A real neural network state for quantum chemistry
The restricted Boltzmann machine (RBM) has been successfully applied to solve
the many-electron Schr$\ddot{\text{o}}$dinger equation. In this work we propose
a single-layer fully connected neural network adapted from RBM and apply it to
study ab initio quantum chemistry problems. Our contribution is two-fold: 1)
our neural network only uses real numbers to represent the real electronic wave
function, while we obtain comparable precision to RBM for various prototypical
molecules; 2) we show that the knowledge of the Hartree-Fock reference state
can be used to systematically accelerate the convergence of the variational
Monte Carlo algorithm as well as to increase the precision of the final energy.
Yangjun Wu, Xiansong Xu, Dario Poletti, Yi Fan, Chu Guo, Honghui Shang
# A real neural network state for quantum chemistry

###### Abstract

The restricted Boltzmann machine (RBM) has been successfully applied to solve the many-electron Schrodinger equation. In this work we propose a single-layer fully connected neural network adapted from RBM and apply it to study ab initio quantum chemistry problems. Our contribution is two-fold: 1) our neural network only uses real numbers to represent the real electronic wave function, while we obtain comparable precision to RBM for various prototypical molecules; 2) we show that the knowledge of the Hartree-Fock reference state can be used to systematically accelerate the convergence of the variational Monte Carlo algorithm as well as to increase the precision of the final energy.

## I Introduction

Ab initio electronic structure calculations based on quantum-chemical approaches (Hartree-Fock theory and post-Hartree-Fock methods) have been successfully applied in molecular systems [1]. For strongly correlated many-electron systems, the exponentially growing Hilbert space size limits the application scale of most numerical algorithms. For example, the full configuration interaction (FCI) which takes the whole Hilbert space into account, is currently limited within around \(24\) orbitals and \(24\) electrons [2]. The density matrix renormalization group (DMRG) algorithm [3; 4] has been used to solve larger chemical systems of several tens of electrons [5; 6], however it is essentially limited by the expressive power of its underlying variational ansatz: the matrix product state (MPS) which is a special instance of the one-dimensional tensor network state [7], therefore DMRG could also be extremely difficult to approach even larger systems. The coupled cluster (CC) [8; 9] method expresses the exact wave function in terms of an exponential form of a variational wave function ansatz, and higher level of accuracy can be obtained by considering electronic excitations up to doublets in CCSD or triplets in CCSD(T). In practice, it is often accurate with a durable computational cost, thus considered as the "gold standard" in electronic structure calculations. However, the accuracy of the CC method is only restricted in studying weakly correlated systems [10]. The multi-configuration self-consistent field (MCSCF) [11; 12; 13] method is crucial for describing molecular systems containing nearly degenerate orbitals. It introduces a small number of (active) orbitals, then the configuration interaction coefficients and the orbital coefficients are optimized to minimize the total energy of the MCSCF state. It has been applied to systems with around 50 active orbitals [14], but they are still limited by the exponential complexity that grows with the system size.

In recent years the variational Monte Carlo (VMC) method in combination with a neural network ansatz for the underlying quantum state (wave function) [15], referred to as the neural network quantum states (NNQS), has been demonstrated to be a scalable and accurate tool for many-spin systems [16; 17; 18] and many-fermion systems [19]. NNQS allow very flexible choices of the neural network ansatz, and with an appropriate variational ansatz, it could often achieve comparable or higher accuracy compared to existing methods. NNQS has also been applied to solve ab-initio quantum chemistry systems in real space with up to \(30\) electrons [20; 21; 22], as well as in discrete basis after second quantization [23; 24; 25]. Up to now various neural networks have been used, such as the restricted Boltzmann machine (RBM) [15], convolutional neural network [16], recurrent neural networks [26] and variational auto-encoder [25]. In all those neural networks, the RBM is a very special instance in that: 1) it has a very simple structure which contains only a fully connected dense layer plus a nonlinear activation; 2) with such a simple structure, RBM can be more expressive than MPS [27], in fact it is equivalent to certain two-dimensional tensor network states [28], and can even represent certain quantum state with volume-law entanglement [29]. In practice RBM achieves comparable accuracy to other more sophisticated neural networks for complicated applications such as frustrated many-spin systems [30; 31].

For the ground state of molecular systems, the wave function is real. However, if one uses a real RBM as the variational ansatz for the wave function, then all the amplitudes of the wave function will be positive, which means that it may be good for ferromagnetic states but will be completely wrong for anti-ferromagnetic states. Therefore even for real wave functions one would have to use complex RBMs or two RBMs [32] in general. In this work we propose a neural network with real numbers which is slightly modified from the RBM such that its output can be both positive and negative, and use it as the neural network ansatz to solve quantum chemistry problems. To accelerate convergence of the VMC iterations, we explicitly use the Hartree-Fock reference state as the starting point for the Monte Carlo sampling af