  By no fast-forwarding theorem, the simulation time for the Hamiltonian
evolution needs to be $O(\|H\| t)$, which essentially states that one can not
go across the multiple scales as the simulation time for the Hamiltonian
evolution needs to be strictly greater than the physical time. We demonstrated
in the context of the semiclassical Schr\"odinger equation that the
computational cost for a class of observables can be much lower than the
state-of-the-art bounds. In the semiclassical regime (the effective Planck
constant $h \ll 1$), the operator norm of the Hamiltonian is $O(h^{-1})$. We
show that the number of Trotter steps used for the observable evolution can be
$O(1)$, that is, to simulate some observables of the Schr\"odinger equation on
a quantum scale only takes the simulation time comparable to the classical
scale. In terms of error analysis, we improve the additive observable error
bounds [Lasser-Lubich 2020] to uniform-in-$h$ observable error bounds. This is,
to our knowledge, the first uniform observable error bound for semiclassical
Schr\"odinger equation without sacrificing the convergence order of the
numerical method. Based on semiclassical calculus and discrete microlocal
analysis, our result showcases the potential improvements taking advantage of
multiscale properties, such as the smallness of the effective Planck constant,
of the underlying dynamics and sheds light on going across the scale for
quantum dynamics simulation.
