High-precision simulation of finite-size thermalizing systems at long times

To simulate thermalizing systems at long times, the most straightforward approach is to calculate the thermal properties at the corresponding energy. In a quantum many-body system of size N, for local observables and many initial states, this approach has an error of $O(1/N)$, which is reminiscent of the finite-size error of the equivalence of ensembles. In this paper, we propose a simple and efficient numerical method so that the simulation error is of higher order in $1/N$. This finite-size error scaling is proved by assuming the eigenstate thermalization hypothesis.