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arXiv:1906.11216

Finding Quantum Many-Body Ground States with Artificial Neural Network

Jiaxin Wu, Wenjuan Zhang

Published Jun 26, 2019• 0 citations
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Abstract

Solving ground states of quantum many-body systems has been a long-standing problem in condensed matter physics. Here, we propose a new unsupervised machine learning algorithm to find the ground state of a general quantum many-body system utilizing the benefits of artificial neural network. Without assuming the specific forms of the eigenvectors, this algorithm can find the eigenvectors in an unbiased way with well controlled accuracy. As examples, we apply this algorithm to 1D Ising and Heisenberg models, where the results match very well with exact diagonalization.

DOI
10.48550/arxiv.1906.11216

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The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the nontrivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with a variable number of hidden neurons. A reinforcement-learning scheme we demonstrate is capable of both finding the ground state and describing the unitary time evolution of complex interacting quantum systems. Our approach achieves high accuracy in describing prototypical interacting spins models in one and two dimensions.
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Over the past few years, machine learning has emerged as a powerful computational tool to tackle complex problems in a broad range of scientific disciplines.In particular, artificial neural networks have been successfully used to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built from a large number of interacting particles.In this article, we review some applications of neural networks in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick start for a newcomer to the field.The prerequisites of this tutorial are basic probability theory and calculus, linear algebra, basic notions of neural networks, statistical physics, and quantum mechanics.The reader is introduced to supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and the variational Monte Carlo method with recurrent neural networks for approximating the ground state of a many-body Hamiltonian.For each algorithm, we briefly review the key ingredients and their corresponding neuralnetwork implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.
Solving the Schrödinger equation for interacting many-body quantum systems faces computational challenges due to exponential scaling with system size. This complexity limits the study of important phenomena in materials science and physics. We develop an Artificial Neural Network (ANN)-driven algorithm to simulate fermionic systems on lattices. Our method uses Pauli matrices to represent quantum states, incorporates Markov Chain Monte Carlo sampling, and leverages an adaptive momentum optimizer. We demonstrate the algorithm's accuracy by simulating the Heisenberg Hamiltonian on a one-dimensional lattice, achieving results with an error in the order of 10−410^{-4}10−4 compared to exact diagonalization. Furthermore, we successfully model a magnetic phase transition in a two-dimensional lattice under an applied magnetic field. Importantly, our approach avoids the sign problem common to traditional Fermionic Monte Carlo methods, enabling the investigation of frustrated systems. This work demonstrates the potential of ANN-based algorithms for efficient simulation of complex quantum systems, opening avenues for discoveries in condensed matter physics and materials science.