanglement structure. To ensure fairness across all comparisons, all models are trained under identical conditions, using the same optimizer, GPU hardware, hyperparameters, and framework implementation. The quantum circuits in QU-Net are executed on the PennyLane default.qubit simulator. To evaluate the expressiveness of the new embeddings, we compare them to the embeddings extracted from a classical U-Net. Theoretically, if the new embeddings preserve the information from the lesion images, we should obtain relatively high or similar classification metrics when distinguishing between benign and malignant cases. If this is the case, it would indicate that a significant amount of the original information represented in the form of lesion images has been compressed into a lower-dimensional representation while maintaining comparable classification performance with reduced computational cost during inference.

Typically, an end-to-end approach obtains superior inference performance, as it enables the model to establish a direct mapping function between input data and output predictions. However, the ISIC-2017 dataset includes additional patient metadata in a structured format (CSV files). To incorporate this multimodal information [9], we adopt a two-phase prediction pipeline. In the first phase, image features are extracted through the self-supervised learning process. In the second phase, the quantum embeddings [10] obtained from the PQC are concatenated with the structured patient metadata. This approach based on feature representation learning [11] allows the usage of multiple data modalities in the prediction process.

Our experiments show that amplitude encoding stabilizes training, while angle encoding often introduces fluctuations. Among all configurations, the best performance is obtained with amplitude encoding and a single rotation layer. For reconstruction, QU-Net with entanglement converges faster than U-Net (25 epochs vs. 44) and reaches a lower MSE (0.00015 vs. 0.00017). For classification, QU-Net with no entanglement achieves an F1-score of 79% compared with 74.14% for U-Net, despite compressing images into a smaller latent space (7 vs. 128).

These results suggest that incorporating a quantum layer as the bottleneck enhances the U-Net with faster convergence and improved generalization. It also allows for compressing the data into a more efficient and expressive embedding.

2. Related Works

Classical deep learning approaches have demonstrated significant progress in automated skin cancer diagnosis, though challenges in data efficiency, model interpretability, and generalization remain active areas of research [12].

For the ISIC 2017 dataset, numerous studies employed deep learning architectures for skin lesion classification. For instance, ref. [13] proposed an ensemble framework combining a modified Inception-ResNet-v2 with EfficientNet-B4, enhanced with soft-attention. They achieved an F1-score of 79%. Ref. [14] introduced a ViT model with an optimized weighted cross-entropy loss function to address the dataset imbalance. They also added a regularization term to encourage the model to focus on the lesion area. They compared their ViT with well-established vision models. They reported an F1-score of 88.4% for ViT, in contrast to 85.2%, 83.5%, 86%, and 87% F1-scores for ResNet50, VGG19, ResNeXt, and standard ViT, respectively. Other approaches rely on transfer learning, such as the modified VGG-16 model by ref. [15], which obtained an F1-score of 71% on ISIC-2017.

These results are summarized in

Table 1. Deep learning models for skin lesion classification on ISIC-2017.

References	Method	Description	F1-Score
[13]	Inception-ResNet-v2 + EfficientNet-B4	Ensemble with Soft-Attention	79%
[14]	Improved ViT	Weighted loss + lesion-focused regularization	88.4%
[14]	ResNet50/VGG19/ResNeXt/ViT	Baseline comparison models	85.2%, 83.5%, 86%, 87%
[15]	Modified VGG-16	Transfer learning approach	71%

:

