Generator of an abstract quantum walk

Abstract

We give an explicit formula of the generator of an abstract Szegedy evolution operator in terms of the discriminant operator of the evolution. We also characterize the asymptotic behavior of a quantum walker through the spectral property of the discriminant operator by using the discrete analog of the RAGE theorem.

Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise

“…In the past fifteen years, quantum walks with a finite number of internal degrees of freedom have been intensively studied and many deep results have been obtained of them (see, e.g., [2,[4][5][6][7][8][9] and references therein). One typical result in this aspect is the finding that those walks have quite different asymptotic behavior, compared to their classical counterparts.…”

A Note on the Spectral Mapping Theorem of Quantum Walk Models

“…As a first step in this direction, we try to apply our new method to the problems whose spectral structures have been well developed, that is, Szegedy walks [7,10] and abstract quantum walks [6,8,9]. We obtain the following new observation of U by this method which has not discussed well before.…”

“…Remark that there are preceding works of quantum walks in such an abstract setting: [8,9]. There quantum walks on infinite dimensional Hilbert spaces are considered.…”

Unitary equivalence classes of one-dimensional quantum walks II

“…A quantum walk is defined by a pair (U, {H v } v∈V ), in which V is a countable set, {H v } v∈V is a family of separable Hilbert spaces, and U is a unitary operator on H = v∈V H v [20]. In this paper, we discuss one-dimensional (twostate) quantum walks, in which V = Z and H v = C 2 .…”

“…We consider unitary equivalence of quantum walks in the sense of [19,20]. If two quantum walks are unitarily equivalent, then their digraphs and dimensions of their Hilbert spaces are the same.…”