  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.

Computational Physics;Chemical Physics