Recently, many machine learning applications in the field of healthcare tried to integrate quantum components in their model design, mostly in the form of a dressed quantum circuit [16]. For instance, in their effort to improve personalized medical treatment, ref. [17] tried to predict the drug response for cancer patients by employing a graph convolutional network (GCN) to extract the drug features and a convolutional neural network (CNN) to extract patient cell line features. They input the combined features to a deep quantum neural network (QNN) where predictions are made. The proposed model outperforms its classical counterpart based on a fully connected layer (FCL) by a 15% margin. Ref. [18] adopt a hybrid quantum-classical transfer learning for the detection of Alzheimer’s, they employ a pretrained ResNet34 to extract 512 features from brain MRI images and classify them using a quantum support vector machine (QSVM) [19], which obtained an accuracy of 97.2% compared with its classical counterpart, which obtained 92.2%. Also for brain MRI images, a quantum convolutional neural network (QCNN) was tested on the binary classification of instances from different datasets by [20], a 98.72% accuracy score was obtained by the QCNN outperforming the classical CNN 94.23% accuracy. In another work, ref. [21] use a multimodal approach where the different types of data transmitted using the internet of medical things (IoMT) to the cloud are combined in the prediction task. A QCNN is employed to extract important features from medical images that are fused with other modality features. The combined information is used to train a variational quantum circuit (VQC). The proposed strategy was tested on two tasks, breast cancer diagnosis and COVID-19 diagnosis, where the model obtained an accuracy of 97.07% and 97.61%, respectively. In their hybrid quantum model, ref. [22] integrates what they call a modified hardware efficient ansatz (MHEA) with the goal of classifying the multi-class cardiac pathologies. They also adopt a multimodal approach where the discriminative features are extracted from the heart MRI images and combined with the patient and the clinical data. They benchmark their proposed model with other classical and quantum models, incorporating a well-established ansatz-like variational quantum eigen-solver (VQE). The authors select the best features using recursive feature elimination with only 30 features. The test was conducted using different simulators and optimization techniques. Their proposed model, incorporating MHEA, outperforms all its counterparts with a maximum average improvement of 7.77% accuracy difference. It also speeds up the testing process by up to 60%. For the classification of thyroid cancers, ref. [23] employs random quantum circuits in a QCNN for discriminative features extraction from thyroid ultrasound images. The proposed QCNN obtained an accuracy of 97.63%, outperforming the classical models, which only obtained a 93.87% accuracy rate. For skin cancer, using the HAM10000 dataset [24], ref. [25] conducted an extensive benchmarking comparing various configurations of qubit rotations and Pauli gates in their proposed quanvolutional neural network (QuanvNN). Additionally, they evaluated the performance of several pretrained models, both in conjunction with quantum support vector machine (QSVM) and independently. They found that combining the Y-axis qubit rotation RY with the Pauli-Z gate in the quantum convolutional layer produced the optimal classification accuracy of 82.86%, outpacing the best results obtained with a classical pretrained MobileNet that obtained an accuracy of 73.42%.

These results are summarized in

Table 2. Quantum-enhanced models across different tasks.

References	Tasks	Method	Results
[17]	Patient drug response prediction	GCN for drugs + CNN for cell lines + QNN	15% better than FCL
[18]	Alzheimer detection	ResNet34 + QSVM	97.2% (QSVM) vs. 92.2% (classical)
[20]	Brain MRI binary classification	QCNN	98.72% (QCNN) vs. 94.23% (CNN)
[21]	Breast cancer and COVID-19 diagnosis	QCNN + VQC	97.07% (breast), 97.61% (COVID)
[22]	Cardiac pathology classification	QHEA + multimodal data	3.19–7.77% better than other models
[23]	Thyroid cancer classification	Quantum filter + QCNN	97.63% vs. 93.87% (classical)
[25]	Skin cancer classification (HAM10000)	QuanvNN + QSVM	82.86% (QNN) vs. 73.42% (MobileNet)

.

To investigate quantum advantage, ref. [4] conducted a large-scale study that tested 12 quantum models in 6 binary classification tasks and compared their efficiencies with out-of-the-box classical models. Their findings show that vanilla classic models outperform quantum models. Moreover, they show that removing entanglement from the design of these quantum models leads to an improvement in their performance, suggesting that quantumness may not always lead to improvement. ref. [4] shows that out of 55 publicly available papers with the terms quantum machine learning and outperform, 40% claim that quantum models outperform classical models and 50% claim that improving a quantum method such as optimizers [26,27], the strategy of pre-training [28] or the used ansatz [29,30] lead to better performance than the original models. Only 4% [31,32,33] found that the quantum models did not outperform the classical ones. They attribute this to the noise levels in current quantum hardware, suggesting that hardware limitations rather than the quantum models themselves were responsible for the lack of superior performance.

3. Proposed Method

In this section, we will provide more details about the proposed approach and methodology for QU-Net.

3.1. Data

The ISIC 2017 [6] dataset, released by the International Skin Imaging Collaboration (ISIC), is an open-source collection of high pixel-density dermoscopic images aimed at advancing melanoma detection through automated image analysis. All images in this dataset are dermoscopic RGB photographs captured using standardized optical devices. It contains the following:

• 2750 skin cancer images with different sizes.
• Classification labels
• Segmentation masks
• Metadata (age and sex)

The dataset comprises 2000 training images, 150 validation images and 600 test images, each accompanied by corresponding ground truth data and additional information about the patient. It is structured to support two primary tasks: lesion segmentation and disease classification. The dataset includes images from various international clinical centers, ensuring a diverse representation of skin lesion types and imaging conditions.

As shown in Figure 1, it is challenging to visually distinguish between benign and malignant skin cancer due to their similar appearances.

After conducting exploratory data analysis on the ISIC-2017 dataset, we identified two major issues. Firstly, the dataset includes images from various sources, resulting in inconsistent image sizes. The second issue is the severe imbalance of the distribution of the skin labels, 81.3% of the cases present in the dataset are benign while 18.7% are malignant. This means that the benign class outnumbers the malignant class approximately 4 to 1.

