Taming the expressiveness of neural-network wave functions for robust convergence to quantum many-body states

Abstract

Neural networks are emerging as a powerful tool for determining the quantum states of interacting many-body fermionic systems. The standard approach trains a neural-network ansatz by minimizing the mean local energy estimated from Monte Carlo samples. However, this typically results in large sample-to-sample fluctuations in the estimated mean energy and thus slow convergence of the energy minimization. We propose that minimizing a logarithmically compressed variance of the local energies can dramatically improve convergence. Moreover, this loss function can be adapted to systematically obtain the energy spectrum across multiple runs. We demonstrate these ideas for spin-1/2 particles in a 2D harmonic trap with attractive Poschl-Teller interactions between opposite spins.

Figures (6)

• Figure 1: PE and PBE properties of NN wave functions. a. Spatial distributions of up- and down-spin particles and the distributions of $\sigma_L$ and $\bar{E}_L$ for $s_{I}$=0.002. The distributions are obtained from 200 sets of 200,000 Monte Carlo samples. The spatial distributions are taken from one representative set. b. Same as a but for $s_{I}$=0.4. c. Median $\sigma_L$, spread $s_\sigma$, median $\bar{E}_L$, and the fraction of sets with negative $\bar{E}_L$ as a function of $s_{I}$. Each gray dot is obtained from one trial of 200 sets. There are 20 trials, and the thick line connects the median values.
• Figure 2: Comparison of log-variance minimization and mean-energy minimization for $N_\uparrow = 1$, $N_\downarrow = 1$. $(L, n_H, h_H) = (2, 2, 2)$. The neural-network wave functions are randomly initialized with $s_{I} = 0.002$ (red, blue) or $s_{I} = 0.4$ (orange, cyan). a. Examples of $\sigma_L$ as a function of iteration number for log-variance minimization (red, orange) and mean-energy minimization (blue, cyan). b. Number of iterations required to reach the stopping criterion. Runs for which $\sigma_L$ does not decrease below 0.1 are excluded. c. Minimum $\sigma_L$ achieved in the runs. d. Mean values of $\sigma_L$ over the last 20 iterations before stopping. e. Energy levels obtained in the runs. The numbers indicate the median values of $|\Delta E|$ relative to those obtained by exact diagonalization. The black lines in the violin plots in b– d show the medians.
• Figure 3: Obtaining the energy spectrum for $N_\uparrow = 1$, $N_\downarrow = 1$. $(L, n_H, h_H) = (3, 5, 5)$, and $S_I = 0.2$. Different runs converge to different energy levels. Excluding energies obtained in previous runs leads to new energy levels.
• Figure 4: Increasing system size. Here, $N = N_\uparrow + N_\downarrow$ with $N_\uparrow = N_\downarrow$. Particle density, $\bar{E}_L$, and $\sigma_L$ are shown for $N=6, 8, 10, 12$. The network structure was $(L, n_H, h_H)=(5, 5, 5)$ for $N=6$ and $(L, n_H, h_H)=(5, 5, 10)$ for the other system sizes.
• Figure S1: Particle densities and distributions of $\bar{E}_L$ and $\sigma_L$ for 100 sample sets with $N_\uparrow = 1$ and $N_\downarrow = 1$, obtained from two trial wave functions defined by a Eq. \ref{['eqn-PE']} and b Eq. \ref{['eqn-PBE']}. The insets show enlarged views of the low-$\bar{E}_L$ and low-$\sigma_L$ regions. The red line denotes the ground-state energy.