3,073 research outputs found
Limit theorems for quantum walks with memory
Recently Mc Gettrick [1] introduced and studied a discrete-time 2-state
quantum walk (QW) with a memory in one dimension. He gave an expression for the
amplitude of the QW by path counting method. Moreover he showed that the return
probability of the walk is more than 1/2 for any even time. In this paper, we
compute the stationary distribution by considering the walk as a 4-state QW
without memory. Our result is consistent with his claim. In addition, we obtain
the weak limit theorem of the rescaled QW. This behavior is striking different
from the corresponding classical random walk and the usual 2-state QW without
memory as his numerical simulations suggested.Comment: Quantum Information and Computation, Vol.10, No.11&12, pp.1004-1017
(2010
Probability distributions and weak limit theorems of quaternionic quantum walks in one dimension
The discrete-time quantum walk (QW) is determined by a unitary matrix whose
component is complex number. Konno (2015) extended the QW to a walk whose
component is quaternion.We call this model quaternionic quantum walk (QQW). The
probability distribution of a class of QQWs is the same as that of the QW. On
the other hand, a numerical simulation suggests that the probability
distribution of a QQW is different from the QW. In this paper, we clarify the
difference between the QQW and the QW by weak limit theorems for a class of
QQWs.Comment: 11 pages, 2 figures, Interdisciplinary Information Sciences (in
press
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa
Limit theorems for a localization model of 2-state quantum walks
We consider 2-state quantum walks (QWs) on the line, which are defined by two
matrices. One of the matrices operates the walk at only half-time. In the usual
QWs, localization does not occur at all. However, our walk can be localized
around the origin. In this paper, we present two limit theorems, that is, one
is a stationary distribution and the other is a convergence theorem in
distribution.Comment: International Journal of Quantum Information, Vol.9, No.3, pp.863-874
(2011
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