SPATIALLY-DISCRETIZED HIGH-TEMPERATURE APPROXIMATIONS AND THEIR O(N) IMPLEMENTATION ON A GRID

Abstract

We consider the problem of performing imaginary-time propagation of wavefunctions on a grid. We demonstrate that spatially-continuous high-temperature approximations can be discretized in such a way that their convergence order is preserved. Requirements of minimal computational work and reutilization of data then uniquely determine the optimal grid, quadrature technique, and propagation method. It is shown that the optimal propagation technique is O(N), with respect to the grid size. The grid technique is utilized to compare the Monte Carlo efficiency of the Trotter–Suzuki approximation against a recently introduced fourth-order high-temperature approximation, while circumventing the issue of statistical noise, which usually prevents such comparisons from being carried out. We document the appearance of a systematic bias in the Monte Carlo estimators that involve temperature differentiation of the density matrix, bias that is due to the dependence of the eigenvalues on the inverse temperature. This bias is negotiated more successfully by the short-time approximations having higher convergence order, which leads to non-trivial computational savings.