To address the issues of data heterogeneous sizes while maintaining a high resolution, we standardize the images by resizing them to a uniform shape of (512, 512, 3) using bilinear interpolation. Additionally, we crop images to (400, 400, 3) to focus on regions of interest where the tumors are centered, while excluding areas of normal skin, which reduces the model-trainable parameters. We also augment training data to improve model performance and reduce the effect of imbalance on class learning. Figure 2 shows the image processing steps.

3.2. Proposed Model

In 2015, ref. [34] introduced the U-Net architecture. While its original goal was image segmentation, where it reduced reliance on handcrafted techniques [35], it became widely used for many purposes like image de-noising, generative AI and image embedding, which will be the use-case of our work.

As shown in Figure 3, U-Net is composed of four main parts:

• Contracting Path (Encoder): The encoder is responsible for capturing context by progressively down-sampling the input image. This is achieved by a series of convolutional and max-pooling layers that reduce the spatial dimensions of the feature maps while increasing the depth.
• Bottleneck: The bottleneck serves as a bridge between the contracting and the expansive paths. Its design compresses the data flow into a limited set of values to capture the most essential information.
• Expansive Path (Decoder): The decoder is responsible for producing the desired output by expanding the feature maps. It consists of a series of up-sampling techniques (transposed convolutions, unpooling, …) and information fusion operations (concatenation, pixel-wise addition, …) with corresponding high-resolution features from the contracting path.
• Skip Connections: They serve to fuse up-sampled information from the bottleneck with high-resolution feature maps extracted by the encoder. They connect each layer in the encoder with its corresponding symmetric layer on the decoder.

Given the current limitations of quantum hardware and the noisy intermediate-scale quantum (NISQ) [36,37], Appendix A imposes constraints such as a limited number of qubits. Using a quantum layer as the bottleneck in the U-Net architecture aligns with the concept of a bottleneck, which is designed to compress the data flow into a limited set of values, making efficient use of the available quantum resources.

To test the efficiency of QU-Net, we compare it with a classical U-Net architecture. We define the two models with the same architecture, hyperparameters, optimizer, loss function, and training procedure. The only difference is that the QU-Net’s bottleneck layer is a parameterized quantum circuit (PQC), while the classical U-Net uses a fully connected layer.

3.2.1. Parametrized Quantum Circuits

Parameterized Quantum Circuits (PQCs) [38] are the building blocks of Quantum Neural Networks (QNNs). They consist of quantum circuits with tunable parameters, which can be optimized similarly to the weights in classical neural networks. These parameters are adjusted iteratively to minimize a predefined cost function, which enables the quantum model to learn and generalize from data. Figure 4 shows a simple PQC of two qubits parametrized by a Y-axis rotation angle:

A PQC [38] consists of a sequence of quantum gates, which can be classified into two types:

• Fixed gates: These are predefined operations that establish entanglement or transformations that do not change during training (CNOT, Hadamard, etc.).
• Parameterized gates: These are quantum operations that depend on adjustable parameters, usually represented as rotation angles (Rx, Ry, etc.).

The most fundamental quantum gates are listed in

Table 3. Quantum Gates: Symbols, Matrices, and Descriptions.

Gate Name	Qubits	Circuit Symbol	Unitary Matrix	Description
Pauli-X (NOT)	1		$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$	Analogous to classical NOT gate: switches $|0\rangle$ to $|1\rangle$ and vice versa
Pauli-Y	1		$\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$	Rotation through $\pi$ radians around Bloch sphere Y-axis
Pauli-Z (phase flip)	1		$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$	Rotation through $\pi$ radians around Bloch sphere Z-axis
X-Rotation	1		$\left(\begin{array}{cc}cos\frac{\theta }{2}& -isin\frac{\theta }{2}\\ -isin\frac{\theta }{2}& cos\frac{\theta }{2}\end{array}\right)$	Rotates state vector about the Bloch sphere X-axis by $\theta$
Y-Rotation	1		$\left(\begin{array}{cc}cos\frac{\theta }{2}& -sin\frac{\theta }{2}\\ sin\frac{\theta }{2}& cos\frac{\theta }{2}\end{array}\right)$	Rotates state vector about the Bloch sphere Y-axis by $\theta$
Z-Rotation	1		$\left(\begin{array}{cc}{e}^{-i\frac{\theta }{2}}& 0\\ 0& e^{i\frac{\theta }{2}}\end{array}\right)$	Rotates state vector about the Bloch sphere Z-axis by $\theta$
Hadamard	1		${\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$	Transforms a basis state into an even superposition of the two basis states
CNOT (Controlled-NOT)	2		$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$	Applies Pauli-X to target qubit if control qubit is $|1\rangle$
SWAP	2		$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$	Swaps the states of two qubits
Rot ($R(\varphi ,\theta ,\omega )$)	1		$\left(\begin{array}{cc}{e}^{-i(\varphi +\omega )/2}cos(\theta /2)& e^{-i(\varphi -\omega )/2}sin(\theta /2)\\ -e^{i(\varphi -\omega )/2}sin(\theta /2)& e^{i(\varphi +\omega )/2}cos(\theta /2)\end{array}\right)$	A general rotation gate that combines rotations around Z and Y axes: $R_Z\left(\omega \right)R_Y\left(\theta \right)R_Z\left(\varphi \right)$. Provides flexible state preparation and transformation.

