  We address the dynamics of continuous-time quantum walks on one-dimensional
disordered lattices inducing dynamical noise in the system. Noise is described
as time-dependent fluctuations of the tunneling amplitudes between adjacent
sites, and attention is focused on non-Gaussian telegraph noise, going beyond
the usual assumption of fast Gaussian noise. We observe the emergence of two
different dynamical behaviors for the walker, corresponding to two opposite
noise regimes: slow noise (i.e. strong coupling with the environment) confines
the walker into few lattice nodes, while fast noise (weak coupling) induces a
transition between quantum and classical diffusion over the lattice. A phase
transition between the two dynamical regimes may be observed by tuning the
ratio between the autocorrelation time of the noise and the coupling between
the walker and the external environment generating the noise. We also address
the non-Markovianity of the quantum map by assessing its memory effects, as
well as evaluating the information backflow to the system. Our results suggest
that the non-Markovian character of the evolution is linked to the dynamical
behavior in the slow noise regime, and that fast noise induces a Markovian
dynamics for the walker.
