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arXiv:1710.03213 · Physical review. E

Machine-learning quantum mechanics: Solving quantum mechanics problems using radial basis function networks

Peiyuan Teng

Published Sep 14, 2018• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.98.033305

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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.
Carleo and Troyer [3] have recently pointed out the possibility of solving quantum many-body problems by using Artificial Neural Networks (ANN). Their work is based on minimizing a variational wave function to obtain the ground states for various spin-dependent systems. This work is primarily focused on developing efficient method using ANN to solve the ground state wave function for atomic systems. We have developed a theoretical groundwork to represent the wave function of a many-electron atom by using artificial neural network while still preserving its antisymmetric property. By using the Metropolis algorithm, Variational Monte Carlo (VMC), and Stochastic Reconfiguration (SR) methods for minimization, we were able to obtain a highly accurate ground state wave function for the He atom. To verify our optimization algorithm, we reproduced the results for the ground state of a three dimensional Simple Harmonic Oscillator (SHO) given by Teng [18].
Abstract An overview is presented of recent advances in applying variational, projection, and diffusion Monte Carlo methods (VMC, PMC, and DMC, respectively). The prospects for using the Monte Carlo method to calculate path integrals (PIMC) when modeling molecular systems are also considered. It is emphasized that modern machine learning effectively meets the requirements of parameterizing wave functions of the target space of solutions for a wide range of problems in quantum molecular modeling and functionals of the corresponding observables.
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.
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).
To analyze quantum many-body Hamiltonians, recently, machine learning techniques have been shown to be quite useful and powerful. However, the applicability of such machine learning solvers is still limited. Here, we propose schemes that make it possible to apply machine learning techniques to analyze fermion-boson coupled Hamiltonians and to calculate excited states. As for the extension to fermion-boson coupled systems, we study the Holstein model as a representative of the fermion-boson coupled Hamiltonians. We show that the machine-learning solver achieves highly accurate ground-state energy, improving the accuracy substantially compared to that obtained by the variational Monte Carlo method. As for the calculations of excited states, we propose a different approach than that proposed in K. Choo et al., Phys. Rev. Lett. 121 (2018) 167204. We discuss the difference in detail and compare the accuracy of two methods using the one-dimensional S=1/2S=1/2S=1/2 Heisenberg chain. We also show the benchmark for the frustrated two-dimensional S=1/2S=1/2S=1/2 J1J_1J1​-J2J_2J2​ Heisenberg model and show an excellent agreement with the results obtained by the exact diagonalization. The extensions shown here open a way to analyze general quantum many-body problems using machine learning techniques.
The construction of trial wave functions based on neural networks combined with the variational Monte Carlo method is discussed. The mathematical formulation for representing quantum states as neural networks is introduced. The advantages of employing such trial states and how machine learning works are considered. It is shown that the variational method is a kind of unsupervised learning algorithm, where the multiple minima landscape is used as an asset that leads to a stable optimization procedure. The feature representation plays an important role on interpretability and on extracting physical insights from nontrivial trial wave functions. The algorithm is illustrated for the Yukawa potential and the hydrogen molecule.