.

A PQC is generally represented as a sequence of unitary transformations, parameterized by a set of angles $\theta$:

$$U\left(\theta \right)=U_L\left({\theta }_L\right)\cdots U_2\left({\theta }_2\right)U_1\left({\theta }_1\right)$$

(1)

where

• $U_i\left({\theta }_i\right)$ represents the unitary transformation at the i-th layer. It may contain multiple parameters or be parameter-free.
• L is the total number of layers.
• $\theta =\{{\theta }_1,{\theta }_2,\dots ,{\theta }_L\}$ denotes the set of all trainable parameters.

The optimization of these parameters is typically performed using a hybrid quantum-classical approach, where a classical optimizer updates the parameters based on a quantum-measured cost function.

3.2.2. Quantum State Preparation and Encoding

Since quantum circuits operate exclusively on quantum states, an essential step before applying PQCs is the encoding of classical data into quantum states [39]. This encoding process determines how classical information is mapped onto the Hilbert space. Various encoding techniques exist, including:

• Basis Encoding: Directly encoding classical bits into qubit states.
• Amplitude Encoding: Encoding classical data as quantum amplitudes, allowing compact representation.
• Angle Encoding: Encoding data using rotation angles of quantum gates.

The choice of encoding method impacts the expressiveness and efficiency of the quantum model. Once the data are encoded, PQCs manipulate the quantum states through entanglement and transformation operations to extract meaningful patterns for tasks such as classification, regression, or generative modeling. In this context, we will experiment with amplitude and angle encoding.

In amplitude encoding, classical data values are encoded into the probability amplitudes of a quantum state. Given a classical data vector $\mathbf{x}=[x_1,x_2,\dots ,x_N]$, n qubits are required to encode $\mathbf{x}$ onto a quantum state, where $N\le 2^n$, it is encoded as follows:

$$\left|\psi \right.〉=\sum _{i=0}^{2^n-1}{\displaystyle \frac{x_i}{\parallel \mathbf{x}\parallel }}\left|i\right.〉$$

(2)

Here, $|i\rangle$ is the computational basis state corresponding to the binary representation of i, expressed as $|b_1{b}_2\dots b_n\rangle$, where $b_j\in \{0,1\}$. ${\displaystyle \frac{x_i}{\parallel \mathbf{x}\parallel }}$ are the normalized data amplitudes, where $\parallel \mathbf{x}\parallel$ is the Euclidean norm of the vector $\mathbf{x}$, defined as follows:

$$\parallel \mathbf{x}\parallel =\sqrt{\sum _{i=0}^{2^n-1}{\left|x_i\right|}^2}$$

(3)

The normalization ensures that this condition is satisfied:

$$\sum _{i=0}^{2^n-1}{\left|{\displaystyle \frac{x_i}{\parallel \mathbf{x}\parallel }}\right|}^2=1$$

(4)

In amplitude encoding, the number of qubits n required to encode the data is logarithmic to the size of the data vector $\mathbf{x}$. Specifically, for a data vector of size N, the number of qubits needed to perform amplitude encoding of the data is as follows:

$$n_{qubits}=\lceil {log}_{2}N\rceil$$

(5)

This means that n qubits can represent $2^n$ data points, enabling the encoding of an exponentially large dataset into a quantum state. We use amplitude encoding to prepare quantum states, allowing us to load up to $2^7=128$ features from the QU-Net encoder onto the quantum layer:

$$|\psi \rangle =\sum _{i=0}^{127}{c}_i|i\rangle ={\displaystyle \frac{x_0}{\parallel \mathbf{x}\parallel }}\underset{7\mathrm{qubits}}{\underbrace{|0000000\rangle }}+{\displaystyle \frac{x_1}{\parallel \mathbf{x}\parallel }}|0000001\rangle +\cdots +{\displaystyle \frac{x_{127}}{\parallel \mathbf{x}\parallel }}|1111111\rangle$$

(6)

In angle encoding, each classical feature is encoded as a rotation angle applied to a single qubit. Given a data vector $\mathbf{x}=[x_1,x_2,\dots ,x_n]$, one qubit is used per feature, and the quantum state is prepared as follows:

$$|\psi \left(\mathbf{x}\right)\rangle =\underset{j=1}{\overset{n}{⨂}}{R}_Y\left(x_j\right)|0\rangle =\underset{j=1}{\overset{n}{⨂}}\left(cos{\displaystyle \frac{x_j}{2}}|0\rangle +sin{\displaystyle \frac{x_j}{2}}|1\rangle \right).$$

(7)

Each feature $x_j$ is mapped to a rotation angle (typically after scaling), and no global normalization is required. The number of qubits scales linearly with the number of features:

$$n_{\mathrm{qubits}}=n.$$

(8)

Figure 5 shows the used parametrized quantum circuits. It employs 7 qubits simulated using PennyLane’s default.qubit backend.

