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Machine Learning Science and Technology

Learning quantum kernels with continuous-variable optical circuits

Aikaterini Mandilara, Aristeides D. Papadopoulos, Dimitris Syvridis

Published Jun 1, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.1088/2632-2153/ae7baf

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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.
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.
The popular qubit framework has dominated recent work on quantum kernel machine learning, with results characterising expressivity, learnability and generalisation. As yet, there is no comparative framework to understand these concepts for continuous variable (CV) quantum computing platforms. In this paper we represent CV quantum kernels as closed form functions and use this representation to provide several important theoretical insights. We derive a general closed form solution for all CV quantum kernels and show every such kernel can be expressed as the product of a Gaussian and an algebraic function of the parameters of the feature map. Furthermore, in the multi-mode case, we present quantification of a quantum-classical separation for all quantum kernels via a hierarchical notion of the “stellar rank" of the quantum kernel feature map. We then prove kernels defined by feature maps of infinite stellar rank, such as GKP-state encodings, can be approximated arbitrarily well by kernels defined by feature maps of finite stellar rank. Finally, we simulate learning with a single-mode displaced Fock state encoding and show that (i) accuracy on our specific task (an annular data set) increases with stellar rank, (ii) for underfit models, accuracy can be improved by increasing a bandwidth hyperparameter, and (iii) for noisy data that is overfit, decreasing the bandwidth will improve generalisation but does so at the cost of effective stellar rank.
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Continuous-variable (CV) quantum computing offers a promising framework for scalable quantum machine learning, leveraging optical systems with infinite-dimensional Hilbert spaces. While discrete-variable (DV) quantum neural networks have shown remarkable progress in various computer vision tasks, CV quantum models remain comparatively underexplored. In this work, we present a feasibility study of continuous-variable quantum neural networks (CV-QCNNs) applied to biomedical image classification. Utilizing photonic circuit simulation frameworks, we construct CV quantum circuits composed of Gaussian gates, such as displacement, squeezing, rotation, and beamsplitters to emulate convolutional behavior. Our experiments are conducted on the MedMNIST dataset collection, a set of annotated medical image benchmarks for multiple diagnostic tasks. We evaluate CV-QCNN's performance in terms of classification accuracy, model expressiveness, and resilience to Gaussian noise, comparing against classical CNNs and equivalent DV quantum circuits. This study aims to identify trade-offs between DV and CV paradigms for quantum-enhanced medical imaging. Our results highlight the potential of continuous-variable models and their viability for future computer-aided diagnosis systems.
A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.