Phase measurement of quantum walks: Application to structure theorem of the positive support of the Grover walk

We obtain a structure theorem of the positive support of the n-th power of the time evolution 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 time evolution of the Grover walk is obtained by lifting up that of the adjacency matrix to the complex plain.