This is followed by the parametrized part, which incorporates three rotational gates per qubit. Each qubit is transformed by a Rot gate ($R(\varphi ,\theta ,\omega )$) gate. For a single qubit, the rotation matrix is as follows:

$$R(\varphi ,\theta ,\omega )=\left(\begin{array}{cc}{e}^{-i(\varphi +\omega )/2}cos(\theta /2)& e^{-i(\varphi -\omega )/2}sin(\theta /2)\\ -e^{i(\varphi -\omega )/2}sin(\theta /2)& e^{i(\varphi +\omega )/2}cos(\theta /2)\end{array}\right)$$

(9)

For a single qubit in the state $|\psi \rangle =\alpha |0\rangle +\beta |1\rangle$, applying $R(\varphi ,\theta ,\omega )$ results in the following:

$$|{\psi }^{\prime }\rangle =R(\varphi ,\theta ,\omega )|\psi \rangle =\left(\begin{array}{cc}{e}^{-i(\varphi +\omega )/2}cos(\theta /2)& e^{-i(\varphi -\omega )/2}sin(\theta /2)\\ -e^{i(\varphi -\omega )/2}sin(\theta /2)& e^{i(\varphi +\omega )/2}cos(\theta /2)\end{array}\right)\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)$$

(10)

These parameterized rotations allow for manipulating the qubits’ states by tunable angles $\varphi$, $\theta$, and $\omega$.

After applying the rotational gates, entanglement can be introduced using CNOT gates. The CNOT gate acts on two qubits and is represented by the following matrix:

$$\mathrm{CNOT}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$$

(11)

For example, if the control qubit is in a superposition:

$$|{\psi }_{\mathrm{control}}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\right|0\rangle +|1\rangle )$$

(12)

and the target qubit is in the $|0\rangle$ state:

$$|{\psi }_{\mathrm{target}}\rangle =|0\rangle$$

(13)

the combined state is as follows:

$$|{\psi }_{\mathrm{combined}}\rangle =|{\psi }_{\mathrm{control}}\rangle \otimes |{\psi }_{\mathrm{target}}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ 1\end{array}\right)\otimes \left(\begin{array}{c}1\\ 0\end{array}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right)$$

(14)

Applying the CNOT gate results in the final entangled state:

$$|{\psi }_{\mathrm{entangled}}\rangle =\mathrm{CNOT}·|{\psi }_{\mathrm{combined}}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\right|00\rangle +|11\rangle )$$

(15)

When all qubits are connected with CNOT gates, a change in one qubit state influences the states of the remaining qubits.

The circuit is concluded with Pauli-Z measurements on each qubit, extracting the expectation values of the qubits in the Z-basis. The Pauli-Z operator is defined as follows:

$$Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

(16)

For a single qubit in the state:

$$|\psi \rangle =\alpha |0\rangle +\beta |1\rangle$$

(17)

where $\alpha$ and $\beta$ are complex amplitudes, the probabilities of observing $|0\rangle$ and $|1\rangle$ are given by the following:

$$P\left(0\right)={\left|\alpha \right|}^2,P\left(1\right)={\left|\beta \right|}^2$$

(18)

The expectation value of the Pauli-Z operator for this state is calculated as follows:

$$\langle Z\rangle =\langle \psi \left|Z\right|\psi \rangle$$

(19)

Substituting $|\psi \rangle =\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)$, we have the following:

$$\langle Z\rangle =\left(\begin{array}{cc}{\alpha }^{*}& {\beta }^{*}\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)$$

(20)

Performing the matrix multiplication:

$$\langle Z\rangle =\left(\begin{array}{cc}{\alpha }^{*}& {\beta }^{*}\end{array}\right)\left(\begin{array}{c}\alpha \\ -\beta \end{array}\right)$$

(21)

This simplifies to the following:

$$\langle Z\rangle ={\left|\alpha \right|}^2-{\left|\beta \right|}^2$$

(22)

Thus, the expectation value of the Pauli-Z operator corresponds to the difference in probabilities of the qubit being in the $|0\rangle$ and $|1\rangle$ states:

$$\langle Z\rangle =P\left(0\right)-P\left(1\right)$$

(23)

Hence, the Pauli-Z measurement is analogous to the ArgMax operation in classical machine learning, where the index of the highest predicted probability, corresponding to the predicted class, is returned. These measurements convert the quantum state back into a classical representation, which is subsequently passed to the decoder part of the model.

