  The Grover walk, which is related to the Grover's search algorithm on a
quantum computer, is one of the typical discrete time quantum walks. However, a
localization of the two-dimensional Grover walk starting from a fixed point is
striking different from other types of quantum walks. The present paper
explains the reason why the walker who moves according to the degree-four
Grover's operator can remain at the starting point with a high probability. It
is shown that the key factor for the localization is due to the degeneration of
eigenvalues of the time evolution operator. In fact, the global time evolution
of the quantum walk on a large lattice is mainly determined by the degree of
degeneration. The dependence of the localization on the initial state is also
considered by calculating the wave function analytically.
