Optimal energy estimation in path-integral Monte Carlo simulations

We investigate the properties of two standard energy estimators used in path-integral Monte Carlo simulations. By disentangling the variance of the estimators and their autocorrelation times we analyze the dependence of the performance on the update algorithm and present a detailed comparison of refined update schemes such as multigrid and staging techniques. We show that a proper combination of the two estimators leads to a further reduction of the statistical error of the estimated energy with respect to the better of the two without extra cost.