Spectral mapping theorem of an abstract quantum walk

Abstract

Given two Hilbert spaces, H and K, we introduce an abstract unitary operator U on H and its discriminant T on K induced by a coisometry from H to K and a unitary involution on 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.

Generator of an abstract quantum walk

“…In this section, we treat a specific class of abstract QWs, an extension of the Szegedy walks. Let us recall some notations and facts from [13]. Let H and K be complex Hilbert spaces.…”

“…We state the basic properties of these subspaces without proof. For the proof, one can consult [13], where we used the notations L, L 1 , and L 0 with D = L, D 1 = L 1 , and D 0 = L.…”

“…Using the theorem, they studied the spectral and asymptotic properties of the Grover walks on crystal lattices. In our previous paper [13], we studied an abstract evolution of the form…”

“…Section 3 contains a brier review of the abstract Szegedy walk. We summarize the results from [13] without proofs. In Section 4, we state the main results of this paper and prove (7) and (8).…”

“…general, since our analysis proposed here essentially works well when the time evolution is decomposed into two involution operators U = E 2 E 1[13]. Applying our abstract QW to the time-dependent QW effectively is an open problem.…”