100 research outputs found
Propagation in quantum walks and relativistic diffusions
Propagation in quantum walks is revisited by showing that very general 1D
discrete-time quantum walks with time- and space-dependent coefficients can be
described, at the continuous limit, by Dirac fermions coupled to
electromagnetic fields. Short-time propagation is also established for
relativistic diffusions by presenting new numerical simulations of the
Relativistic Ornstein-Uhlenbeck Process. A geometrical generalization of Fick's
law is also obtained for this process. The results suggest that relativistic
diffusions may be realistic models of decohering or random quantum walks. Links
with general relativity and geometrical flows are also mentioned.Comment: 3 figure
Quantum walks and non-Abelian discrete gauge theory
A new family of discrete-time quantum walks (DTQWs) on the line with an exact
discrete gauge invariance is introduced. It is shown that the continuous
limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac
fermion coupled to usual gauge fields in spacetime. A discrete
generalization of the usual curvature is also constructed. An alternate
interpretation of these results in terms of superimposed Maxwell fields
and gauge fields is discussed in the Appendix. Numerical simulations
are also presented, which explore the convergence of the DTQWs towards their
continuous limit and which also compare the DTQWs with classical (i.e.
non-quantum) motions in classical fields. The results presented in this
article constitute a first step towards quantum simulations of generic
Yang-Mills gauge theories through DTQWs.Comment: 7 pages, 2 figure
Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks
Gauge invariance is one of the more important concepts in physics. We discuss
this concept in connection with the unitary evolution of discrete-time quantum
walks in one and two spatial dimensions, when they include the interaction with
synthetic, external electromagnetic fields. One introduces this interaction as
additional phases that play the role of gauge fields. Here, we present a way to
incorporate those phases, which differs from previous works. Our proposal
allows the discrete derivatives, that appear under a gauge transformation, to
treat time and space on the same footing, in a way which is similar to standard
lattice gauge theories. By considering two steps of the evolution, we define a
density current which is gauge invariant and conserved. In the continuum limit,
the dynamics of the particle, under a suitable choice of the parameters,
becomes the Dirac equation, and the conserved current satisfies the
corresponding conservation equation
Self-truncation and scaling in Euler-Voigt- and related fluid models
A generalization of the Euler-Voigt- model is obtained by
introducing derivatives of arbitrary order (instead of ) in the
Helmholtz operator. The limit is shown to correspond to
Galerkin truncation of the Euler equation. Direct numerical simulations (DNS)
of the model are performed with resolutions up to and Taylor-Green
initial data. DNS performed at large demonstrate that this simple
classical hydrodynamical model presents a self-truncation behavior, similar to
that previously observed for the Gross-Pitaeveskii equation in Krstulovic and
Brachet [Phys. Rev. Lett. 106, 115303 (2011)]. The self-truncation regime of
the generalized model is shown to reproduce the behavior of the truncated Euler
equation demonstrated in Cichowlas et al. [Phys. Rev. Lett. 95, 264502 (2005)].
The long-time growth of the self-truncation wavenumber appears to
be self-similar.
Two related -Voigt versions of the EDQNM model and the Leith model
are introduced. These simplified theoretical models are shown to reasonably
reproduce intermediate time DNS results. The values of the self-similar
exponents of these models are found analytically.Comment: 14 figure
Quantum Walks in artificial electric and gravitational Fields
The continuous limit of quantum walks (QWs) on the line is revisited through
a recently developed method. In all cases but one, the limit coincides with the
dynamics of a Dirac fermion coupled to an artificial electric and/or
relativistic gravitational field. All results are carefully discussed and
illustrated by numerical simulations.Comment: 13 pages, 3 figures. Submitted to Physica A. arXiv admin note: text
overlap with arXiv:1212.582
Nonlinear Optical Galton Board: thermalization and continuous limit
The nonlinear optical Galton board (NLOGB), a quantum walk like (but
nonlinear) discrete time quantum automaton, is shown to admit a complex
evolution leading to long time thermalized states. The continuous limit of the
Galton Board is derived and shown to be a nonlinear Dirac equation (NLDE). The
(Galerkin truncated) NLDE evolution is shown to thermalize toward states
qualitatively similar to those of the NLOGB. The NLDE conserved quantities are
derived and used to construct a stochastic differential equation converging to
grand canonical distributions that are shown to reproduce the (micro canonical)
NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are
thus demonstrated to exhibit spontaneous thermalization.Comment: 8 pages, 14 figures, accepted on PRE as Regular Articl
Quantum Bandits
We consider the quantum version of the bandit problem known as {\em best arm
identification} (BAI). We first propose a quantum modeling of the BAI problem,
which assumes that both the learning agent and the environment are quantum; we
then propose an algorithm based on quantum amplitude amplification to solve
BAI. We formally analyze the behavior of the algorithm on all instances of the
problem and we show, in particular, that it is able to get the optimal solution
quadratically faster than what is known to hold in the classical case.Comment: All your comments are very welcome
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