  We propose an efficient quantum algorithm for simulating the dynamics of
general Hamiltonian systems. Our technique is based on a power series expansion
of the time-evolution operator in its off-diagonal terms. The expansion
decouples the dynamics due to the diagonal component of the Hamiltonian from
the dynamics generated by its off-diagonal part, which we encode using the
linear combination of unitaries technique. Our method has an optimal dependence
on the desired precision and, as we illustrate, generally requires considerably
fewer resources than the current state-of-the-art. We provide an analysis of
resource costs for several sample models.
