  We study a model of free fermions on a chain with dynamics generated by
random unitary gates acting on nearest neighbor bonds and present an exact
calculation of time-ordered and out-of-time-ordered correlators. We consider
three distinct cases: the random circuit with spatio-temporal disorder (i) with
and (ii) without particle number conservation and (iii) the particle
non-conserving case with purely temporal disorder. In all three cases, temporal
disorder causes diffusive operator spreading and $\sim\sqrt{t}$ entanglement
growth. This is in sharp contrast to Anderson localization for the case of
static disorder and with the ballistic behavior observed in both the clean case
of Hamiltonian evolution and in fully random unitary quantum circuits.
