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arXiv:2502.19970 · Quantum Science and Technology

Quantum generative classification with mixed states

Diego H. Useche, Sergio Quiroga-Sandoval, Sebastián Leonardo L Molina Díaz, Vladimir Vargas-Calderón, Juan E. Ardila-García, Fabio Augusto Gonzalez

Published Jul 23, 2025• 0 citations
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Abstract

Abstract Classification can be performed using either a discriminative or a generative learning approach. Discriminative learning consists of constructing the conditional probability of the outputs given the inputs, while generative learning consists of constructing the joint probability density of the inputs and outputs. Although most classical and quantum methods are discriminative, there are some advantages of the generative learning approach. For instance, it can be applied to unsupervised learning, statistical inference, uncertainty estimation, and synthetic data generation. In this article, we present a quantum generative multiclass classification strategy, called quantum generative classification (QGC). This model uses a variational quantum algorithm to estimate the joint probability density function of features and labels of a data set by means of a mixed quantum state. We also introduce a quantum map called quantum-enhanced Fourier features, which leverages quantum superposition to prepare high-dimensional data samples in quantum hardware using a small number of qubits. We show that the QGC algorithm can be viewed as a Gaussian mixture that reproduces a kernel Hilbert space of the training data. In addition, we developed a hybrid quantum–classical neural network that shows that it is possible to perform generative classification on high-dimensional data sets. The method was tested on various low- and high-dimensional data sets including the 10-class MNIST and Fashion-MNIST data sets, illustrating that the generative classification strategy is competitive against other previous quantum models.

DOI
10.1088/2058-9565/adf350

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Abstract Classification can be performed using either a discriminative or a generative learning approach. Discriminative learning consists of constructing the conditional probability of the outputs given the inputs, while generative learning consists of constructing the joint probability density of the inputs and outputs. Although most classical and quantum methods are discriminative, there are some advantages of the generative learning approach. For instance, it can be applied to unsupervised learning, statistical inference, uncertainty estimation, and synthetic data generation. In this article, we present a quantum generative multiclass classification strategy, called quantum generative classification (QGC). This model uses a variational quantum algorithm to estimate the joint probability density function of features and labels of a data set by means of a mixed quantum state. We also introduce a quantum map called quantum-enhanced Fourier features, which leverages quantum superposition to prepare high-dimensional data samples in quantum hardware using a small number of qubits. We show that the QGC algorithm can be viewed as a Gaussian mixture that reproduces a kernel Hilbert space of the training data. In addition, we developed a hybrid quantum–classical neural network that shows that it is possible to perform generative classification on high-dimensional data sets. The method was tested on various low- and high-dimensional data sets including the 10-class MNIST and Fashion-MNIST data sets, illustrating that the generative classification strategy is competitive against other previous quantum models.
This paper presents a novel framework for Quantum Generative Adversarial Networks (QGANs) that integrates Quantum Kernel Learning into the discriminator architecture. By replacing the classical discriminator with a Quantum Kernel Discriminator (QKD), the proposed model leverages high-dimensional Hilbert space mappings to enhance separability between real and generated data. The quantum discriminator employs Pauli Feature Maps and Fidelity Quantum Kernels to evaluate sample similarity in the quantum space, leading to improved training stability and reduced mode collapse. Empirical results on the MNIST dataset demonstrate faster convergence, higher fooling scores, and improved sample diversity compared to classical GANs. Additionally, the application of Quantum Kernel Methods to the Iris dataset highlights the generalization capabilities of the QKD approach. These findings establish the potential of hybrid quantum-classical architectures in advancing generative modeling and open new avenues for applying quantum machine learning in low-data and high-complexity domains.
Quantum computers have the unique ability to operate relatively quickly in high-dimensional spaces-this is sought to give them a competitive advantage over classical computers. In this work, we propose a novel quantum machine learning model called the Quantum Discriminator, which leverages the ability of quantum computers to operate in the high-dimensional spaces. The quantum discriminator is trained using a quantum-classical hybrid algorithm in [Formula: see text] time, and inferencing is performed on a universal quantum computer in [Formula: see text] time. The quantum discriminator takes as input the binary features extracted from a given datum along with a prediction qubit, and outputs the predicted label. We analyze its performance on the Iris and Bars and Stripes data sets, and show that it can attain 99% accuracy in simulation.
Multiclass classification is of great interest for various applications, for example, it is a common task in computer vision, where one needs to categorize an image into three or more classes. Here we propose a quantum machine learning approach based on quantum convolutional neural networks for solving the multiclass classification problem. The corresponding learning procedure is implemented via TensorFlowQuantum as a hybrid quantum-classical (variational) model, where quantum output results are fed to the softmax activation function with the subsequent minimization of the cross entropy loss via optimizing the parameters of the quantum circuit. Our conceptional improvements here include a new model for a quantum perceptron and an optimized structure of the quantum circuit. We use the proposed approach to solve a 4-class classification problem for the case of the MNIST dataset using eight qubits for data encoding and four ancilla qubits; previous results have been obtained for 3-class classification problems. Our results show that the accuracy of our solution is similar to classical convolutional neural networks with comparable numbers of trainable parameters. We expect that our findings will provide a new step toward the use of quantum neural networks for solving relevant problems in the NISQ era and beyond.
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Quantum Machine Learning (QML) has emerged as a paradigm that combines the computational advantages of quantum computing with the predictive capabilities of machine learning to address complex data classification problems. As data dimensionality increases rapidly and classical learning algorithms face scalability constraints, QML leverages quantum parallelism, entanglement, and high-dimensional Hilbert space representations to enhance learning performance. This paper reviews and analyses advanced QML algorithms, including Quantum Support Vector Machines (QSVM), Variational Quantum Circuits (VQC), Quantum Neural Networks (QNN), and Quantum Kernel Estimation, for optimizing both binary and multi-class classification tasks under high-complexity conditions. The proposed QML framework is evaluated on benchmark datasets like MNIST, the Breast Cancer Wisconsin (BCW) dataset, and synthetic nonlinear datasets, and is compared against classical machine learning baselines, including Support Vector Machines (SVM), Random Forests (RF), and Deep Neural Networks (DNN). The results demonstrate notable improvements in classification accuracy (up to 96.8%), decision margins, and computational efficiency in quantum-suitable data regimes, while also highlighting current limitations related to noise, circuit depth, and hardware constraints. Overall, the study presents a unified QML framework, theoretical formulations, and experimental evaluations that illustrate the potential of quantum algorithms for next-generation classification tasks.
Quantum machine learning is considered one of the current research fields with great potential. In recent years, Havl\'{\i}\ifmmode \check{c}\else \v{c}\fi{}ek et al. [Nature (London) 567, 209 (2019)] have proposed a quantum machine learning algorithm with quantum-enhanced feature spaces, which effectively addressed a binary classification problem and offered a potential pathway to achieving quantum advantage. However, a straightforward binary classification algorithm falls short in solving multiclass classification problems. In this paper, we propose a quantum algorithm that rigorously demonstrates that quantum kernel methods enhance the efficiency of multiclass classification in real-world applications, providing quantum-enhanced performance. To demonstrate this, we design six distinct quantum kernels within the quantum algorithm to map input data into quantum state spaces and estimate the corresponding quantum kernel matrices. The results from quantum simulations reveal that the quantum algorithm outperforms its classical counterpart in handling six real-world multiclass classification problems. Furthermore, we leverage a variety of performance metrics to comprehensively evaluate the classification and generalization performance of the quantum algorithm. The results demonstrate that the quantum algorithm achieves superior classification and better generalization performance relative to classical counterparts.