ANALYSIS OF THE STATISTICAL ERRORS IN CONDITIONED REAL-TIME PATH-INTEGRAL METHODS

An analysis is provided of the statistical errors in the Monte Carlo evaluation of the conditioned real time discretized path integral propagator, The analysis considers the case of a harmonic potential. For this case, all the required integrals can be performed analytically. This analysis is also relevant to a semiclassical evaluation of the integrals in more general problems. It is found (in the simplest case) that the optimal relative statistical error per independent sampling is proportional to D(D/2), where D is the dimensionality of the integrand. Therefore, the number of Monte Carlo samplings must scale as D(D) in order to achieve a desired level of accuracy. Since D is proportional to the number of time steps in the discretized path integral, this analysis demonstrates that the length of the calculations required increases very rapidly as the number of time steps is increased.