  We obtain a structure theorem of the positive support of the $n$-th power of
the Grover walk on $k$-regular graph whose girth is greater than $2(n-1)$. This
structure theorem is provided by the parity of the amplitude of another quantum
walk on the line which depends only on $k$. The phase pattern of this quantum
walk has a curious regularity. We also exactly show how the spectrum of the
$n$-th power of the Grover walk is obtained by lifting up that of the adjacency
matrix to the complex plain.
