  We propose a new algorithm based on the Metropolis sampling method to perform
Monte Carlo integration for path integrals in the recently proposed formulation
of quantum field theories on the Lefschetz thimble. The algorithm is based on a
mapping between the curved manifold defined by the Lefschetz thimble of the
full action and the flat manifold associated with the corresponding quadratic
action. We discuss an explicit method to calculate the residual phase due to
the curvature of the Lefschetz thimble. Finally, we apply this new algorithm to
a simple one-plaquette model where our results are in perfect agreement with
the analytic integration. We also show that for this system the residual phase
does not represent a sign problem.
