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arXiv:2610.24403

Prototype-Guided Optimization of Quantum Data Embeddings with Linear Cost

Seokhoon Jeong, Daniel K. Park

quant-phcs.LGSubmitted Oct 17, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

In supervised quantum machine learning with classical data, classification performance critically depends on the geometry induced by quantum data embedding. Existing data-adaptive approaches rely on pairwise fidelity evaluations between samples, which scale as $O(B^2)$ for $B$ samples and create a computational bottleneck in iterative embedding optimization. To address this issue, we propose a prototype-guided framework for efficient data-adaptive quantum embedding. Using class-prototype separation and sample-to-prototype compactness, we derive a measurable lower certificate for class-ensemble distinguishability and construct a prototype-based objective requiring only $O(B)$ fidelity evaluations. We integrate this objective into an end-to-end pipeline that combines task-specific quantum circuit architecture optimization with continuous refinement of the data-embedding parameters. Across five benchmark datasets with up to 20 qubits, our method retains competitive quantum kernel support vector machine (SVM) performance while reducing fidelity-evaluation cost, with particularly strong gains over budget-matched pairwise optimization on T-MNIST. Geometric analyses examine how the learned embeddings relate to kernel alignment and classification margins, revealing dataset-dependent benefits and limitations. We further derive a prototype-based classifier in the reproducing kernel Hilbert space that provides an explicit witness for the SVM objective and a low-cost proxy for model selection, and extend the framework to multiple prototypes for multimodal class structures.

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26 pages, 5 figures, 5 tables

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In supervised quantum machine learning with classical data, classification performance critically depends on the geometry induced by quantum data embedding. Existing data-adaptive approaches rely on pairwise fidelity evaluations between samples, which scale as O(B2)O(B^2)O(B2) for BBB samples and create a computational bottleneck in iterative embedding optimization. To address this issue, we propose a prototype-guided framework for efficient data-adaptive quantum embedding. Using class-prototype separation and sample-to-prototype compactness, we derive a measurable lower certificate for class-ensemble distinguishability and construct a prototype-based objective requiring only O(B)O(B)O(B) fidelity evaluations. We integrate this objective into an end-to-end pipeline that combines task-specific quantum circuit architecture optimization with continuous refinement of the data-embedding parameters. Across five benchmark datasets with up to 20 qubits, our method retains competitive quantum kernel support vector machine (SVM) performance while reducing fidelity-evaluation cost, with particularly strong gains over budget-matched pairwise optimization on T-MNIST. Geometric analyses examine how the learned embeddings relate to kernel alignment and classification margins, revealing dataset-dependent benefits and limitations. We further derive a prototype-based classifier in the reproducing kernel Hilbert space that provides an explicit witness for the SVM objective and a low-cost proxy for model selection, and extend the framework to multiple prototypes for multimodal class structures.
Many practically relevant applications of quantum machine learning involve classical data, for which performance depends critically on how inputs are embedded into quantum states. Yet the use of a fixed embedding circuit ansatz remains standard practice. We propose an energy-based generative learning framework that synthesizes gate sequences to optimize embedding structures and refine data-tailored parameters, using a fidelity-based surrogate objective to guide the search toward improved class distinguishability. Empirically, the method improves classification performance across diverse settings, while also revealing datasets where architecture search within the present embedding family yields only limited additional gains. We explain this saturation by deriving bounds on the achievable empirical risk in terms of the Wasserstein distance in the input space, showing that classical data geometry provides an \emph{a priori} diagnostic for regimes in which substantial gains from embedding optimization are unlikely. The results establish a practically useful and theoretically motivated framework for searching effective quantum data embeddings through generative optimization, with the attainable gains diagnosed through the geometry of the underlying classical data.
Quantum machine learning has emerged as a promising frontier, leveraging the principles of quantum computing to enhance classical algorithms. This paper proposes an optimized Quantum Support Vector Classifier (QSVC) for the classification of the MNIST dataset (a benchmark in handwritten digit recognition). We employ a Chebyshev Parameterized Quantum Circuit (PQC) with an increased number of qubits and layers, and utilize an existing projected quantum kernel method to effectively capture the complex patterns in the data. To enhance model robustness, data augmentation techniques are applied, and dimensionality reduction is performed using Principal Component Analysis (PCA). The proposed QSVC demonstrates competitive accuracy on both the training and testing datasets, highlighting the potential of quantum kernel methods in practical machine learning applications. A comparative analysis with classical Support Vector Machines (SVM) demonstrates the feasibility of quantum methods for high-dimensional classification tasks. Additionally, we investigate the impact of varying the number of qubits in the quantum circuit, providing in-depth insights into the scalability and performance trade-offs of quantum kernel methods.
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(n2))(O(n^2))(O(n2)) to (O(nm))(O(nm))(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).
Neural quantum kernels (NQKs) construct quantum kernels by pretraining a quantum neural network (QNN) and subsequently reusing the trained circuit as a task-adapted embedding. Extending this framework to qudits, with local unitaries in SU(d)\mathrm{SU}(d)SU(d), provides a natural route to richer data embeddings through the increased local degrees of freedom and a direct interface for multiclass classification via intrinsically multi-level quantum systems. In this work, focusing on qutrits (d=3d=3d=3), we extend NQKs to the qudit setting and perform a systematic study of key design choices, including the number of encoded features, the number of qutrits, the kernel construction (1-to-nnn and nnn-to-nnn), and the parameterization of SU(3)\mathrm{SU}(3)SU(3) unitaries. Across binary and three-class tasks on four benchmark datasets, qutrit NQKs improve over the corresponding QNN baselines in nearly all settings considered and can benefit from scaling both the feature budget and the system size, although the magnitude of these gains may saturate, is dataset-dependent, and depends on the chosen parameterization. In particular, an ablation over SU(3)\mathrm{SU}(3)SU(3) parameterizations shows that the unitary representation can substantially impact both optimization behaviour and classifier performance. These findings highlight the potential of qudit-based quantum models not only as a straightforward generalization of qubit-based architectures, but also as a promising means to better exploit complex data structures in quantum machine learning.
The expressivity and scalability of fixed feature embeddings and shallow quantum circuits for Quantum Support Vector Machines (QSVMs) are restricted due to the limitations of near-term hardware. In this work, we propose a novel QSVM framework consisting of Low-Rank Adaptation (LoRA)-modified VisionTransformer embeddings as well as trainable variational quantum circuits. The architecture enhances the representational power by the joint optimization of embedding and circuit parameters. Empirical evaluation on distilled benchmark datasets shows significant performance gains, with up to 38.7% absolute accuracy improvement on MNIST (61.3% to 100.0%). The model also achieves 91% to 100% test accuracies on Fashion-MNIST, KMNIST and CIFAR-10 datasets with only 200 training samples for each dataset. The results show that the performance improvements are due to the joint design between the feature embeddings and the quantum circuits, rather than just the quantum kernels. Cross-dataset evaluations and circuit-depth analyses provide experimental evidence that the approach is suitable for scalable quantum machine learning in image classification tasks.
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.