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Heavy-tailed random error in quantum Monte Carlo.

J R Trail1

  • 1University of Cambridge, Cambridge, CB3 0HE, United Kingdom. jrt32@cam.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 21, 2008
PubMed
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The central limit theorem is often invalid for variational Monte Carlo methods. This study shows that a generalized central limit theorem is needed to accurately assess statistical errors in quantum physics calculations.

Area of Science:

  • Quantum physics
  • Computational physics
  • Statistical mechanics

Background:

  • Continuum many-body quantum physics combined with Monte Carlo methods is a standard approach for first-principles calculations.
  • Statistical estimates require a measure of random error for reliability, typically assuming the central limit theorem (CLT).

Purpose of the Study:

  • To investigate the validity of the central limit theorem (CLT) in the context of variational Monte Carlo (VMC) methods.
  • To demonstrate the need for a generalized central limit theorem (GCLT) when CLT assumptions are violated in VMC calculations.

Main Methods:

  • Analysis of the statistical errors in estimates of total energy and local energy variance within the VMC framework.
  • Derivation of the probability distribution for random errors to explicitly show deviations from CLT.

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Main Results:

  • The central limit theorem (CLT) has limited validity or is invalid for the most common variational Monte Carlo (VMC) implementations.
  • Uncontrolled statistical errors are identified in estimates of total energy and local energy variance due to CLT invalidity.
  • Examples are provided where the CLT does not hold for estimated quantities in VMC.

Conclusions:

  • The standard assumption of the central limit theorem (CLT) is often inappropriate for variational Monte Carlo (VMC) methods.
  • A generalized central limit theorem (GCLT) is necessary for accurately characterizing random errors in VMC and other quantum Monte Carlo methods.
  • This work provides a general framework for assessing the reliability of statistical estimates in quantum Monte Carlo simulations.