361 research outputs found
Thermodynamic formalism for dissipative quantum walks
We consider the dynamical properties of dissipative continuous-time quantum
walks on directed graphs. Using a large-deviation approach we construct a
thermodynamic formalism allowing us to define a dynamical order parameter, and
to identify transitions between dynamical regimes. For a particular class of
dissipative quantum walks we propose a quantum generalization of the the
classical PageRank vector, used to rank the importance of nodes in a directed
graph. We also provide an example where one can characterize the dynamical
transition from an effective classical random walk to a dissipative quantum
walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page
Optimal Covariant Measurement of Momentum on a Half Line in Quantum Mechanics
We cannot perform the projective measurement of a momentum on a half line
since it is not an observable. Nevertheless, we would like to obtain some
physical information of the momentum on a half line. We define an optimality
for measurement as minimizing the variance between an inferred outcome of the
measured system before a measuring process and a measurement outcome of the
probe system after the measuring process, restricting our attention to the
covariant measurement studied by Holevo. Extending the domain of the momentum
operator on a half line by introducing a two dimensional Hilbert space to be
tensored, we make it self-adjoint and explicitly construct a model Hamiltonian
for the measured and probe systems. By taking the partial trace over the newly
introduced Hilbert space, the optimal covariant positive operator valued
measure (POVM) of a momentum on a half line is reproduced. We physically
describe the measuring process to optimally evaluate the momentum of a particle
on a half line.Comment: 12 pages, 3 figure
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
Oomyzus sokolowskii (Hymenoptera: Eulophidae) Joins the Small Complex of Parasitoids Known to Attack the Diamondback Moth on Kauai
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
Coined quantum walks on percolation graphs
Quantum walks, both discrete (coined) and continuous time, form the basis of
several quantum algorithms and have been used to model processes such as
transport in spin chains and quantum chemistry. The enhanced spreading and
mixing properties of quantum walks compared with their classical counterparts
have been well-studied on regular structures and also shown to be sensitive to
defects and imperfections in the lattice. As a simple example of a disordered
system, we consider percolation lattices, in which edges or sites are randomly
missing, interrupting the progress of the quantum walk. We use numerical
simulation to study the properties of coined quantum walks on these percolation
lattices in one and two dimensions. In one dimension (the line) we introduce a
simple notion of quantum tunneling and determine how this affects the
properties of the quantum walk as it spreads. On two-dimensional percolation
lattices, we show how the spreading rate varies from linear in the number of
steps down to zero, as the percolation probability decreases to the critical
point. This provides an example of fractional scaling in quantum walk dynamics.Comment: 25 pages, 14 figures; v2 expanded and improved presentation after
referee comments, added extra figur
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
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
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