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arXiv:1910.08798 · Physical Review A

Data classification by quantum radial-basis-function networks

Changpeng Shao

Published Oct 30, 2020• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.102.042418

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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.
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 log⁡M\log MlogM two-dimensional rotations to introduce log⁡M\log MlogM weights. Here MMM 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 log⁡M\log MlogM. Numerical tests indicate that the performance of this neural network in solving classification problems improves when MMM 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 MMM, to log⁡M\log MlogM.
Radial basis function (RBF) network is a simple but useful neural network model that contains wide applications in machine learning. The training of an RBF network reduces to solve a linear system, which is time consuming classically. Based on HHL algorithm, we propose two quantum algorithms to train RBF networks. To apply the HHL algorithm, we choose using the Hamiltonian simulation algorithm proposed in [P. Rebentrost, A. Steffens, I. Marvian and S. Lloyd, Phys. Rev. A 97, 012327, 2018]. However, to use this result, an oracle to query the entries of the matrix of the network should be constructed. We apply the amplitude estimation technique to build this oracle. The final results indicate that if the centers of the RBF network are the training samples, then the quantum computer achieves exponential speedup at the number and the dimension of training samples over the classical computer; if the centers are determined by the K-means algorithm, then the quantum computer achieves quadratic speedup at the number of samples and exponential speedup at the dimension of samples.
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.
A non-linear activation function is an integral component of neural network algorithms used for various tasks such as data classification and pattern recognition. In neu-romorphic/emerging-hardware-based implementations of neural network algorithms, the non-linear activation function is often implemented through dedicated analog electronics. This enables faster execution of the activation function during training and inference of neural networks compared to conventional digital implementation. Here, with a similar motivation, we propose a novel non-linear activation function that can be used in a neural network for data classification. Our activation function can be implemented by taking advantage of the inherent nonlinearity in qubit preparation and SU(2) operation in quantum mechanics. These operations are directly implementable on quantum hardware through single-qubit quantum gates as we show here. In addition, the SU(2) parameters are adjustable here making the activation function adaptable; we adjust the parameters through classical feedback like in a variational algorithm in quantum machine learning. Using our proposed quantum activation function, we report accurate classification using popular machine learning data sets like Fisher's Iris, Wisconsin's Breast Cancer (WBC), Abalone, and MNIST on three different platforms: simulations on a classical computer, simulations on a quantum simulation framework like Qiskit, and experimental implementation on quantum hardware (IBM-Q). Then we use a Bloch-sphere-based approach to intuitively explain how our proposed quantum activation function, with its adaptability, helps in data classification.
Artificial neural networks are at the heart of modern deep learning algorithms. We describe how to embed and train a general neural network in a quantum annealer without introducing any classical element in training. To implement the network on a state-of-the-art quantum annealer, we develop three crucial ingredients: binary encoding the free parameters of the network; polynomial approximation of the activation function; and reduction of binary higher-order polynomials into quadratic ones. Together, these ideas allow encoding the loss function as an Ising model Hamiltonian. The quantum annealer then trains the network by finding the ground state. We implement this for an elementary network and illustrate the advantages of quantum training: its consistency in finding the global minimum of the loss function and the fact that the network training converges in a single annealing step, which leads to short training times while maintaining a high classification performance. After training the network using a quantum annealer, one can then use the quantum network weights in a classical network algorithm of identical design for inference. Our approach opens an avenue for the quantum training of general machine learning models.