A Note on the Spectral Mapping Theorem of Quantum Walk Models

Abstract

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.

Localization of a multi-dimensional quantum walk with one defect

“…Let D ± = ker d ∩ ker(S ± 1). The subspace D B = D + ⊕ D − ⊂ H is called the birth eigenspace of U and its orthogonal complement D I the inherited subspace of U (see [17,26]). As shown elsewhere [34], the restriction U I := U| D I to the inherited subspace is unitarily equivalent to exp(+i arccos T ) ⊕ exp(−i arccos T ) and the restriction U B := U| D B to the birth eigenspace is I D + ⊕(−I D − ).…”

Eigenvalues of quantum walk induced by recurrence properties of the underlying birth and death process: application to computation of an edge state

“…The following lemma is obtained by tracing the proofs of spectral mapping theorems of quantum walks in [7,11] with some modifications of the settings.…”

Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

“…where ϕ(z) = (z + z −1 )/2 and M ± = dim B ± denotes the cardinality of the set {±1} with the convention {±1} M ± = ∅ when M ± = 0. This statement is called the spectral mapping theorem of quantum walks [34,11] and ϕ −1 (σ(T )) is called the inherited part [29,13]. In the case of the Grover walk, the discriminant operator T is unitarily equivalent to the transition probability operator P of the symmetric random walk on the graph where the Grover walk itself is defined.…”