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Quantum walks with a one-dimensional coin

Mar 24, 2016

8 pages

Published in:

Phys.Rev.A 93 (2016) 6, 062334

e-Print:

1603.07666 [quant-ph]

DOI:

10.1103/PhysRevA.93.062334

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Abstract: (arXiv)

Quantum walks (QWs) describe particles evolving coherently on a lattice. The internal degree of freedom corresponds to a Hilbert space, called coin system. We consider QWs on Cayley graphs of some group

G

. In the literature, investigations concerning infinite

G

have been focused on graphs corresponding to

G=\mathbb{Z}^d

with coin system of dimension 2, whereas for one-dimensional coin (so called scalar QWs) only the case of finite

G

has been studied. Here we prove that the evolution of a scalar QW with

G

infinite Abelian is trivial, providing a thorough classification of this kind of walks. Then we consider the infinite dihedral group

D_\infty

, that is the unique non-Abelian group

G

containing a subgroup

H\cong\mathbb{Z}

with two cosets. We characterize the class of QWs on the Cayley graphs of

D_\infty

and, via a coarse-graining technique, we show that it coincides with the class of spinorial walks on

\mathbb{Z}

which satisfies parity symmetry. This class of QWs includes the Weyl and the Dirac QWs. Remarkably, there exist also spinorial walks that are not coarse-graining of a scalar QW, such as the Hadamard walk.

Note:

8 pages, 4 figures

03.67.Ac
02.20.-a

References(32)

Figures(6)

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