Figure 6 shows the architecture of the proposed QU-Net and the evolution of the image shape throughout the model. Initially, we employ a self-supervised learning approach, wherein the objective of QU-Net is to reconstruct the input images. The mean squared error is computed between the output of the model and the image itself and optimized to reduce the loss between the predictions and the identity. This method enables the model to learn a general representation of the images while preserving task independence. Consequently, the bottleneck layer generates a compressed representation of the image that remains uninfluenced by the optimization of any specific task. This ensures that the features captured are general and not biased toward a particular objective; therefore, they can represent the original data.

The rationale behind this approach is to take advantage of the compressed representations learned for the classification of skin cancer, where we try to distinguish between benign and malignant cases. As illustrated in Figure 7, by concatenating these representations with metadata features, we can implement a multi-modal approach that integrates both image data and relevant metadata for skin cancer classification. This approach not only structures the new data representation and allows space-efficient storage, but also enables the use of various classifiers such as Random Forest, XGBoost, and MLP to classify the images.

Similarly, we train a classical U-Net architecture as illustrated in Figure 8 to compare its reconstruction performance with that of QU-Net.

Additionally, we evaluate the learned embedding from the classical U-Net against the quantum embedding in the classification task to assess their respective capabilities in representing the image data. As shown in Figure 9, the output of the encoder is concatenated with other relevant features to create a structured dataset that can be categorized using classical models such as SVM, MLP, random forest…

In QU-Net, the use of 7 qubits in the quantum layer results in the skin images being embedded in a vector of 7 values. The classical U-Net generates an image embedding vector of 128 values. Approximately, all models have 3.6M Parameters.

4. Experimental Results

In this section, we compare the results of the different PQC configurations. The combination with the best results will be compared with a classic U-Net on the reconstruction and the classification of the images represented by the bottleneck embedding, thereby assessing their capability of compressing the information in a meaningful way.

4.1. Reconstruction

Figure 10 show, respectively, images of skin cancer, their reconstructed versions generated by the different models, and the error.

Although all models achieved comparable results in image reconstruction (Figure 10), the loss evolution across experiments reveals notable differences in generalization capability among the architectures.

While training curves showed similar convergence behavior overall (Figure 11), inconsistencies became more apparent on unseen data (Figure 12). Angle encoding-based QU-Nets showed more fluctuations on unseen data. Classical U-Net exhibited more oscillations during validation, whereas QU-Net variants based on amplitude encoding showed smoother and more stable learning dynamics despite operating with a smaller latent representation (7 latent values vs. 128 for U-Net).

In particular, the best reconstruction performance on unseen data was achieved by the Amplitude encoding QU-Net with a single fully entangled layer, converging in 25 epochs and producing the lowest reconstruction error among all the models evaluated, as shown in Figure 13 and Figure 14.

More broadly, amplitude-based PQCs displayed faster and more stable convergence trends compared with angle-based configurations (13, 25, 29 epochs vs. 55, 69, 77 epochs), with the classical U-Net falling in between (44 epochs). These results suggest that amplitude encoding may act as a stabilizing and generalizing factor, particularly in scenarios where the latent bottleneck is extremely compressed.

Table 4. Average train, validation, and test MSE per image of QU-Net variants and U-Net.

	Amplitude			Angle
Loss (MSE)	1-Ent	2-Ent	1-NoEnt	1-Ent	2-Ent	1-NoEnt	U-Net
Train	0.000106	0.000079	0.000075	0.000084	0.000071	0.000072	0.000115
Validation	0.000133	0.000160	0.000155	0.000170	0.000146	0.000149	0.000147
Test	0.000151	0.000172	0.000172	0.000190	0.000165	0.000163	0.000172

Bold values indicate the best (lowest) result in each row.

compares the average mean squared error (MSE) per image of U-Net and QU-Net variants on train, validation and test sets. The results suggest that QU-Net with amplitude encoding and single entangled layer gives the best reconstruction of skin cancer images.

4.2. Classification

We quantify the quality of the learned embeddings with the performance of simple models such as logistic regression, k-nearest neighbors, and tree-based classifiers, trained on these embeddings to classify instances as either benign or malignant. Logistic Regression (LR) serves to measure the linear separability of the classes. K-Nearest Neighbors (KNN) is used to examine whether instances from similar classes are grouped close to each other based on the Euclidean distance. Tree-based models such as Random Forest (RF) are also utilized to verify the purity of leaf splits. Taking into consideration the severe imbalance of the classes, we utilize the F1-score as our main metric to measure the model’s performance. We also use stratified cross-validation with 10 folds, where class distributions in the train and test sets are the same across all folds.

