Radial basis function (RBF) networks are expanded to incorporate quantum kernel functions enabling a new type of hybrid quantum-classical machine learning algorithm. Using this approach, synthetic examples are introduced which allow for proof of concept on interpolation and classification applications. Quantum kernels have primarily been applied to support vector machines (SVMs), however the quantum kernel RBF network offers potential benefit over quantum kernel based SVMs due to the RBF networks ability to perform multi-class classification natively compared to the standard implementation of the SVM.
Radial basis function (RBF) networks are expanded to incorporate quantum kernel functions enabling a new type of hybrid quantum-classical machine learning algorithm. Using this approach, synthetic examples are introduced which allow for proof of concept on interpolation and classification applications. Quantum kernels have primarily been applied to support vector machines (SVMs), however the quantum kernel RBF network offers potential benefit over quantum kernel based SVMs due to the RBF networks ability to perform multi-class classification natively compared to the standard implementation of the SVM.
We give an explicit simple method to build quantum neural networks (QNNs) to solve classification problems. Besides the input (state preparation) and output (amplitude estimation), it has one hidden layer which uses a tensor product of logM two-dimensional rotations to introduce logM weights. Here M is the number of training samples. We also have an efficient method to prepare the quantum states of the training samples. By the quantum-classical hybrid method or the variational method, the training algorithm of this QNN is easy to accomplish in a quantum computer. The idea is inspired by the kernel methods and the radial basis function (RBF) networks. In turn, the construction of QNN provides new findings in the design of RBF networks. As an application, we introduce a quantum-inspired RBF network, in which the number of weight parameters is logM. Numerical tests indicate that the performance of this neural network in solving classification problems improves when M increases. Since using exponentially fewer parameters, more advanced optimization methods (e.g. Newton's method) can be used to train this network. Finally, about the convex optimization problem to train support vector machines, we use a similar idea to reduce the number of variables, which equals M, to logM.
Abstract Quantum kernel approaches have been touted as a possible way to gain representational advantages over classical machine learning techniques however most comparative work only looks at classification accuracy leaving the geometric structure of the kernels unexplored. We carry out a comprehensive kernel-level comparison between classical support vector machines (SVMs) with radial basis function (RBF) kernels and quantum support vector classifiers that use ZZFeatureMap and PauliFeatureMap fidelity kernels, tested on two synthetic and two real-world two-dimensional datasets. After applying centred kernel alignment(CKA), the Hilbert-Schmidt Independence Criterion (HSIC), eigenvalue spectrum analysis, spectral entropy, and effective rank. Firstly, shallow quantum kernels form a moderately aligned band with RBF across all datasets and conditions that is quite stable (CKA : 0.41 → 0.57), whereas going deeper in the circuit to two repetitions largely mismatches both alignment and accuracy with no representational gain (e.g. Iris cross-validation accuracy: 0.920 → 0.670, p = 0.0004). Secondly, depolarising noise surprisingly increases the quantum-classical kernel alignment, which means that near-term hardware kernels are structurally closer to RBF than noiseless simulations suggest. Thirdly, the scenarios that lead to maximum apparent quantum-classical divergence, small sample sizes and low noise, are precisely those most used to benchmark quantum kernel advantage. Kernel-level structural analysis is essential for meaningful evaluation of quantum machine learning methods.
Quantum machine learning (QML) has emerged as an important area for Quantum applications, although useful QML applications would require many qubits. Therefore our paper is aimed at exploring the successful application of the Quantum Support Vector Machine (QSVM) algorithm while balancing several practical and technical considerations under the Noisy Intermediate-Scale Quantum (NISQ) assumption. For the quantum SVM under NISQ, we use quantum feature maps to translate data into quantum states and build the SVM kernel out of these quantum states, and further compare with classical SVM with radial basis function (RBF) kernels. As data sets are more complex or abstracted in some sense, classical SVM with classical kernels leads to less accuracy compared to QSVM, as classical SVM with typical classical kernels cannot easily separate different class data. Similarly, QSVM should be able to provide competitive performance over a broader range of data sets including ``simpler'' data cases in which smoother decision boundaries are required to avoid any model variance issues (i.e., overfitting). To bridge the gap between ``classical-looking'' decision boundaries and complex quantum decision boundaries, we propose to utilize general shallow unitary transformations to create feature maps with rotation factors to define a tunable quantum kernel, and added regularization to smooth the separating hyperplane model. We show in experiments that this allows QSVM to perform equally to SVM regardless of the complexity of the data sets and outperform in some commonly used reference data sets.
Radial-basis-function (RBF) networks are third-layered neural networks that are widely used in function approximation and data classification. Here we propose a quantum model for RBF networks. As in the classical case, we use radial basis functions as activation functions; quantum linear algebraic techniques and coherent states can be applied to implement these functions. We define the state of the weight as a tensor product of single-qubit states. This gives a simple approach to implement the quantum RBF network in quantum circuits. In this model we prove that training is almost quadratically faster than in the classical case. We demonstrate numerically that the quantum RBF network can solve binary classification problems as effectively as the classical RBF network, while the time and number of iterations used for training are much smaller.
Abstract Quantum computers have the potential to speed up certain computational tasks. A possibility this opens up within the field of machine learning is the use of quantum techniques that may be inefficient to simulate classically but could provide superior performance in some tasks. Machine learning algorithms are ubiquitous in particle physics and as advances are made in quantum machine learning technology there may be a similar adoption of these quantum techniques. In this work a quantum support vector machine (QSVM) is implemented for signal-background classification. We investigate the effect of different quantum encoding circuits, the process that transforms classical data into a quantum state, on the final classification performance. We show an encoding approach that achieves an average Area Under Receiver Operating Characteristic Curve (AUC) of 0.848 determined using quantum circuit simulations. For this same dataset the best classical method tested, a classical Support Vector Machine (SVM) using the Radial Basis Function (RBF) Kernel achieved an AUC of 0.793. Using a reduced version of the dataset we then ran the algorithm on the IBM Quantum ibmq_casablanca device achieving an average AUC of 0.703. As further improvements to the error rates and availability of quantum computers materialise, they could form a new approach for data analysis in high energy physics.
Quantum kernel methods are a promising branch of quantum machine learning, but their per- formance across data types and encoding choices remains incompletely understood. To address this gap, we developed an algorithmic framework for variational quantum kernels utilizing resource- efficient ansätze for complex classification tasks and introduced a parameter scaling technique to accelerate convergence. We conducted a comprehensive benchmark of this framework on eight chal- lenging, real-world and high-dimensional datasets covering tabular, image, time series, and graph data. Our results show that the proposed quantum kernel demonstrated competitive classification accuracy compared to standard classical kernels in classical simulation, such as the radial basis function (RBF) kernel. This work demonstrates that properly designed quantum kernels can func- tion as versatile, high-performance tools, laying a foundation for quantum-enhanced applications in machine learning. Further research is needed to fully assess the practical performance of quantum methods.