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arXiv:2509.10190 · EPJ Quantum Technology

Quantum kernel and HHL-based support vector machines for multi-class classification

Gabriela Pinheiro, Donovan Slabbert, Luis Antônio Brasil Kowada, Francesco Petruccione

Published Mar 13, 2026• 3 citations

Abstract

Abstract We compare two quantum approaches that use support vector machines for multi-class classification on a reduced Sloan Digital Sky Survey () dataset: the quantum kernel-based QSVM and the Harrow-Hassidim-Lloyd least-squares SVM (HHL LS-SVM). Both one-vs-rest and two-step hierarchical classification schemes were implemented. The QSVM involves angle encoding of ten features, two unitary operator blocks consisting of rotational operator gates, and a projective measurement that projects the final state to the zero state. The HHL-based method involves solving a system of linear equations using the HHL algorithm and using the solution in a support vector machine approach. The results indicate that the QSVM outperforms HHL LS-SVM in most cases. HHL LS-SVM performs somewhat competitively in selected cases, particularly when isolating galaxies (majority), however, it also performs poorly in others, especially when isolating QSOs (minority). Comparisons with classical SVMs confirm that quantum and classical methods achieve broadly similar performance, with classical models performing slightly ahead overall. Scaling analysis reveals a trade-off: QSVM performance suffers from quadratic scaling with the number of samples and features, but benefits from explicit feature representation during training, while HHL LS-SVM scales essentially constantly, with moderate fluctuations, but suffers from limited representative elements. The HHL-based method is also highly noise-sensitive. These results suggest that QSVM performs better overall under the experimental conditions considered here and is likely to be more amenable to near-term hardware execution in terms of circuit depth and structure, while the more efficient scaling of HHL LS-SVM makes it a useful option for larger datasets beyond the NISQ era.

DOI: 10.1140/epjqt/s40507-026-00484-0

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Kernel-based one-class classification (OCC) methods leverage kernel functions and are widely used for anomaly detection due to their ability to map data into high-dimensional feature spaces. One such model is Least Squares Support Vector Data Description (LS-SVDD), which aims to construct a minimum volume hypersphere that encompasses the target class observations. A pivotal component in the construction of the hypersphere is the choice of kernel. In this study, the Radial Basis Function (RBF) kernel is compared to a quantum kernel constructed through quantum feature maps. The two kernel methods are compared across four distributional shift scenarios—mean shift, rotating covariance, correlation drift, and Gaussian mixture—at three dimensionalities through simulations. Additionally, the two kernel methods are also evaluated using real-world data through an illustrative example. Furthermore, the Harrow-Hassidim-Lloyd (HHL) quantum algorithm was examined as a potential speedup for solving linear systems of equations within the LS-SVDD framework in quantum space.

Abstract Support vector machines (SVMs) are widely used for classification, but become computationally expensive as datasets grow and data become high-dimensional or nonlinearly separable. Quantum support vector machines (QSVMs) enhance the traditional SVMs by replacing conventional feature mappings with quantum-implemented embeddings that project data into high-dimensional Hilbert spaces, potentially improving expressive power and efficiency. However, training QSVMs on current noisy intermediate-scale quantum (NISQ) devices remains challenging, as constructing the quantum kernel matrix requires repeated circuit executions and scales quadratically with dataset size. We propose NQ-SVM , a hybrid quantum-classical method that incorporates the Nystr\"om approximation to enhance the scalability of QSVMs. By selecting a small set of landmark points, the Nystr\"om method yields a low-rank kernel approximation, reducing complexity from

(O(n^2))

to

(O(nm))

, while preserving the quantum advantages in handling high-dimensional, non-linear data. Extensive simulations demonstrate that NQ-SVM achieves comparable classification accuracy to classical SVMs and full QSVMs, while substantially reducing training time and kernel evaluation costs. These results underscore the potential of NQ-SVM as a scalable and practical hybrid framework for near-term quantum machine learning (QML).

The current development of Support Vector Machines (SVM) has reached a bottleneck, with issues such as long training times and weak interpretability when dealing with large-scale, multi-dimensional data. This paper introduces the concept of Quantum Support Vector Machines (QSVM) and achieves efficient solutions through quantum algorithms such as the HHL algorithm. The research designs a computational architecture that combines classical and quantum computing, utilizing the Pauli decomposition of Hermitian matrices to simulate the quantum simulation of Hamiltonian quantities and implementing the quantum simulation of unitary operators. Based on the conclusions, a complete quantum linear solver circuit is designed, achieving exponential growth in computational complexity. Experiments using the Iris dataset demonstrate the excellent classification performance of QSVM, which outperform classical SVM in classification accuracy, computational time complexity, and memory requirements. More efficient quantum mapping algorithms and quantum circuit optimization methods are implemented, providing new ideas and methods for the application of SVM to large-scale datasets.

Background Quantum support vector machines and quantum kernel methods have emerged as promising approaches within quantum machine learning, with the goal of leveraging quantum computing to enhance classification performance and computational efficiency. This review systematically surveys recent advances in QSVM and kernel‐based quantum classifiers, and analyzes their algorithmic frameworks, experimental implementations, and practical challenges. Methods We systematically examine QSVM approaches, including three schemes based on HHL algorithm, Hadamard test, and variational optimization, alongside multi‐class extension and quantum feature mapping. Result Findings indicate that QSVM can offer theoretical speed‐ups in specific settings, particularly when combined with quantum feature mappings that encode data into high‐dimensional Hilbert spaces. However, current implementations remain constrained by hardware limitations, and a lack of large‐scale general validation. Most studies focus on proof‐of‐principle experiments with limited real‐world applicability. Conclusion While promising, QSVM requires further work on scalability, noise resilience, and real‐world integration. Future research should focus on robust algorithms and empirical studies.

Abstract Well-known quantum machine learning techniques, specifically quantum kernel-assisted support vector machines (QSVMs) and quantum convolutional neural networks (QCNNs), are applied to the binary classification of pulsars. In this comparative study, it is illustrated with simulations that both quantum methods successfully achieve effective classification of the HTRU-2 data set that connects pulsar class labels to eight separate features. While QCNNs are superior in terms of training and prediction speed compared to QSVMs, the preference shifts toward QSVMs when the present noisy NISQ-era devices are incorporated into the comparison. QSVMs demonstrate superior overall performance compared to QCNNs when assessed using binary classification performance metrics. Classical methods are implemented to serve as a benchmark for comparison with the quantum approaches.