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

TL;DR

This paper presents a structure theorem for the positive support of the n-th power of the Grover walk on certain regular graphs, revealing regular phase patterns and spectral relationships with the adjacency matrix.

Contribution

It introduces a novel structure theorem linking the positive support of the Grover walk to a parity-dependent quantum walk on the line, enhancing understanding of quantum walk spectra.

Findings

Established a structure theorem for the positive support of the Grover walk.

Identified regular phase patterns in the quantum walk amplitudes.

Connected the spectrum of the Grover walk to the adjacency matrix spectrum.

Abstract

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.