19,354 research outputs found
Decoherence vs entanglement in coined quantum walks
Quantum versions of random walks on the line and cycle show a quadratic
improvement in their spreading rate and mixing times respectively. The addition
of decoherence to the quantum walk produces a more uniform distribution on the
line, and even faster mixing on the cycle by removing the need for
time-averaging to obtain a uniform distribution. We calculate numerically the
entanglement between the coin and the position of the quantum walker and show
that the optimal decoherence rates are such that all the entanglement is just
removed by the time the final measurement is made.Comment: 11 pages, 6 embedded eps figures; v2 improved layout and discussio
Decoherence in quantum walks - a review
The development of quantum walks in the context of quantum computation, as
generalisations of random walk techniques, led rapidly to several new quantum
algorithms. These all follow unitary quantum evolution, apart from the final
measurement. Since logical qubits in a quantum computer must be protected from
decoherence by error correction, there is no need to consider decoherence at
the level of algorithms. Nonetheless, enlarging the range of quantum dynamics
to include non-unitary evolution provides a wider range of possibilities for
tuning the properties of quantum walks. For example, small amounts of
decoherence in a quantum walk on the line can produce more uniform spreading (a
top-hat distribution), without losing the quantum speed up. This paper reviews
the work on decoherence, and more generally on non-unitary evolution, in
quantum walks and suggests what future questions might prove interesting to
pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work
since first posted and corrections from comments received; some non-trivial
typos fixed. Comments now limited to changes that can be applied at proof
stag
Decoherence in Discrete Quantum Walks
We present an introduction to coined quantum walks on regular graphs, which
have been developed in the past few years as an alternative to quantum Fourier
transforms for underpinning algorithms for quantum computation. We then
describe our results on the effects of decoherence on these quantum walks on a
line, cycle and hypercube. We find high sensitivity to decoherence, increasing
with the number of steps in the walk, as the particle is becoming more
delocalised with each step. However, the effect of a small amount of
decoherence can be to enhance the properties of the quantum walk that are
desirable for the development of quantum algorithms, such as fast mixing times
to uniform distributions.Comment: 15 pages, Springer LNP latex style, submitted to Proceedings of DICE
200
Decoherence can be useful in quantum walks
We present a study of the effects of decoherence in the operation of a
discrete quantum walk on a line, cycle and hypercube. We find high sensitivity
to decoherence, increasing with the number of steps in the walk, as the
particle is becoming more delocalised with each step. However, the effect of a
small amount of decoherence is to enhance the properties of the quantum walk
that are desirable for the development of quantum algorithms. Specifically, we
observe a highly uniform distribution on the line, a very fast mixing time on
the cycle, and more reliable hitting times across the hypercube.Comment: (Imperial College London) 6 (+epsilon) pages, 6 embedded eps figures,
RevTex4. v2 minor changes to correct typos and refs, submitted version. v3
expanded into article format, extra figure, updated refs, Note on "glued
trees" adde
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
Localization of the Grover walks on spidernets and free Meixner laws
A spidernet is a graph obtained by adding large cycles to an almost regular
tree and considered as an example having intermediate properties of lattices
and trees in the study of discrete-time quantum walks on graphs. We introduce
the Grover walk on a spidernet and its one-dimensional reduction. We derive an
integral representation of the -step transition amplitude in terms of the
free Meixner law which appears as the spectral distribution. As an application
we determine the class of spidernets which exhibit localization. Our method is
based on quantum probabilistic spectral analysis of graphs.Comment: 32 page
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