Generalized Lanczos method for systematic optimization of neural-network quantum states

TL;DR

This paper introduces the NQS Lanczos method, combining supervised learning, variational Monte Carlo, and the Lanczos algorithm to systematically enhance neural-network quantum states for many-body systems, especially in frustrated regimes.

Contribution

It presents a novel systematic approach that improves neural-network quantum states using a combination of supervised learning and VMC, with linear computational scaling.

Findings

Systematic energy improvement in the 2D Heisenberg J1-J2 model.

Addresses underfitting issues in neural-network quantum states.

Achieves comparable or better accuracy than traditional methods.

Abstract

Recently, artificial intelligence for science has made significant inroads into various fields of natural science research. In the field of quantum many-body computation, researchers have developed numerous ground state solvers based on neural-network quantum states (NQSs), achieving ground state energies with accuracy comparable to or surpassing traditional methods such as variational Monte Carlo methods, density matrix renormalization group, and quantum Monte Carlo methods. Here, we combine supervised learning, variational Monte Carlo (VMC), and the Lanczos method to develop a systematic approach to improving the NQSs of many-body systems, which we refer to as the NQS Lanczos method. The algorithm mainly consists of two parts: the supervised learning part and the VMC optimization part. Through supervised learning, the Lanczos states are represented by the NQSs. Through VMC, the NQSs are further optimized. We analyze the reasons for the underfitting problem and demonstrate how the NQS Lanczos method systematically improves the energy in the highly frustrated regime of the two-dimensional Heisenberg $J_1$-$J_2$ model. Compared to the existing method that combines the Lanczos method with the restricted Boltzmann machine, the primary advantage of the NQS Lanczos method is its linearly increasing computational cost.