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Artificial neural networks for quantum mechanics

Takeshi Kawato

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

The objective of this project is to solve for the energies and wave functions of quantum systems using an artificial neural network. The artificial neural network is explained and is applied to solve ordinary differential equations and eigenvalue problems. The method of Largaris, Likas and Fotiadis is extended to treat the problem of piecewise-constant potential problems in quantum mechanics.

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The objective of this project is to solve for the energies and wave functions of quantum systems using an artificial neural network. The artificial neural network is explained and is applied to solve ordinary differential equations and eigenvalue problems. The method of Largaris, Likas and Fotiadis is extended to treat the problem of piecewise-constant potential problems in quantum mechanics.
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].
Artificial neural networks (ANNs) have become important in quantum chemistry. Herein, applications to nuclear quantum effects, such as zero-point energy, vibrationally excited states, and hydrogen tunneling, are explored. ANNs are used to solve the time-independent Schrödinger equation for single- and double-well potentials representing hydrogen-bonded molecular systems capable of proton transfer. ANN mappings are trained to predict the lowest five proton vibrational energies, wave functions, and densities from the proton potentials and to predict the excited state proton vibrational energies and densities from the proton ground state density. For the inverse problem, ANN mappings are trained to predict the proton potential from the proton vibrational energy levels or the proton ground state density. This latter mapping is theoretically justified by the first Hohenberg-Kohn theorem establishing a one-to-one correspondence between the external potential and the ground state density. ANNs for two- and three-dimensional systems are also presented to illustrate the straightforward extension to higher dimensions.
Solving a multivariable static Schr\"odinger equation for a quantum system, to produce multiple excited-state energy eigenvalues and wave functions, is one of the basic tasks in mathematical and computational physics. Here we propose a neural-network-based solver, which enables us to cover the high-dimensional variable space for this purpose. The efficiency of the solver is analyzed by examples aimed at demonstrating the concept and various aspects of the task: the simultaneous finding of multiple excited states of lowest energies, the computation of energy-degenerate states with orthogonalized wave functions, the scalability to handle a multivariable problem, and the self-consistent determination and automatic adjustment of the imbedded Monte Carlo procedure. The solver adheres to the computational techniques developed in machine learning and is vastly different from traditional numerical methods.
Artificial neural networks (ANNs) and quantum computing (QC) have both received rapidly increasing interest in recent years for several reasons. Their roles in the mechanics community remain a subject of ongoing debate and research. ANNs have successfully been employed on problems previously considered unsolvable ( Silver et al. 2016 ), as well as on practical and illustrative problems especially in the image recognition domain ( Szegedy et al. 2013 , among others). The most straightforward way to interpret the behavior of ANNs is focusing on their ability to approximate any continuous function to arbitrary precision ( Kreinovich 1991 ). The adjustment to a given function happens automatically except for hyperparameters, such as the number of hidden layers or the learning rate, which remain constant during training. This makes them a perfect tool for reducing computational cost in otherwise time-consuming simulations. Quantum computing on the other hand has received some publicity in 2018 mainly because of a paper published by Google Quantum A.I. Lab scientists defining quantum supremacy ( Boixo & Isakov 2018 ), where it is shown that a fully working universal quantum computer with only 50 qubits will outperform any near-term realizable classical supercomputer on certain tasks. This publication was followed up by Google revealing Bristlecone ( Kelly 2018 ), a quantum processor consisting of 72 qubits. While many interesting quantum algorithms have proven to run polynomially or even exponentially faster than their classical analogons ( Montanaro 2016 ), these results only hold true for a decoherence-free machine. Therefore, to actually achieve quantum supremacy , quantum error correction schemes need to be implemented, requiring several physical qubits to make up a single logical qubit suitable for computation.
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.