Classification using quantum kernels in a radial basis function network

TL;DR

The paper proposes a hybrid quantum–classical radial basis function network (Q-RBF) that uses a quantum kernel $\kappa_Q({\bf x},{\bf y}) = |\langle \psi({\bf y})|\psi({\bf x})\rangle|^2$ to perform kernel evaluations within an otherwise classical RBF framework. Data are encoded into quantum states on the Bloch sphere via $R_x$ rotations and entangled with $\hat{U}$ (CNOT for two qubits or Haar unitary for more) to form the quantum feature map, yielding a kernel matrix $\hat{\boldsymbol{\phi}}$ that is then used to solve for coefficients with Moore–Penrose inversion. The authors demonstrate proof-of-concept results for interpolation of sinusoidal, polynomial, and chaotic logistic map data, and for multiclass classification on Spiral and Iris datasets, showing competitive performance against classical RBF, SVM, and MLP baselines. They discuss practical considerations for near-term hardware, including entanglement requirements and data-encoding fidelity, and call for further benchmarking and hardware demonstrations to assess potential advantages of quantum kernels in multiclass, kernel-based learning tasks.

Abstract

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.

Figures (8)

• Figure 1: (a) Interpolation of a sine wave with an increasing number of centres. It is observed that increasing the number of data centres results in an improved fit. This is consistently seen throughout this paper. (b) Interpolation of a sine wave using Gaussian SVM, MLP, C-RBF and Q-RBF models. This subfigure demonstrates the RBF's ability to successfully interpolate data points, along with the Gaussian SVM. The MSE for this plot is shown in Table 1.
• Figure 2: (a) Interpolation of the polynomial function in Eq. (\ref{['polyeq']}) with an increasing number of centres. Similar trends are observed to Fig. 1 highlighting this model can be generalised. (b) Interpolation of the polynomial function in Eq. (\ref{['polyeq']}) using Gaussian SVM, MLP, C-RBF and Q-RBF models. As before, the RBF's and Gaussian SVM are interpolating the data well. The MSE for this plot is shown in Table 1.
• Figure 3: (a) Interpolation of the Logistic map in Eq. (\ref{['logisticeq']}) with an increasing number of centres. In this subfigure, the axes are $x(t)$ against $x(t+1)$ to better demonstrate the fit, given the time series of the function appears as $\delta$-correlated random noise. (b) Interpolation of the Logistic map in Eq. (\ref{['logisticeq']}) using Gaussian SVM, MLP, C-RBF and Q-RBF models. Similar trends to before are observed, highlighting the consistency with which the RBF's and SVM's interpolate successfully. The MSE for this plot is shown in Table 1.
• Figure 4: Interpolation of the Logistic map in Eq. (\ref{['logisticeq']}), viewed as a timeseries. Despite appearing as random data, the RBF models are able to infer the underlying generating model.
• Figure 5: This highlights the 3 classes of spirals which are generated using Eq. (\ref{['spiral_eq']}). Determining which class each datapoint is generated from is the aim of the classification task.