LATTICE CONVOLUTIONAL NETWORKS FOR LEARNING GROUND STATES OF QUANTUM MANY-BODY SYSTEMS

Abstract

Deep learning methods have been shown to be effective in representing ground-state wave functions of quantum many-body systems. Existing methods use convolutional neural networks (CNNs) for square lattices due to their image-like structures. For non-square lattices, existing method uses graph neural network (GNN) in which structure information is not precisely captured, thereby requiring additional hand-crafted sublattice encoding. In this work, we propose lattice convolutions in which a set of proposed operations are used to convert non-square lattices into grid-like augmented lattices on which regular convolution can be applied. Based on the proposed lattice convolutions, we design lattice convolutional networks (LCN) that use self-gating and attention mechanisms. Experimental results show that our method achieves performance on par or better than the GNN method on spin 1/2 J 1 -J 2 Heisenberg model over the square, honeycomb, triangular, and kagome lattices while without using hand-crafted encoding.

1. INTRODUCTION

Study of quantum many-body problems is of fundamental interests in physics. It is crucial for theoretical modeling and simulation of complex quantum systems, materials and molecules (Carleo et al., 2019) . For instance, graphene, arguably the most famous 2D material, is made of carbon atoms on a honeycomb lattice. Solving quantum many-body problems remains to be very challenging because of the exponential growth of Hilbert space dimensions with the number of particles in quantum systems. Only approximation solutions are available in most cases. Tensor network (White, 1992; Schollwöck, 2011; Orús, 2014; Biamonte and Bergholm, 2017) is one of the popular techniques to model quantum many-body systems but suffers entanglement problems (Choo et al., 2018) . Variational Monte Carlo (VMC) (McMillan, 1965) is a more general methodology to obtain quantum many-body wave functions by optimizing a compact parameterized variational ansatz with data sampled from itself. But how to design variational ansatz with high expressivity to represent real quantum states is still an open problem. Recently traditional machine learning models, such as restricted Boltzmann machine (RBM) (Smolensky, 1986), has been used as variational ansatz (Carleo and Troyer, 2017; Nomura et al., 2017; Choo et al., 2018; Kaubruegger et al., 2018; Choo et al., 2020; Nomura, 2021; Chen et al., 2022) . Following this direction, some studies explore deep Boltzmann machines (Gao and Duan, 2017; Carleo et al., 2018; Pastori et al., 2019) and fully-connected neural networks to represent quantum states (Saito and Kato, 2018; Cai and Liu, 2018; Saito, 2017; 2018; Saito and Kato, 2018) . Most recent studies also use CNN as variational ansatz for square lattice systems (Liang et al., 2018; Choo et al., 2019; Zheng et al., 2021; Liang et al., 2021; Roth and MacDonald, 2021) . And GNN has been applied to non-square lattices and random graph systems (Yang et al., 2020a; Kochkov et al., 2021) . In this work, we explore the potential of using CNN as variational anstaz for non-square lattice quantum spin systems. We propose lattice convolutions that use a set of proposed operations to convert non-square lattices into grid-like augmented lattices on which any existing CNN architectures can be applied. Based on proposed lattice convolution, we design highly expressive lattice convolutional networks (LCN) by leveraging self-gating and attention mechanisms. Experimental results show that our method achieves performance on par or better than the GNN method over the square, honeycomb, triangular, and kagome lattice quantum systems on spin 1/2 J 1 -J 2 Heisenberg model, a prototypical

