16,417 research outputs found
Recurrence of biased quantum walks on a line
The Polya number of a classical random walk on a regular lattice is known to
depend solely on the dimension of the lattice. For one and two dimensions it
equals one, meaning unit probability to return to the origin. This result is
extremely sensitive to the directional symmetry, any deviation from the equal
probability to travel in each direction results in a change of the character of
the walk from recurrent to transient. Applying our definition of the Polya
number to quantum walks on a line we show that the recurrence character of
quantum walks is more stable against bias. We determine the range of parameters
for which biased quantum walks remain recurrent. We find that there exist
genuine biased quantum walks which are recurrent.Comment: Journal reference added, minor corrections in the tex
Connecting the discrete and continuous-time quantum walks
Recently, quantized versions of random walks have been explored as effective
elements for quantum algorithms. In the simplest case of one dimension, the
theory has remained divided into the discrete-time quantum walk and the
continuous-time quantum walk. Though the properties of these two walks have
shown similarities, it has remained an open problem to find the exact relation
between the two. The precise connection of these two processes, both quantally
and classically, is presented. Extension to higher dimensions is also
discussed.Comment: 5 pages, 1 figur
Multigraph models for causal quantum gravity and scale dependent spectral dimension
We study random walks on ensembles of a specific class of random multigraphs
which provide an "effective graph ensemble" for the causal dynamical
triangulation (CDT) model of quantum gravity. In particular, we investigate the
spectral dimension of the multigraph ensemble for recurrent as well as
transient walks. We investigate the circumstances in which the spectral
dimension and Hausdorff dimension are equal and show that this occurs when rho,
the exponent for anomalous behaviour of the resistance to infinity, is zero.
The concept of scale dependent spectral dimension in these models is
introduced. We apply this notion to a multigraph ensemble with a measure
induced by a size biased critical Galton-Watson process which has a scale
dependent spectral dimension of two at large scales and one at small scales. We
conclude by discussing a specific model related to four dimensional CDT which
has a spectral dimension of four at large scales and two at small scales.Comment: 30 pages, 3 figures, references added, minor changes in the abstract
to match the published versio
Implementing the one-dimensional quantum (Hadamard) walk using a Bose-Einstein Condensate
We propose a scheme to implement the simplest and best-studied version of
quantum random walk, the discrete Hadamard walk, in one dimension using
coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC).
Implementation of quantum walk using BEC gives access to the familiar quantum
phenomena on a macroscopic scale. This paper uses rf pulse to implement
Hadamard operation (rotation) and stimulated Raman transition technique as
unitary shift operator. The scheme suggests implementation of Hadamard
operation and unitary shift operator while the BEC is trapped in long Rayleigh
range optical dipole trap. The Hadamard rotation and a unitary shift operator
on BEC prepared in one of the internal state followed by a bit flip operation,
implements one step of the Hadamard walk. To realize a sizable number of steps,
the process is iterated without resorting to intermediate measurement. With
current dipole trap technology it should be possible to implement enough steps
to experimentally highlight the discrete quantum random walk using a BEC
leading to further exploration of quantum random walks and its applications.Comment: 7 pages, 3 figure
Weak limits for quantum random walks
We formulate and prove a general weak limit theorem for quantum random walks
in one and more dimensions. With denoting position at time , we show
that converges weakly as to a certain distribution which
is absolutely continuous and of bounded support. The proof is rigorous and
makes use of Fourier transform methods. This approach simplifies and extends
certain preceding derivations valid in one dimension that make use of
combinatorial and path integral methods
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