  We provide numerical evidence that the nonlinear searching algorithm
introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of
quantum walks with effective nonlinear phase, can be extended to the finite
2-dimensional grid, keeping the same computational advantage \BHg{with} respect
to the classical algorithms. For this purpose, we have considered the free
lattice Hamiltonian, with linear dispersion relation introduced by Childs and
Ge \cite{Childs_2014}. The numerical simulations showed that the walker finds
the marked vertex in $O(N^{1/4} \log^{3/4} N) $ steps, with probability
$O(1/\log N)$, for an overall complexity of $O(N^{1/4}\log^{7/4}N)$. We also
proved that there exists an optimal choice of the walker parameters to avoid
that the time measurement precision affects the complexity searching time of
the algorithm.
