  We study the role played by noise on the QW introduced in [1], a 1D model
that is inspired by a two particle interacting QW. The noise is introduced by a
random change in the value of the phase during the evolution, from a constant
probability distribution within a given interval. The consequences of
introducing such kind of noise depend on both the center value and the width of
that interval: a wider interval manifests as a higher level of noise. For some
range of parameters, one obtains a quasi-localized state, with a diffusive
speed that can be controlled by varying the parameters of the noise. The
existence of this (approximately) localized state for such kind of
time-dependent noise is, to the best of our knowledge, totally new, since
localization (i.e., Anderson localization) is linked in the literature to a
spatial random noise.
