11 research outputs found
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
Alternate two-dimensional quantum walk with a single-qubit coin
We have recently proposed a two-dimensional quantum walk where the
requirement of a higher dimensionality of the coin space is substituted with
the alternance of the directions in which the walker can move [C. Di Franco, M.
Mc Gettrick, and Th. Busch, Phys. Rev. Lett. {\bf 106}, 080502 (2011)]. For a
particular initial state of the coin, this walk is able to perfectly reproduce
the spatial probability distribution of the non-localized case of the Grover
walk. Here, we present a more detailed proof of this equivalence. We also
extend the analysis to other initial states, in order to provide a more
complete picture of our walk. We show that this scheme outperforms the Grover
walk in the generation of - spatial entanglement for any initial
condition, with the maximum entanglement obtained in the case of the particular
aforementioned state. Finally, the equivalence is generalized to wider classes
of quantum walks and a limit theorem for the alternate walk in this context is
presented.Comment: 9 pages, 9 figures, RevTeX
Measurement-induced generation of spatial entanglement in a two-dimensional quantum walk with single-qubit coin
One of the proposals for the exploitation of two-dimensional quantum walks
has been the efficient generation of entanglement. Unfortunately, the
technological effort required for the experimental realization of standard
two-dimensional quantum walks is significantly demanding. In this respect, an
alternative scheme with less challenging conditions has been recently studied,
particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc
Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend
the investigation to a scenario where a measurement is performed on the coin
degree of freedom after the evolution, allowing a further comparison with the
standard two-dimensional Grover walk.Comment: 9 pages, 4 figures, RevTeX
Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin
Multi-dimensional quantum walks usually require large coin spaces. Here we
show that the non-localized case of the spatial density probability of the
two-dimensional Grover walk can be obtained using only a two-dimensional coin
space and a quantum walk in alternate directions. We present a formal proof of
this correspondence and analyze the behavior of the coin-position entanglement
as well as the x-y spatial entanglement in our scheme with respect to the
Grover one. We show that our experimentally simpler scheme allows to entangle
the two orthogonal directions of the walk more efficiently.Comment: 5 pages, 2 figures, RevTeX
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
Alternate two-dimensional quantum walk with a single-qubit coin
We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of x-y spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented