Population control bias and importance sampling in full configuration interaction quantum Monte Carlo

Population control is an essential component of any projector Monte Carlo algorithm. This control mechanism usually introduces a bias in the sampled quantities that is inversely proportional to the population size. In this paper, we investigate the population control bias in the full configuration interaction quantum Monte Carlo method. We identify the precise origin of this bias and quantify it in general. We show that it has different effects on different estimators and that the shift estimator is particularly susceptible. We derive a reweighting technique, similar to the one used in diffusion Monte Carlo, for correcting this bias and apply it to the shift estimator. We also show that by using importance sampling, the bias can be reduced substantially. We demonstrate the necessity and the effectiveness of applying these techniques for sign-problem-free systems where this bias is especially notable. Specifically, we show results for large one-dimensional Hubbard models and the two-dimensional Heisenberg model, where corrected FCIQMC results are comparable to the other high-accuracy results.

Figure 1

FCIQMC results for the 14-site one-dimensional Hubbard model for different numbers of walkers. The projected energy $E^{\pm }$ is computed using a uniform trial wave function as defined by Eq. (10). The set target numbers of walkers are $100,200,\cdots ,3200$, while their corresponding average number of walkers $\overline{N}$ are always larger due to the overshoot in the constant-shift mode.

Figure 2

A comparison of the two sides of Eq. (28) for the 14-site one-dimensional Hubbard model of Fig. 1. This relation allows numerically estimating the uniform projected energy $E^{\pm }$ from the average shift $\overline{S}$ using the average number of walkers $\overline{N}$ and its covariance with the shift $Cov(S,N)$.

Figure 3

Correction weights for the 14-site Hubbard model using different correction orders $n$. The data corresponds to an average number of walkers $\overline{N}=167$.

Figure 4

Corrected energy estimates of the 14-site Hubbard model. All points have error bars, but most of them are too small to be seen on this plot. Within these error bars, the corrected shift $\overline{S}$ and the corrected uniform projected energy $E^{\pm }$ agree with each other and with the exact energy $E_0$.

Figure 5

Corrected shift estimates with increasing orders of correction $n$ for the 14-site Hubbard model. The data corresponds to the average number of walkers $\overline{N}=167$.

Figure 6

Corrected shift estimates of the 14-site Hubbard model using different orders of correction $n$ and different average number of walkers $\overline{N}$.

Figure 7

Comparison of the energy estimates of the $6\times 6$ Heisenberg model with and without importance sampling (labeled “Guided” and “Original”, respectively). The importance sampling uses a Gutzwiller wave function with parameter $g=3$.

Figure 8

FCIQMC energy estimates of the $6\times 6$ Heisenberg model using and 1000 walkers and different values of the Gutzwiller wave function parameter $g$.