Heavy-tailed random error in quantum Monte Carlo

The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.

Figure 1

Constant energy surfaces in the large $E$ limit. (a) shows the surface in terms of the electron-nucleus vector as coalescence is approached. The same geometry arises for electron-electron coalescence where the electrons possess different spin. (b) shows the surface in terms of the electron-electron vector for electrons of like spin for the case where no singularity is present in the local kinetic energy. (c) shows the constant energy surface for singularities at the nodal surface due to the local kinetic energy $T_L$.

Figure 2

Variation of the local energy in the presence of singularities of all four types, with an electronic coordinate ${\mathbf{r}}_1$ passing through singular hypersurfaces. $I$, $II$, $III$, and $IV$ denote singularities due to $e\text{−}n$ interaction, $e\text{−}e$ interaction, the nodal surface, and incorrect asymptotic behavior (shown here for the Gaussian case), respectively. Units are arbitrary.

Figure 3

The seed probability density function estimated by a histogram of $r={10}^7$ sampled local energies (black). These are results for an accurate all-electron carbon trial wave function, as described in the text. Shown in gray is the model distribution of Eq. (23) that reproduces the mean and variance of the samples, and the dotted line is the normal distribution that reproduces the same mean and variance.

Figure 4

Probability density function for the random error in the estimated total energy. Results shown are for a kernel estimate of the PDF resulting from ${10}^4$ estimates with $r={10}^3$ for each estimate (black). Gray lines show the predicted distribution, including leptokurtotic tails, for different ${\lambda }_3$ values. For comparison, the normal distribution that emerges in the large $r$ limit is also shown (dotted line).

Figure 5

Probability density function for the random error in the estimated residual variance. Results shown are for a kernel estimate of the PDF resulting from ${10}^4$ estimates with $r={10}^3$ for each estimate (black). Gray lines show the predicted large $r$ limit (a stable PDF). For comparison, a normal distribution with a mean and variance taken to be the sample mean and variance of the data is also shown (dotted line).