Figure 15 shows the performance of the classification models on the embeddings of the different QU-Net variants. Best results are obtained with XGBoost.

A comparison of the classification models on the QU-Net variants embeddings is visualized using a heatmap in Figure 16. Best classification results for each of the embeddings are shown in Figure 17.

Figure 18 shows the average F1-score and accuracy in all models for each of the encoding techniques. Overall, amplitude encoding versions led to slightly better measures. Figure 19 illustrates the mean impact of the entanglement Strategy and the number of layers on the F1-score. On average, the PQC with a single layer and no entanglement obtained better results. A trend can be observed, as the PQC becomes more complex, the results tend to decrease, suggesting that, unlike classical ANNs, increasing PQC depth does not necessarily lead to performance improvement.

Table 5. Comparison of classification performance on U-Net and QU-Net embeddings.

Classifier	U-Net		QU-Net
	F1 Score (%)	Accuracy (%)	F1 Score (%)	Accuracy (%)
Random Forest (RF)	74.14	81.33	75.86	80.17
LightGBM (LGBM)	72.93	79.33	76.68	80.17
XGBoost (XGB)	77.38	81.33	78.27	80.50
Logistic Regression	70.78	79.33	73.07	80.17
K-Nearest Neighbors	70.43	74.00	76.05	78.33
Soft-Voting (RF-XGB-LGBM)	73.33	80.00	79.03	81.83

Bold values indicate the best (highest) result in each column.

shows a comparison of the classification results of the selected models using the embeddings from U-Net and QU-Net variant with amplitude encoding, no entanglement and one layer. Both were concatenated with metadata.

The results presented in Table 5 demonstrate that QU-Net embeddings consistently outperform U-Net embeddings across all the evaluated classifiers, both in terms of F1 Score (79.03% vs. 74.14%)and Accuracy (81.83% vs. 81.33%). This indicates that the QU-Net representations are more discriminative and informative, although they have significantly fewer dimensions compared with the U-Net embedding (7 vs. 128).

An observation emerges from the performance of the soft voting classifier. Soft voting works by averaging the class probabilities predicted by each base classifier and selecting the class with the highest average probability. When using U-Net embeddings, the soft voting strategy underperforms relatively to some of its individual models. This suggests that probability magnitudes that reflect the confidence of the base classifiers were not sufficient to achieve an improvement through averaging. In contrast, the soft voting strategy achieves the highest performance metrics with QU-Net embeddings compared with individual classifiers. This indicates that models trained on QU-Net embeddings were more confident in their predictions, allowing the ensemble to benefit from probabilistic consensus.

Table 6. Comparison of deep learning models for skin lesion classification on ISIC-2017. Our results represent baseline classifiers performance on quantum embeddings.

References	Method	Description	F1-Score
[15]	Modified VGG-16	Transfer learning approach	71%
[13]	Inception-ResNet-v2 + EfficientNet-B4	Ensemble with Soft-Attention	79%
Ours	QU-Net (quantum-enhanced U-Net)	Quantum embeddings + metadata + baseline classifier	79.03%
[14]	ResNet50/VGG19/ ResNeXt/ViT	Baseline comparison models	85.2%, 83.5%, 86%, 87%
[14]	Improved ViT	Weighted loss + lesion-focused regularization	88.4%

shows a comparison of existing methods with our approach. Although no hyperparameter tuning, feature selection, or classification improvement techniques were used in this study, our results seem competitive with works focused on classification.

An additional experiment was conducted where metadata features were not appended to the latent embeddings of U-Net and QU-Net.

Table 7. F1-score Comparison of U-Net vs. QU-Net (No Metadata).

Classifier	U-Net F1	QU-Net F1	Difference (Q − U)
Random Forest	71.11	72.97	+1.86
LightGBM	72.29	73.23	+0.94
XGBoost	69.77	73.73	+3.96
Logistic Regression	70.78	71.80	+1.02
K-Nearest Neighbors	72.39	71.87	−0.52
Soft-Voting (RF–XGB–LGBM)	71.00	72.78	+1.78

Bold values indicate the best (highest) result in each row.

shows the difference in classification results on the embeddings without metadata. We can observe that generally, QU-Net embeddings obtained better F1-scores.

Table 8. Comparison of classification performance on U-Net and QU-Net embeddings with and without metadata.

