The Penalty Method for Random Walks with Uncertain Energies

We generalize the Metropolis et al. random walk algorithm to the situation where the energy is noisy and can only be estimated. Two possible applications are for long range potentials and for mixed quantum-classical simulations. If the noise is normally distributed we are able to modify the acceptance probability by applying a penalty to the energy difference and thereby achieve exact sampling even with very strong noise. When one has to estimate the variance we have an approximate formula, good in the limit of large number of independent estimates. We argue that the penalty method is nearly optimal. We also adapt an existing method by Kennedy and Kuti and compare to the penalty method on a one dimensional double well.