156 research outputs found
Correlated Markov Quantum Walks
We consider the discrete time unitary dynamics given by a quantum walk on
performed by a particle with internal degree of freedom, called coin
state, according to the following iterated rule: a unitary update of the coin
state takes place, followed by a shift on the lattice, conditioned on the coin
state of the particle. We study the large time behavior of the quantum
mechanical probability distribution of the position observable in for
random updates of the coin states of the following form. The random sequences
of unitary updates are given by a site dependent function of a Markov chain in
time, with the following properties: on each site, they share the same
stationnary Markovian distribution and, for each fixed time, they form a
deterministic periodic pattern on the lattice.
We prove a Feynman-Kac formula to express the characteristic function of the
averaged distribution over the randomness at time in terms of the nth power
of an operator . By analyzing the spectrum of , we show that this
distribution posesses a drift proportional to the time and its centered
counterpart displays a diffusive behavior with a diffusion matrix we compute.
Moderate and large deviations principles are also proven to hold for the
averaged distribution and the limit of the suitably rescaled corresponding
characteristic function is shown to satisfy a diffusion equation.
An example of random updates for which the analysis of the distribution can
be performed without averaging is worked out. The random distribution displays
a deterministic drift proportional to time and its centered counterpart gives
rise to a random diffusion matrix whose law we compute. We complete the picture
by presenting an uncorrelated example.Comment: 37 pages. arXiv admin note: substantial text overlap with
arXiv:1010.400
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
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
Green function approach for scattering quantum walks
In this work a Green function approach for scattering quantum walks is
developed. The exact formula has the form of a sum over paths and always can be
cast into a closed analytic expression for arbitrary topologies and position
dependent quantum amplitudes. By introducing the step and path operators, it is
shown how to extract any information about the system from the Green function.
The method relevant features are demonstrated by discussing in details an
example, a general diamond-shaped graph.Comment: 13 pages, 6 figures, this article was selected by APS for Virtual
Journal of Quantum Information, Vol 11, Iss 11 (2011
Random Time-Dependent Quantum Walks
We consider the discrete time unitary dynamics given by a quantum walk on the
lattice performed by a quantum particle with internal degree of freedom,
called coin state, according to the following iterated rule: a unitary update
of the coin state takes place, followed by a shift on the lattice, conditioned
on the coin state of the particle. We study the large time behavior of the
quantum mechanical probability distribution of the position observable in
when the sequence of unitary updates is given by an i.i.d. sequence of
random matrices. When averaged over the randomness, this distribution is shown
to display a drift proportional to the time and its centered counterpart is
shown to display a diffusive behavior with a diffusion matrix we compute. A
moderate deviation principle is also proven to hold for the averaged
distribution and the limit of the suitably rescaled corresponding
characteristic function is shown to satisfy a diffusion equation. A
generalization to unitary updates distributed according to a Markov process is
also provided. An example of i.i.d. random updates for which the analysis of
the distribution can be performed without averaging is worked out. The
distribution also displays a deterministic drift proportional to time and its
centered counterpart gives rise to a random diffusion matrix whose law we
compute. A large deviation principle is shown to hold for this example. We
finally show that, in general, the expectation of the random diffusion matrix
equals the diffusion matrix of the averaged distribution.Comment: Typos and minor errors corrected. To appear In Communications in
Mathematical Physic
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
Implementation of Clifford gates in the Ising-anyon topological quantum computer
We give a general proof for the existence and realizability of Clifford gates
in the Ising topological quantum computer. We show that all quantum gates that
can be implemented by braiding of Ising anyons are Clifford gates. We find that
the braiding gates for two qubits exhaust the entire two-qubit Clifford group.
