  The numerical simulation of quantum many-body dynamics is typically limited
by the linear growth of entanglement with time. Recently numerical studies have
shown, however, that for 1D Bethe-integrable models the simulation of local
operators in the Heisenberg picture can be efficient as the corresponding
operator-space entanglement grows only logarithmically. Using the spin-1/2 XX
chain as generic example of an integrabel model that can be mapped to free
particles, we here provide a simple explanation for this. We show furthermore
that the same reduction of complexity applies to operators that have a
high-temperature auto correlation function which decays slower than
exponential, i.e., with a power law. This is amongst others the case for models
where the Blombergen-De Gennes conjecture of high-temperature diffusive
dynamics holds. Thus efficient simulability may already be implied by a single
conservation law (like that of total magnetization), as we will illustrate
numerically for the spin-1 XXZ model.
