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arXiv:2509.12072

Learning kernels with quantum optical circuits

Aikaterini Mandilara, Aristeides D. Papadopoulos, Dimitris Syvridis

Published Sep 15, 2025• 0 citations
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Abstract

Support Vector Machines (SVMs) are a cornerstone of supervised learning, widely used for data classification. A central component of their success lies in kernel functions, which enable efficient computation of inner products in high-dimensional feature spaces. Recent years have seen growing interest in leveraging quantum circuits -- both qubit-based and quantum optical -- for computing kernel matrices, with ongoing research exploring potential quantum advantages. In this work, we investigate two classical techniques for enhancing SVM performance through kernel learning -- the Fisher criterion and quasi-conformal transformations -- and translate them into the framework of quantum optical circuits. Conversely, using the example of the displaced squeezed vacuum state, we demonstrate how established concepts from quantum optics can inspire novel perspectives and enhancements in SVM methodology. This cross-disciplinary approach highlights the potential of quantum optics to both inform and benefit from advances in machine learning.

DOI
10.48550/arxiv.2509.12072

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Support Vector Machines (SVMs) are a cornerstone of supervised learning, widely used for data classification. A central component of their success lies in kernel functions, which enable efficient computation of inner products in high-dimensional feature spaces. Recent years have seen growing interest in leveraging quantum circuits -- both qubit-based and quantum optical -- for computing kernel matrices, with ongoing research exploring potential quantum advantages. In this work, we investigate two classical techniques for enhancing SVM performance through kernel learning -- the Fisher criterion and quasi-conformal transformations -- and translate them into the framework of quantum optical circuits. Conversely, using the example of the displaced squeezed vacuum state, we demonstrate how established concepts from quantum optics can inspire novel perspectives and enhancements in SVM methodology. This cross-disciplinary approach highlights the potential of quantum optics to both inform and benefit from advances in machine learning.
Abstract Kernel-based learning methods are central to classical machine learning, yet their quantum counterparts have largely remained limited to fixed circuit constructions. Most existing approaches aim to improve the performance of kernel-based support vector machines by heuristically tailoring kernel functions to specific tasks. In this work, we introduce a principled framework for adapting quantum kernels by translating two well-established classical techniques—the Fisher criterion and quasi-conformal kernel transformations—into the setting of continuous-variable (CV) quantum circuits. We show that the classical Fisher criterion is equivalent to maximizing the Hilbert–Schmidt distance in the quantum feature space induced by a quantum circuit. We further demonstrate that quasi-conformal kernel transformations admit a direct and natural implementation at the circuit level, realized by augmenting the original quantum circuit with additional modes. Using experimentally accessible optical circuits composed solely of displacement and squeezing operations, we numerically demonstrate that each method independently leads to a systematic enhancement of learning performance. Taken together, our theoretical and numerical results establish a concrete route for importing powerful classical kernel-learning strategies into quantum machine learning, and position CV quantum optics as a flexible and experimentally viable platform for adaptive, task-driven quantum kernel design.
The quantum support vector machines (QSVMs) offer a paradigm change that may leverage quantum mechanical processes to outperform traditional support vector machines (SVMs) in high-dimensional and nonlinear classification tasks. These machines are located at the crossroads of supervised machine learning and quantum computing. The purpose of this chapter is to provide a comprehensive technical analysis of QSVMs, with a particular emphasis on the use of variation quantum circuits for implicit feature space transformations and the development of these circuits within the context of quantum kernel estimation. Through the use of quantum state preparation and interference, we provide a comprehensive explanation of the possible fabrication of quantum kernels that are based on inner product evaluations in exponentially huge Hilbert spaces. When it comes to boosting model expressivity beyond classical limits, the importance that entanglement and quantum feature mapping play is specifically stressed. In addition to this, we investigate algorithmic implementations on noisy intermediate-scale quantum (NISQ) hardware, focusing on problems associated with quantum decoherence, gate fidelity, and loop depth optimization. Several fields, including quantum-enhanced drug discovery, anomaly detection in quantum sensor networks, fault classification in quantum error correction, and portfolio optimization, are now the focus of research that is being conducted in order to develop real-world applications. This chapter comes to a close with an analysis of the existing theoretical limits, resource needs, and future perspectives for the use of QSVMs in machine learning applications in order to achieve a quantum advantage. In order to facilitate the development of scalable quantum machine learning models that are founded on strong mathematical and computational notions, this synthesis is geared at academics and practitioners.
Support Vector Machines (SVMs) are powerful machine learning models widely used for classification and regression tasks. However, their performance can be significantly impacted by the choice of the kernel function, which defines how data points are mapped into a higher-dimensional space. Traditional kernels often struggle to capture complex, non-linear relationships effectively, leading to suboptimal results. This paper proposes a novel approach to kernel design inspired by quantum mechanical principles. Specifically, we leverage concepts of quantum entanglement and superposition to generate kernels that can potentially capture richer and more nuanced data patterns. Our core claim is that employing these quantum-inspired designs can improve SVM performance and scalability. We present a theoretical framework outlining how quantum mechanical effects can be translated into kernel functions. The proposed method is based on constructing kernel functions that implicitly encode correlations between data points, mimicking the inherent interconnectedness observed in quantum systems. We explore specific kernel formulations based on these principles and discuss their potential advantages and challenges. The ultimate goal is to demonstrate that quantum inspiration can unlock new avenues for kernel design, leading to more effective SVM models.
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).
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.
This paper explores a novel approach to kernel design for machine learning, drawing inspiration from the principles of quantum entanglement. Traditional kernel functions in Support Vector Machines (SVMs) frequently necessitate extensive manual tuning and often fail to achieve optimal performance. We introduce a kernel design algorithm that mimics the correlated behavior observed in entangled quantum systems. The algorithm constructs kernel functions based on the correlated features of data points, effectively simulating entanglement effects to potentially enhance classification accuracy. This work presents a fundamental shift in kernel design, moving away from hand-crafted kernels towards a more adaptive and potentially more powerful approach guided by quantum-inspired concepts. The core contribution lies in the methodology for generating kernels that dynamically capture complex relationships within data, leading to improved classification outcomes. We demonstrate the potential of this approach through a theoretical framework and outline a design strategy that can be readily applied to various machine learning tasks.