Confidence and efficiency scaling in variational quantum Monte Carlo calculations

Based on the central limit theorem, we discuss the problem of evaluation of the statistical error of Monte Carlo calculations using a time-discretized diffusion process. We present a robust and practical method to determine the effective variance of general observables and show how to verify the equilibrium hypothesis by the Kolmogorov-Smirnov test. We then derive scaling laws of the efficiency illustrated by variational Monte Carlo calculations on the two-dimensional electron gas.

Figure 1

$C_{N,k}/C_{N,0}$ as a function of $k$ (black line) and ${\sigma }_{\mathrm{eff}}^2\left(k\right)/C_{N,0}$ (red line). The blue line is ${\mathrm{\Sigma }}_{N,k}/\left(\sqrt{N}{C}_{N,0}\right)$; The crossing of the black and blue curves occurs for $k=31$ giving ${\sigma }_{\mathrm{eff}}^2=1.90C_0$.

Figure 2

${\sigma }_{\mathrm{eff}}^2\left(k\right)/C_0$ for three independent samplings.

Figure 3

Distributions of $x=\frac{{\overline{X}}_{\alpha }-\overline{X}}{{\sigma }_{\mathrm{eff}}}$ for 40 independent samples of 45 000 trials. The black line is the distribution function of the normalized Gaussian.

Figure 4

Distribution functions of $D_{\alpha }$. The red curve stands for the original sampling. The blue curve is obtained with a stride of 15 (see text).

Figure 5

(a) Inverse efficiency, ${\mathrm{Q}}^{-1}$, and (b) acceptance probability, ${\mathsf{R}}_a$, for different values of the Monte Carlo diffusion time step, $\tau$, and the dimension of configuration space, $\nu$ $\left(r_s=35\right)$.

Figure 6

${\sigma }^2/{\sigma }_{\mathrm{eff}}^2$ as a function of the time discretization $\left(r_s=35\right)$. Different patterns correspond to different values of $\nu$ $\left({\sigma }^2=0.04,0.02,0.01\right)$.

Figure 7

Efficiency $\mathrm{Q}$ as a function of $\tau$ for for different values of $m_G$ $(\nu =58$ and $r_s=1)$.

Figure 8

Repartition function of ${ln\left|\mathrm{\Psi }\right|}^2$ $(\nu =122$ and $r_s=10)$. Black line is the normal law; red line is obtained with 20 000 trials.