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,...
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 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 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.
In this article, machine-learning methods are used to solve quantum mechanics problems. The radial basis function network in a discrete basis is used as the variational wave function for the ground state of a quantum system. Variational Monte Carlo (VMC) calculations are carried out for some simple Hamiltonians. The results are in good agreement with theoretical values. The smallest eigenvalue of a Hermitian matrix can also be acquired using VMC calculations. Results are provided to demonstrate that machine-learning techniques are capable of solving quantum mechanical problems.
Quantum Machine Learning is gaining prominence with the advent of real quantum processors. The quantum analogue of one of the most important and popular model of machine learning, namely Artificial Neural Network is studied and reviewed here in this paper. Various approaches to design and implement a quantum neural network has been studied. Most of the works done in this field involved a translation of classical neural network components into the language of quantum physics. But direct translation cannot be a solution as not every classical component has a quantum analogue. Non-linear activation functions are good examples of it. Quantum operators need to be linear and therefore finding alternate solutions to activation functions is an active area of research. Nevertheless, its advantages over the classical models are profound. Quantum superposition gives exponential storage capacity to a quantum network. Quantum parallelism can be put to use and to train the network with multiple inputs in one go. Despite being a new field, many real world applications of a quantum neural network has also been theorised. These models, their advantages, applications and limitations has been discussed in this paper.
Machine learning is a branch of Artificial Intelligence that seeks to make machines learn from data. It is being applied for solving real world problems with huge amount of data. Though, Machine Learning is receiving wide acceptance, however, execution time is one of the major concerns in practical implementations of Machine Learning techniques. It largely comprises of a set of techniques that trains a model by reducing the error between the desired or actual outcome and an estimated or predicted outcome, which is often called as loss function. Thus, training in machine learning techniques often requires solving a difficult optimization problem, which is the most expensive step in the entire model-building process and its applications. One of the possible solutions in near future for reducing execution time of training process in Machine learning techniques is to implement them on quantum computers instead of classical computers. It is conjectured that quantum computers may be exponentially faster than classical computers for solving problems which involve matrix operations. Some of the machine learning techniques like support vector machines make extensive use of matrices, which can be made faster by implementing them on quantum computers. However, their efficient implementation is non-trivial and requires existence of quantum memories. Thus, another possible solution in near term is to use a hybrid of Classical Quantum approach, where a machine learning model is implemented in classical computer but the optimization of loss function during training is performed on quantum computer instead of classical computer. Several Quantum optimization algorithms have been proposed in recent years, which can be classified as gradient based and gradient free optimization techniques. Gradient based techniques require the nature of optimization problem being solved to be convex, continuous and differentiable otherwise if the problem is non-convex then they can find local optima only whereas gradient free optimization techniques work well even with non-continuous, non-linear and nonconvex optimization problems. This chapter discusses a global optimization technique based on Adiabatic Quantum Computation (AQC) to solve minimization of loss function without any restriction on its structure and the underlying model, which is being learned. Further, it is also shown that in the proposed framework, AQC based approach would be superior to circuit-based approach in solving global optimization problems.