  The accuracy of quantum dynamics simulation is usually measured by the error
of the unitary evolution operator in the operator norm, which in turn depends
on certain norm of the Hamiltonian. For unbounded operators, after suitable
discretization, the norm of the Hamiltonian can be very large, which
significantly increases the simulation cost. However, the operator norm
measures the worst-case error of the quantum simulation, while practical
simulation concerns the error with respect to a given initial vector at hand.
We demonstrate that under suitable assumptions of the Hamiltonian and the
initial vector, 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. In this sense, our result outperforms all
previous error bounds in the quantum simulation literature. Our result extends
that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting.
We also clarify the existence and the importance of commutator scalings of
Trotter and generalized Trotter methods for time-dependent Hamiltonian
simulations.