Classifier	U-Net		QU-Net
	With Metadata (%)	Without Metadata (%)	With Metadata (%)	Without Metadata (%)
Random Forest	F1: 74.14	F1: 71.11	F1: 75.86	F1: 72.97
	Acc: 81.33	Acc: 80.00	Acc: 80.17	Acc: 80.00
LightGBM	F1: 72.93	F1: 72.29	F1: 76.68	F1: 73.23
	Acc: 79.33	Acc: 80.00	Acc: 80.17	Acc: 78.83
XGBoost	F1: 77.38	F1: 69.77	F1: 78.27	F1: 73.73
	Acc: 81.33	Acc: 77.33	Acc: 80.50	Acc: 78.33
Logistic Regression	F1: 70.78	F1: 70.78	F1: 73.07	F1: 71.80
	Acc: 79.33	Acc: 79.33	Acc: 80.17	Acc: 80.50
K-Nearest Neighbors	F1: 70.43	F1: 72.39	F1: 76.05	F1: 71.87
	Acc: 74.00	Acc: 76.00	Acc: 78.33	Acc: 76.00
Soft-Voting (RF-XGB-LGBM)	F1: 73.33	F1: 71.00	F1: 79.03	F1: 72.78
	Acc: 80.00	Acc: 78.00	Acc: 81.83	Acc: 78.83

Bold values indicate the best result between with and without metadata configurations for each model type (U-Net/QU-Net) per row.

shows a comparison of the classification results of QU-Net and U-Net with and without the multi-modal approach.

We can observe that across all classifiers, the inclusion of metadata improved the classification performance for QU-Net embeddings. The multi-modal approach further enhanced the leaf-purity of the tree-based classifiers. Additionally, the improvement of the Logistic Regression F1-score suggests that the metadata increased the linear separability of classes. Although a decrease can be observed in accuracy, it can be explained by the accuracy’s dependency on class balance. Also, KNN inference amelioration suggests that the metadata reinforced the distance structure of the latent space by reducing intra-class distances while increasing inter-class separability, which aligns well with KNN’s reliance on Euclidean proximity for classification. This indicates that the quantum-enhanced latent space can benefit from auxiliary and meaningful information.

The effect of the multi-modal approach on U-Net embeddings was less consistent. While tree-based classifiers did show performance gains, Logistic Regression produced identical results with and without metadata, suggesting that the added metadata did not introduce any new linear separability into the learned feature space. Furthermore, the same metadata degraded the performance of KNN when added to U-Net embeddings. This implies that the additional features reduced the distance-based consistency of the class clusters. These observations suggest that QU-Net embeddings naturally position similar instances close to each other, and metadata further improves this geometric alignment, whereas U-Net embeddings appear more sensitive to such feature concatenation.

Furthermore, this finding underscores the effectiveness of a multi-modal approach that integrates image-based features with structured and meaningful patient data in the classification of skin cancer.

5. Conclusions and Future Perspective

In this work, we proposed a novel architecture illustrated in Figure 6 that incorporates a quantum layer in the form of a parametrized quantum circuit as the bottleneck of U-Net. We refer to it as QU-Net. The results show that this change has led to several improvements. These include enhanced generalization to unseen images in the reconstruction of skin cancer images. Both models had similar training performance Figure 10, but QU-Net was more able to generalize to unseen images Figure 12. This means that adding a quantum layer helped reduce the overfitting of training data. The quantum layer also enabled faster convergence. The required epochs for QU-Net and U-Net to reach the stable zone are, respectively, 25 and 44, as shown in Figure 12. Additionally, QU-Net obtained better reconstruction error in training, validation and test data, Table 4. Furthermore, the quantum layer led to a more representative embedding of skin cancer images and a more efficient data compression (7 vs. 128). This is demonstrated by the difference in results Table 5 where QU-Net embeddings enabled better classification compared with U-Net embeddings. Also, the multi-modal approach where we train QU-Net Figure 7 and U-Net Figure 9 to learn a general and task-independent structured representation of skin images, and combine it with metadata like patient age and sex, further improved the classification results Table 8.

We also evaluated six PQC configurations, varying the encoding method (amplitude vs. angle), circuit depth, and entanglement strategy. The findings show that amplitude encoding stabilizes training, whereas angle encoding introduces fluctuations. The best reconstruction performance was achieved using amplitude encoding with one variational layer and entanglement, while the best classification performance resulted from the same configuration without entanglement. This suggests that lighter PQCs may generalize better.

Based on our results, it appears that the proposed quantum-enhanced U-Net outperformed the classical version in the reconstruction and classification tasks. Due to the lack of quantum computing resources, a hybrid approach like ours, where we combine classical and quantum machine learning techniques, is necessary to make efficient use of available quantum resources.

In the future, we plan to explore the potential of the suggested architecture in the segmentation task. The chosen quantum circuit used as a bottleneck can be changed to a more sophisticated circuit to put more emphasis on the margin between different class embeddings in the quantum Hilbert space. We also plan to explore an end-to-end hybrid quantum classifier to make the gradient wave back-propagate from error computing to input data processing.