  In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment
of one-length path periodically. This small modification to $\mathbb{Z}$
provides localization of the quantum walk. The eigenspace causing this
localization is generated by finite length round trip paths. We find that the
localization is due to the eigenvalues of an underlying random walk. Moreover
we find that the transience of the underlying random walk provides a slow down
of the pseudo velocity of the induced quantum walk and a different limit
distribution from the Konno distribution.
