  We discuss how the bath's memory affects the dynamics of a swap gate. We
present an exactly solvable model that shows various dynamical transitions when
treated beyond the Fermi Golden Rule. By moving continuously a single
parameter, the unperturbed Rabi frequency, we sweep through different analytic
properties of the density of states: I) collapsed resonances that split at an
exceptional point in II) two resolved resonances ; III) out-of-band resonances;
IV) virtual states; and V) pure point spectrum. We associate them with
distinctive dynamical regimes: overdamped, damped oscillations, environment
controlled quantum diffusion, anomalous diffusion and localized dynamics
respectively. The frequency of the swap gate depends differently on the
unperturbed Rabi frequency. In region I) there is no oscillation at all, while
in the regions III) and IV) the oscillation frequency is particularly stable
because it is determined by environment's band width. The anomalous diffusion
could be used as a signature for the presence of the elusive virtual states.
