Trotter formula is one of the most widely used methods for time-dependent Hamiltonian simulation due to its simplicity and efficiency, and there has been remarkable progress in recent years on better and sharper error bounds of Trotter formula. This talk will focus on improved theoretical complexity estimates of Trotter formula for two specific scenarios. One is simulating Schrödinger equation with a time-dependent effective mass, where the corresponding Hamiltonian after spatial discretization is very large. While existing complexity estimates typically use operator norms to measure the accuracy and thus suffer from the large Hamiltonian, we demonstrate that, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. The second example is near adiabatic dynamics, where the Hamiltonian is assumed to have a gap condition. By viewing the Trotter evolution operators as discrete near adiabatic evolution operators, we find it suffices to discretize the dynamics using first-order Trotter formula with O(1) time step size. Furthermore, under suitable assumptions, the complexity of the first order Trotter formula can even scale poly-logarithmically in terms of accuracy.
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