Analyzing the structure of the Clifford group for n \geq 3 qubits we prove that
the the image of the braid group is a non-trivial subgroup of the Clifford
group so that not all Clifford gates could be implemented by braiding in the
Ising topological quantum computation scheme. We also point out which Clifford
gates cannot in general be realized by braiding.Comment: 17 pages, 10 figures, RevTe
Disordered Quantum Walks in one lattice dimension
We study a spin-1/2-particle moving on a one dimensional lattice subject to
disorder induced by a random, space-dependent quantum coin. The discrete time
evolution is given by a family of random unitary quantum walk operators, where
the shift operation is assumed to be deterministic. Each coin is an independent
identically distributed random variable with values in the group of two
dimensional unitary matrices. We derive sufficient conditions on the
probability distribution of the coins such that the system exhibits dynamical
localization. Put differently, the tunneling probability between two lattice
sites decays rapidly for almost all choices of random coins and after arbitrary
many time steps with increasing distance. Our findings imply that this effect
takes place if the coin is chosen at random from the Haar measure, or some
measure continuous with respect to it, but also for a class of discrete
probability measures which support consists of two coins, one of them being the
Hadamard coin.Comment: minor change
Experimental simulation and limitations of quantum walks with trapped ions
We examine the prospects of discrete quantum walks (QWs) with trapped ions.
In particular, we analyze in detail the limitations of the protocol of
Travaglione and Milburn (PRA 2002) that has been implemented by several
experimental groups in recent years. Based on the first realization in our
group (PRL 2009), we investigate the consequences of leaving the scope of the
approximations originally made, such as the Lamb--Dicke approximation. We
explain the consequential deviations from the idealized QW for different
experimental realizations and an increasing number of steps by taking into
account higher-order terms of the quantum evolution. It turns out that these
become dominant after a few steps already, which is confirmed by experimental
results and is currently limiting the scalability of this approach. Finally, we
propose a new scheme using short laser pulses, derived from a protocol from the
field of quantum computation. We show that the new scheme is not subject to the
above-mentioned restrictions, and analytically and numerically evaluate its
limitations, based on a realistic implementation with our specific setup.
Implementing the protocol with state-of-the-art techniques should allow for
substantially increasing the number of steps to 100 and beyond and should be
extendable to higher-dimensional QWs.Comment: 29 pages, 15 figue
Origin and characterization of alpha smooth muscle actin-positive cells during murine lung development
© 2017 The Authors Stem Cells published by Wiley Periodicals, Inc. on behalf of AlphaMed PressACTA2 expression identifies pulmonary airway and vascular smooth muscle cells (SMCs) as well as alveolar myofibroblasts (MYF). Mesenchymal progenitors expressing fibroblast growth factor 10 (Fgf10), Wilms tumor 1 (Wt1), or glioma-associated oncogene 1 (Gli1) contribute to SMC formation from early stages of lung development. However, their respective contribution and specificity to the SMC and/or alveolar MYF lineages remain controversial. In addition, the contribution of mesenchymal cells undergoing active WNT signaling remains unknown. Using Fgf10CreERT2, Wt1CreERT2, Gli1CreERT2, and Axin2CreERT2 inducible driver lines in combination with a tdTomatoflox reporter line, the respective differentiation of each pool of labeled progenitor cells along the SMC and alveolar MYF lineages was quantified. The results revealed that while FGF10+ and WT1+ cells show a minor contribution to the SMC lineage, GLI1+ and AXIN2+ cells significantly contribute to both the SMC and alveolar MYF lineages, but with limited specificity. Lineage tracing using the Acta2-CreERT2 transgenic line showed that ACTA2+ cells labeled at embryonic day (E)11.5 do not expand significantly to give rise to new SMCs at E18.5. However, ACTA2+ cells labeled at E15.5 give rise to the majority (85%–97%) of the SMCs in the lung at E18.5 as well as alveolar MYF progenitors in the lung parenchyma. Fluorescence-activated cell sorting-based isolation of different subpopulations of ACTA2+ lineage-traced cells followed by gene arrays, identified transcriptomic signatures for alveolar MYF progenitors versus airway and vascular SMCs at E18.5. Our results establish a new transcriptional landscape for further experiments addressing the function of signaling pathways in the formation of different subpopulations of ACTA2+ cells. Stem Cells 2017;35:1566–1578
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