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arXiv:1904.10251 · Advanced Theory and Simulations

Artificial Neural Networks as Trial Wave Functions for Quantum Monte Carlo

Jan Kessler, Francesco Calcavecchia, Thomas D. Kühne

Published Jan 25, 2021• 30 citations
arXivHTMLPDF

Abstract

Abstract Inspired by the universal approximation theorem and widespread adoption of artificial neural network techniques in a diversity of fields, feed‐forward neural networks are proposed as a general purpose trial wave function for quantum Monte Carlo simulations of continuous many‐body systems. Whereas for simple model systems the whole many‐body wave function can be represented by a neural network, the antisymmetry condition of non‐trivial fermionic systems is incorporated by means of a Slater determinant. To demonstrate the accuracy of the trial wave functions, an exactly solvable model system of two trapped interacting particles, as well as the hydrogen dimer, is studied.

DOI
10.1002/adts.202000269

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Abstract Inspired by the universal approximation theorem and widespread adoption of artificial neural network techniques in a diversity of fields, feed‐forward neural networks are proposed as a general purpose trial wave function for quantum Monte Carlo simulations of continuous many‐body systems. Whereas for simple model systems the whole many‐body wave function can be represented by a neural network, the antisymmetry condition of non‐trivial fermionic systems is incorporated by means of a Slater determinant. To demonstrate the accuracy of the trial wave functions, an exactly solvable model system of two trapped interacting particles, as well as the hydrogen dimer, is studied.
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 artificial neural networks is introduced. The advantages of employing such trial states and how machine learning works are discussed. 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.
Finding high-quality trial wave functions for quantum Monte Carlo calculations of light nuclei requires a strong intuition for modeling the interparticle correlations as well as large computational resources for exploring the space of variational parameters. Moreover, for systems with three-body interactions, the wave function should account for many-body effects beyond simple pairwise correlations. In this Letter, we design neural networks that efficiently incorporate these factors to generate expressive wave function Ansätze for light nuclei using variational Monte Carlo. Our neural-network approach for A=3 nuclei can capture, already at the level of variational Monte Carlo, the overwhelming majority of the ground-state energy estimated by Green's function Monte Carlo (GFMC). It achieves a ground-state energy within 0.45% of the GFMC result for ^{3}H using the softest chiral interaction, representing a substantial improvement over standard variational Monte Carlo, which exhibits a 3.7% deviation. The result indicates the potential of neural networks to construct effective trial wave functions for quantum Monte Carlo calculations.
We introduce an efficient approach to implement correlated many-body trial wave functions in auxiliary-field quantum Monte Carlo (AFQMC). To control the sign/phase problem in AFQMC, a constraint is derived from an exact gauge condition but is typically imposed approximately through a trial wave function or trial density matrix, whose quality can affect the accuracy of the method. Furthermore, the trial wave function can also affect the efficiency through importance sampling. The most natural form of the trial wave function has been single Slater determinants or their linear combinations. More sophisticated forms, for example, with the inclusion of a Jastrow factor or other explicit correlations, have been challenging to use, and their implementation is often assumed to require a quantum computer. In this work, we demonstrate that a large class of correlated wave functions, written in the general form of multi-dimensional integrals over hidden or auxiliary variables times Slater determinants, can be implemented as a trial wave function by coupling the random walkers to a generalized Metropolis sampling. We discuss the fidelity of AFQMC with stochastically sampled trial wave functions, which are relevant to both quantum and classical algorithms. We illustrate the method and show that an efficient implementation can be achieved, which preserves the computational scaling of AFQMC. We test our method in molecules under bond stretching and in transition metal diatomics. Significant improvements are seen in both accuracy and efficiency over typical trial wave functions, and the method yields total ground-state energies systematically within chemical accuracy. The method can be useful for incorporating other advanced wave functions, for example, neural quantum state wave functions optimized from machine learning techniques, or for other forms of fermion quantum Monte Carlo.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O(N^3)O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N2)O(N^2)O(N2) or less. We show that the accuracy of the approximation can be improved by optimizing the variance of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O({N}^{3})O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost from O(N3)O({N}^{3})O(N3) to O(N2)O({N}^{2})O(N2) or less, while the overall cost depends on the number of Monte Carlo samples and the number of parameters in neural networks. We show that the accuracy of the approximation can be improved by optimizing the ``variance'' of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, data preprocessing, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
In this paper, we demonstrate the expressibility of artificial neural networks (ANNs) in quantum many-body physics by showing that a feed-forward neural network with a small number of hidden layers can be trained to approximate with high precision the ground states of some notable quantum many-body systems. We consider the one-dimensional free bosons and fermions, spinless fermions on a square lattice away from half-filling, as well as frustrated quantum magnetism with a rapidly oscillating ground-state characteristic function. In the latter case, an ANN with a standard architecture fails, while that with a slightly modified one successfully learns the frustration-induced complex sign rule in the ground state and approximates the ground states with high precisions. As an example of practical use of our method, we also perform the variational method to explore the ground state of an antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model.