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

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, reinforcement learning, 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 reinforcement learning part. Through supervised learning, the Lanczos states are represented by the NQSs. Through reinforcement learning, 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.

Variational optimization of the amplitude of neural-network quantum many-body ground states

Neural-network quantum states (NQSs), variationally optimized by combining traditional methods and deep learning techniques, is a new way to find quantum many-body ground states and gradually becomes a competitor of traditional variational methods. However, there are still some difficulties in the optimization of NQSs, such as local minima, slow convergence, and sign structure optimization. Here, we split a quantum many-body variational wave function into a multiplication of a real-valued amplitude neural network and a sign structure, and focus on the optimization of the amplitude network while keeping the sign structure fixed. The amplitude network is a convolutional neural network (CNN) with residual blocks, namely a ResNet. Our method is tested on three typical quantum many-body systems. The obtained ground state energies are lower than or comparable to those from traditional variational Monte Carlo (VMC) methods and density matrix renormalization group (DMRG). Surprisingly, for the frustrated Heisenberg $J_1$-$J_2$ model, our results are better than those of the complex-valued CNN in the literature, implying that the sign structure of the complex-valued NQS is difficult to be optimized. We will study the optimization of the sign structure of NQSs in the future.