Spectral mapping theorem of an abstract quantum walk

Keisuke Asahara, Daiju Funakawa, Etsuo Segawa +2 more

Sho Kubota, Kei Saito, Yusuke Yoshie

Kaname Matsue, Osamu Ogurisu, Etsuo Segawa

We discuss the description of eigenspace of a quantum walk model U with an associating linear operator T in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of U without the spectral decomposition of T. Arguments in this direction reveal the eigenspaces of U characterized by the generalized kernels of linear operators given by T.

Leonid Evgenevich Fedichkin, Fedor Pavlovich Meshchaninov

Yusuke Higuchi, Etsuo Segawa, Akito Suzuki

Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, we introduce an abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on $\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a unitary involution on $\mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.

Jenia Rousseva, Yevgeniy Kovchegov