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arXiv:2407.13809

Enhancing Quantum Machine Learning: The Power of Non-Linear OpticalReproducing Kernels

Shahram Dehdashti, Prayag Tiwari, Kareem H. El-Safty, Peter Bruza, Janis Nötzel

Published Sep 26, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Amidst the array of quantum machine learning algorithms, the quantum kernel method has emerged as a focal point, primarily owing to its compatibility with noisy intermediate-scale quantum devices and its promise to achieve quantum advantage. This method operates by nonlinearly transforming data into feature space constructed with quantum states, enabling classification and regression tasks. In this study, we present a novel feature space constructed using Kerr coherent states, which generalize su(2), su(1,1) coherent states, and squeezed states. Notably, the feature space exhibits constant curvature, comprising both spherical and hyperbolic geometries, depending on the sign of the Kerr parameter. Remarkably, the physical parameters associated with the coherent states, enable control over the curvature of the feature space. Our study employs Kerr kernels derived from encoding data into the phase and amplitude of Kerr coherent states. We analyze various datasets ranging from Moon to breast cancer diagnostics. Our findings demonstrate the robustness of Kerr coherent states, attributed to their flexibility in accommodating different hyperparameters, thereby offering superior performance across noisy datasets and hardware setups.

DOI
10.21203/rs.3.rs-4983681/v1

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Abstract Amidst the array of quantum machine learning algorithms, the quantum kernel method has emerged as a focal point, primarily owing to its compatibility with noisy intermediate-scale quantum devices and its promise to achieve quantum advantage. This method operates by nonlinearly transforming data into feature space constructed with quantum states, enabling classification and regression tasks. In this study, we present a novel feature space constructed using Kerr coherent states, which generalize su(2), su(1,1) coherent states, and squeezed states. Notably, the feature space exhibits constant curvature, comprising both spherical and hyperbolic geometries, depending on the sign of the Kerr parameter. Remarkably, the physical parameters associated with the coherent states, enable control over the curvature of the feature space. Our study employs Kerr kernels derived from encoding data into the phase and amplitude of Kerr coherent states. We analyze various datasets ranging from Moon to breast cancer diagnostics. Our findings demonstrate the robustness of Kerr coherent states, attributed to their flexibility in accommodating different hyperparameters, thereby offering superior performance across noisy datasets and hardware setups.
Among various quantum machine learning (QML) algorithms, the quantum kernel method has especially attracted attention due to its compatibility with noisy intermediate-scale quantum devices and its potential to achieve quantum advantage. This method performs classification and regression by nonlinearly mapping data into quantum states in a higher-dimensional Hilbert space. Thus far, the quantum kernel method has been implemented only on qubit-based systems, but continuous-variable (CV) systems can potentially offer superior computational power by utilizing its infinite-dimensional Hilbert space. Here, we demonstrate the implementation of the classification task with the CV quantum kernel method on a programmable photonic quantum processor. We experimentally prove that the CV quantum kernel method successfully classifies several datasets robustly even under the experimental imperfections, with high accuracies comparable to the classical kernel. This demonstration sheds light on the utility of CV quantum systems for QML and should stimulate further study in other CV QML algorithms.
Abstract In this paper, by mapping datasets to a set of non-linear coherent states, the process of encoding inputs in quantum states as a non-linear feature map is re-interpreted. As a result of the fact that the radial basis function is recovered when data is mapped to a complex Hilbert state represented by coherent states, non-linear coherent states can be considered as a natural generalisation of the associated kernels. In this paper, as an example of kernels based on non-linear coherent states, we propose kernel functions based on generalized hypergeometric functions, as orthogonal polynomial functions. The suggested kernel is implemented with the support vector machine (SVM) on two well known datasets (make_circles, and make_moons) and outperforms the baselines, even when the level of noise is high. In addition, we study the impact of the geometrical properties of the feature space, obtained by the non-linear coherent states, on the SVM classification task, by considering the Fubini–Study metric of the associated coherent states.
Quantum-enhanced machine learning is a rapidly evolving field that aims to leverage the unique properties of quantum mechanics to enhance classical machine learning. However, the practical applicability of these methods remains an open question, particularly beyond the context of specifically crafted toy problems, and given the current limitations of quantum hardware. This study focuses on quantum kernel methods in the context of classification tasks. In particular, it examines the performance of quantum kernel estimation and quantum kernel training (QKT) in connection with two quantum feature mappings, namely, ZZFeatureMap and CovariantFeatureMap. Remarkably, these feature maps have been proposed in the literature under the conjecture of possible near-term quantum advantage and have shown promising performance in ad hoc datasets. This study aims to evaluate their versatility and generalization capabilities in a more general benchmark, encompassing both artificial and established reference datasets. Classical machine learning methods, specifically support vector machines and logistic regression, are also incorporated as baseline comparisons. Experimental results indicate that quantum methods exhibit varying performance across different datasets. Despite outperforming classical methods in ad hoc datasets, mixed results are obtained for the general case among standard classical benchmarks. The experimental data call into question a general added value of applying QKT optimization, for which the additional computational cost does not necessarily translate into improved classification performance. Instead, it is suggested that a careful choice of the quantum feature map in connection with proper hyperparameterization may prove more effective.
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.
Quantum algorithms have become a popular research domain in recent times for discovering quantum-enhanced solutions in machine learning applications. Quantum kernels are one of the directions that establish such quantum-enhanced solutions to some extent. This work presents a detailed analysis of the quantum kernel approach leveraging feature maps and relevant hyperparameters to develop enhanced quantum kernels. The study includes a new high-order feature map and assesses five existing state-of-the-art feature maps for enhanced quantum kernel classifiers. Additionally, the significance of the rotational factor as a hyperparameter is highlighted for improving kernel performance. Also, it is analyzed whether different hyperparameter-tuned feature maps can lead to enhanced decision boundaries, demonstrating kernel expressivity. The analysis is undertaken on classification tasks using four different nonlinear datasets of distinct complexity. Comparative evaluations are also made with traditional machine learning models-Support Vector Machines (Linear and RBF), Naïve Bayes, Linear Discriminant Analysis, Decision Tree, Random Forest, Adaptive Boosting, and MLP. Overall, the study demonstrates that a well-tuned quantum feature map can enhance the generalization ability of quantum kernels, making them more effective for broader quantum-enhanced machine learning applications.
We implement an all-optical setup demonstrating kernel-based quantum machine learning for two-dimensional classification problems. In this hybrid approach, kernel evaluations are outsourced to projective measurements on suitably designed quantum states encoding the training data, while the model training is processed on a classical computer. Our two-photon proposal encodes data points in a discrete, eight-dimensional feature Hilbert space. In order to maximize the application range of the deployable kernels, we optimize feature maps towards the resulting kernels' ability to separate points, i.e., their "resolution," under the constraint of finite, fixed Hilbert space dimension. Implementing these kernels, our setup delivers viable decision boundaries for standard nonlinear supervised classification tasks in feature space. We demonstrate such kernel-based quantum machine learning using specialized multiphoton quantum optical circuits. The deployed kernel exhibits exponentially better scaling in the required number of qubits than a direct generalization of kernels described in the literature.