Localization of Quantum Walks Induced by Recurrence Properties of Random Walks

Abstract

We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization of the corresponding QW. We find the following two fundamental derivations of localization of the QW. The first one is the set of all the ℓ 2 summable eigenvectors of the corresponding RW. The second one is the orthogonal complement, whose eigenvalues are ±1, of the subspace induced by the RW. In particular, as a consequence, for an infinite half line, we show that localization of the QW can be ensured by the positive recurrence of the corresponding RWs, and also that the existence of only one self loop affects localization properties. *

The Witten index for 1D supersymmetric quantum walks with anisotropic coins

“…Type II, II', III coins are defined likewise. That is, we shall always assume that (19) holds true for each ♯ = L, R, whenever the four types of the isotropic coin thus defined.…”

Periodicity of the Discrete-time Quantum Walk on a Finite Graph

“…[17,22]): U sz is a 2m  2m-matrix and We often use the symbol T 0 for the transition matrix for the simple random walk. Let us introduce the notion of ''periodicity'', which is the main theme in this paper.…”

“…Then the following mapping theorem has been obtained; see [9,17]. For more general abstract quantum walks, it is generalized [18].…”

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.…”