  Nonequilibrium time evolution of large quantum systems is a strong candidate
for quantum advantage. Variational quantum algorithms have been put forward for
this task, but their quantum optimization routines suffer from trainability and
sampling problems. Here, we present a classical pre-processing routine for
variational Hamiltonian simulation that circumvents the need of a quantum
optimization by expanding rigorous error bounds in a perturbative regime for
suitable time steps. The resulting cost function is efficiently computable on a
classical computer. We show that there always exists potential for optimization
with respect to a Trotter sequence of the same order and that the cost value
has the same scaling as for Trotter in simulation time and system size. Unlike
previous work on classical pre-processing, the method is applicable to any
Hamiltonian system independent of locality and interaction lengths. Via
numerical experiments for spin-lattice models, we find that our approach
significantly improves digital quantum simulations capabilities with respect to
Trotter sequences for the same resources. For short times, we find accuracy
improvements of more than three orders of magnitude for our method as compared
to Trotter sequences of the same gate number. Moreover, for a given gate number
and accuracy target, we find that the pre-optimization we introduce enables
simulation times that are consistently more than 10 times longer for a target
accuracy of 0.1%.
