Abstract
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 re-weighting 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.
Highlights
Projector Monte Carlo algorithms, including Green’s function Monte Carlo (GFMC) [1], diffusion Monte Carlo (DMC) [2], and full configuration interaction quantum Monte Carlo (FCIQMC) [3], have become indispensable tools in extracting the ground state properties of various quantum systems. These methods employ a stochastic version of the power method; a method that starts from an initial wave function with a nonzero overlap with the ground state and filters out higher excited states by a repeated application of a suitable imaginary-time propagator of the Schrödinger equation
The population bias has been masked in these applications by the initiator bias and did not pose a practical concern. In contrast to these systems, there are some sign-problem-free ones, where no minimum number of walkers is required for getting stable FCIQMC simulation, and it is a matter of obtaining more samples to reduce the statistical error bars
We show that population control leads to a biasing source term in the master equation, which equals the covariance between the shift parameter and the wave function
Summary
Projector Monte Carlo algorithms, including Green’s function Monte Carlo (GFMC) [1], diffusion Monte Carlo (DMC) [2], and full configuration interaction quantum Monte Carlo (FCIQMC) [3], have become indispensable tools in extracting the ground state properties of various quantum systems. The population bias has been masked in these applications by the initiator bias and did not pose a practical concern In contrast to these systems, there are some sign-problem-free ones, where no minimum number of walkers is required for getting stable FCIQMC simulation, and it is a matter of obtaining more samples to reduce the statistical error bars. For such systems, the population control bias is the only systematic bias, making its study most convenient in these cases. We demonstrate the effectiveness of the correction technique and the importance sampling by applying them to the one-dimensional Hubbard model and the two-dimensional Heisenberg model with large lattice sizes
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