  Exploiting multi-dimensional quantum walks as feasible platforms for quantum
computation and quantum simulation is attracting constantly growing attention
from a broad experimental physics community. Here, we propose a two-dimensional
quantum walk scheme with a single-qubit coin that presents, in the considered
regimes, a strong localization-like effect on the walker. The result could
provide new possible directions for the implementation of quantum algorithms or
from the point of view of quantum simulation. We characterize the
localization-like effect in terms of the parameters of a step-dependent qubit
operation that acts on the coin space after any standard coin operation,
showing that a proper choice can guarantee a non-negligible probability of
finding the walker in the origin even for large times. We finally discuss the
robustness to imperfections, a qualitative relation with coherences behavior,
and possible experimental realizations of this model with the current
state-of-the-art settings.
