  We consider the dynamics of strongly localized systems subject to dephasing
noise with arbitrary correlation time. Although noise inevitably induces
delocalization, transport in the noise-induced delocalized phase is
subdiffusive in a parametrically large intermediate-time window. We argue for
this intermediate-time subdiffusive regime both analytically and using
numerical simulations on single-particle localized systems. Furthermore, we
show that normal diffusion is restored in the long-time limit, through
processes analogous to variable-range hopping. With numerical simulations based
on Lanczos exact diagonalization, we demonstrate that our qualitative
conclusions are also valid for interacting systems in the many-body localized
phase.
