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

TL;DR

The paper tackles the difficulty of obtaining accurate ground states in large quantum many-body systems by integrating neural-network quantum states with the Lanczos method. It introduces the NQS Lanczos framework, combining a CNN-based sign and amplitude network representation with a supervised-learning Lanczos (SLL) step and a variational Monte Carlo (VMC) refinement on a superposition of Lanczos states. A core advantage is linear scaling in the number of Lanczos steps, since the method avoids higher-order moment evaluations, enabling more iterations within practical resources. Numerical tests on the frustrated 2D J1-J2 Heisenberg model demonstrate substantial energy improvements and improved sign structures, with VMC providing additional gains, highlighting the approach’s potential for scalable, accurate ground-state calculations in complex quantum systems.

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.

Figures (8)

• Figure 1: Schematic of the structure of the output layers of the sign network and the amplitude network. $v_{\text{out}}$ represents the output of the output layer, and $k_{\text{out}-1}$ denotes the convolutional kernel of the output layer. The flow labeled by symmetry means data argumentation by symmetry operations, which includes the spin-flip and the $\mathcal{C}_{\mathrm{4v}}$ symmetries. The output layer of the amplitude network has a single channel, while that of the sign network has two channels, which are used for the binary classification task of the sign (positive or negative). The outputs of the sign and amplitude network are denoted as $S$ and $A$, respectively. The corresponding wave function can be written as $\psi = S \cdot e^{A}$.
• Figure 2: Illustration of the loss function of the amplitude network. The horizontal axis represents the configurations, while the vertical axis represents the probability density. The curves labeled by 'trg' and 'net' serve as schematic representations of the distributions described by $|\psi^{\text{trg}}_{i}(\sigma)|^2$ and $|\psi^{\text{net}}_{i}(\sigma, \theta)|^2$, respectively. The arrows indicate the expected changes in the 'net' distribution when supervised learning. Both those two distributions are used to generate the samples for supervised learning.
• Figure 3: Flowchart of the NQS Lanczos method. The method begins with $|\psi^\mathrm{net}_0\rangle$ as a starting point and ultimately outputs the improved energy $\tilde{E}$ and state $|\tilde{\Psi}\rangle$. $|\tilde{\Psi}\rangle$ can be used as a new $|\psi^\mathrm{net}_0\rangle$ for the next NQS Lanczos loop. The NQS Lanczos method consists of two parts, the SLL algorithm and the VMC optimization part. The SLL algorithm further consists of two parts: supervised learning of Lanczos states and diagonalization of the Hamiltonian matrix. The output of the SLL algorithm consists of the superposition state $|\Psi\rangle$ and the improved energy $E$. The VMC optimization part further optimizes $|\Psi\rangle$ and gives the final output of the NQS Lanczos method. The number of the Lanczos steps is denoted as $p$. The matrices $M$ and $H$ are used to produce the superposition state, as shown in Appendix \ref{['sec:diagonalization']}.
• Figure 4: Improved energies on the $4 \times 4$ lattice. The horizontal axis represents the number of Lanczos steps. The vertical axis indicates the relative error of the improved energies with respect to the energy obtained from ED, $\epsilon_\text{rel} = (E - E_\text{ED}) / | E_\text{ED}|$.
• Figure 5: Improved energies on the $6 \times 6$ lattice. The model is test with $J_2/J_1 = 0.5$, $0.55$, and $0.6$. The vertical axis indicates the rescaled relative error, $\epsilon_\text{rel} \times 10^{-3}$, of the improved energies with respect to the energy obtained from ED. At $p \neq 0$, the points of SLL indicate the energies obtained by the SLL algorithm. At $p = 0$, the points of SLL indicate the energies of the initial states. The points of VMCL ($p = n$, $\text{ANet}_j$) indicate the energies obtained by the VMC. The superposition state consists of $n+1$ NQSs. $\text{ANet}_j$ indicates that the amplitude network of Lanczos step $j$ is optimized in the VMC.