3,864 research outputs found
Discrete time quantum walks on percolation graphs
Randomly breaking connections in a graph alters its transport properties, a
model used to describe percolation. In the case of quantum walks, dynamic
percolation graphs represent a special type of imperfections, where the
connections appear and disappear randomly in each step during the time
evolution. The resulting open system dynamics is hard to treat numerically in
general. We shortly review the literature on this problem. We then present our
method to solve the evolution on finite percolation graphs in the long time
limit, applying the asymptotic methods concerning random unitary maps. We work
out the case of one dimensional chains in detail and provide a concrete, step
by step numerical example in order to give more insight into the possible
asymptotic behavior. The results about the case of the two-dimensional integer
lattice are summarized, focusing on the Grover type coin operator.Comment: 22 pages, 3 figure
Recurrence for discrete time unitary evolutions
We consider quantum dynamical systems specified by a unitary operator U and
an initial state vector \phi. In each step the unitary is followed by a
projective measurement checking whether the system has returned to the initial
state. We call the system recurrent if this eventually happens with probability
one. We show that recurrence is equivalent to the absence of an absolutely
continuous part from the spectral measure of U with respect to \phi. We also
show that in the recurrent case the expected first return time is an integer or
infinite, for which we give a topological interpretation. A key role in our
theory is played by the first arrival amplitudes, which turn out to be the
(complex conjugated) Taylor coefficients of the Schur function of the spectral
measure. On the one hand, this provides a direct dynamical interpretation of
these coefficients; on the other hand it links our definition of first return
times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte
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
Evaluating the CDMA System Using Hidden Markov and Semi Hidden Markov Models
CDMA is an important and basic part of today’s communications technologies. This technology can be analyzed efficiently by reducing the time, computation burden, and cost by characterizing the physical layer with a Markov Model. Waveform level simulation is generally used for simulating different parts of a digital communication system. In this paper, we introduce two different mathematical methods to model digital communication channels. Hidden Markov and Semi Hidden Markov models’ applications have been investigated for evaluating the DS-CDMA link performance with different parameters. Hidden Markov Models have been a powerful mathematical tool that can be applied as models of discrete-time series in many fields successfully. A semi-hidden Markov model as a stochastic process is a modification of hidden Markov models with states that are no longer unobservable and less hidden. A principal characteristic of this mathematical model is statistical inertia, which admits the generation, and analysis of observation symbol contains frequent runs. The SHMMs cause a substantial reduction in the model parameter set. Therefore in most cases, these models are computationally more efficient models compared to HMMs. After 30 iterations for different Number of Interferers, all parameters have been estimated as the likelihood become constant by the Baum Welch algorithm. It has been demonstrated that by employing these two models for different Numbers of Interferers and Number of symbols, Error sequences can be generated, which are statistically the same as the sequences derived from the CDMA simulation. An excellent match confirms both models’ reliability to those of the underlying CDMA-based physical layer
Convex Relaxations for Permutation Problems
Seriation seeks to reconstruct a linear order between variables using
unsorted, pairwise similarity information. It has direct applications in
archeology and shotgun gene sequencing for example. We write seriation as an
optimization problem by proving the equivalence between the seriation and
combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic
minimization problem over permutations). The seriation problem can be solved
exactly by a spectral algorithm in the noiseless case and we derive several
convex relaxations for 2-SUM to improve the robustness of seriation solutions
in noisy settings. These convex relaxations also allow us to impose structural
constraints on the solution, hence solve semi-supervised seriation problems. We
derive new approximation bounds for some of these relaxations and present
numerical experiments on archeological data, Markov chains and DNA assembly
from shotgun gene sequencing data.Comment: Final journal version, a few typos and references fixe
Rotor walks on general trees
The rotor walk on a graph is a deterministic analogue of random walk. Each
vertex is equipped with a rotor, which routes the walker to the neighbouring
vertices in a fixed cyclic order on successive visits. We consider rotor walk
on an infinite rooted tree, restarted from the root after each escape to
infinity. We prove that the limiting proportion of escapes to infinity equals
the escape probability for random walk, provided only finitely many rotors send
the walker initially towards the root. For i.i.d. random initial rotor
directions on a regular tree, the limiting proportion of escapes is either zero
or the random walk escape probability, and undergoes a discontinuous phase
transition between the two as the distribution is varied. In the critical case
there are no escapes, but the walker's maximum distance from the root grows
doubly exponentially with the number of visits to the root. We also prove that
there exist trees of bounded degree for which the proportion of escapes
eventually exceeds the escape probability by arbitrarily large o(1) functions.
No larger discrepancy is possible, while for regular trees the discrepancy is
at most logarithmic.Comment: 32 page
Non-Markovian decay beyond the Fermi Golden Rule: Survival Collapse of the polarization in spin chains
The decay of a local spin excitation in an inhomogeneous spin chain is
evaluated exactly: I) It starts quadratically up to a spreading time t_{S}. II)
It follows an exponential behavior governed by a self-consistent Fermi Golden
Rule. III) At longer times, the exponential is overrun by an inverse power law
describing return processes governed by quantum diffusion. At this last
transition time t_{R} a survival collapse becomes possible, bringing the
polarization down by several orders of magnitude. We identify this strongly
destructive interference as an antiresonance in the time domain. These general
phenomena are suitable for observation through an NMR experiment.Comment: corrected versio
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