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 quantum many-body model of magnetic materials that captures the exchange interaction between spins. Novelty and Significance. Our work proposes the first pure deep learning approach that does not require any prior knowledge of quantum physics to solve quantum many-body problems on different types of lattice systems. Our approach overcomes the shortcomings of previous neural quantum state methods, which not only require extensive prior knowledge but are also designed for a specific lattice or even a specific regime. However, our method can be seamlessly applied to different lattices and can still achieve competitive or even better performance than existing methods without introducing prior knowledge. As a result, our method possesses great generalizability in practice. This makes our approach of great value in the study of quantum many-body problems. Relations with Prior Work. GNN (Kochkov et al., 2021) is proposed as the first and generic method that can be applied to various lattice shapes. To this end, it is natural that LCN uses the same experiment setting as GNN. While GNN uses different hand-crafted sublattice encoding techniques for different lattice structures, LCN only needs to augment different lattices in a simple and principled way without any prior knowledge. This significantly enhances the generalization capability of LCN in practice. Roth and MacDonald (2021) proposes a general framework called Group-CNN. However, it can only be easily applied to square and triangular lattices. Moreover, it still needs to consider specific symmetry groups for different lattice systems as prior knowledge. Choo et al. ( 2019) applies CNN on square lattice, but it needs to use specific quantum physics knowledge such as point group symmetry and the Marshall sign rule, which is the known sign structure of ground state. However, the Marshal sign rule only works for bipartite graphs (such as square lattice) and non-frustrated regimes.

2. BACKGROUND AND RELATED WORK

In quantum mechanics, a quantum state is represented as a vector in Hilbert space. This vector is a linear combination of observable system configurations {c i }, known as a computational basis. In the context of spin 1/2 systems, each spin can be measured in two states, spin-up or spin-down, which are represented by ↑ and ↓, respectively. All the combinations of spins form a basis. Given N spins, there are in total 2 N configurations in the computational basis. Specifically, a state can be written as |ψ⟩ = 2 N i ψ(c i )|c i ⟩, where |c i ⟩ represents an array of spin configurations of N spins, e.g., ↑↑↓ • • • ↓, and ψ(c i ) is the wave function, which is in general a complex number. The summation is over all possible 2 N spin configurations. The squared norm |ψ(c i )| 2 corresponds to the probability of system collapsing to configuration c i when being measured, and 2 N i |ψ(c i )| 2 = 1 due to normalization.

2.1. GROUND STATES

The ground state of a quantum system is its lowest-energy state. Usually, many physical properties can be determined by the ground state. Particle interactions within a given quantum many-body system are determined by a Hamiltonian, which is an Hermitian matrix H in the Hilbert space. System energy and its corresponding quantum state are governed by the time-independent Schrödinger equation: H|ψ⟩ = E|ψ⟩, which is an eigenvalue equation. The eigenenergy E is the eigenvalue of H and |ψ⟩ is the corresponding eigenvector. In principle, those can be obtained by eigenvalue decomposition given H. The lowest eigenvalue is called the ground state energy, and its associated eigenvector is called the ground state. The ground state and the ground state energy determine the property of the quantum system at zero temperature.

2.2. VARIATIONAL PRINCIPLE IN QUANTUM MECHANICS

Given a system of size N , the dimension of the Hamiltonian matrix is 2 N × 2 N . Since the dimension of the matrix grows exponentially with system size, it is intractable to use eigenvalue decomposition

