  We consider Hamiltonian simulation using the first order Lie-Trotter product
formula under the assumption that the initial state has a high overlap with an
energy eigenstate, or a collection of eigenstates in a narrow energy band. This
assumption is motivated by quantum phase estimation (QPE) and digital adiabatic
simulation (DAS). Treating the effective Hamiltonian that generates the
Trotterized time evolution using rigorous perturbative methods, we show that
the Trotter step size needed to estimate an energy eigenvalue within precision
$\epsilon$ using QPE can be improved in scaling from $\epsilon$ to
$\epsilon^{1/2}$ for a large class of systems (including any Hamiltonian which
can be decomposed as a sum of local terms or commuting layers that each have
real-valued matrix elements). For DAS we improve the asymptotic scaling of the
Trotter error with the total number of gates $M$ from $\mathcal{O}(M^{-1})$ to
$\mathcal{O}(M^{-2})$, and for any fixed circuit depth we calculate an
approximately optimal step size that balances the error contributions from
Trotterization and the adiabatic approximation. These results partially
generalize to diabatic processes, which remain in a narrow energy band
separated from the rest of the spectrum by a gap, thereby contributing to the
explanation of the observed similarities between the quantum approximate
optimization algorithm and diabatic quantum annealing at small system sizes.
Our analysis depends on the perturbation of eigenvectors as well as
eigenvalues, and on quantifying the error using state fidelity (instead of the
matrix norm of the difference of unitaries which is sensitive to an overall
global phase).
