598 research outputs found
Simple digital quantum algorithm for symmetric first order linear hyperbolic systems
This paper is devoted to the derivation of a digital quantum algorithm for
the Cauchy problem for symmetric first order linear hyperbolic systems, thanks
to the reservoir technique. The reservoir technique is a method designed to
avoid artificial diffusion generated by first order finite volume methods
approximating hyperbolic systems of conservation laws. For some class of
hyperbolic systems, namely those with constant matrices in several dimensions,
we show that the combination of i) the reservoir method and ii) the alternate
direction iteration operator splitting approximation, allows for the derivation
of algorithms only based on simple unitary transformations, thus perfectly
suitable for an implementation on a quantum computer. The same approach can
also be adapted to scalar one-dimensional systems with non-constant velocity by
combining with a non-uniform mesh. The asymptotic computational complexity for
the time evolution is determined and it is demonstrated that the quantum
algorithm is more efficient than the classical version. However, in the quantum
case, the solution is encoded in probability amplitudes of the quantum
register. As a consequence, as with other similar quantum algorithms, a
post-processing mechanism has to be used to obtain general properties of the
solution because a direct reading cannot be performed as efficiently as the
time evolution.Comment: 28 pages, 12 figures, major rewriting of the section describing the
numerical method, simplified the presentation and notation, reorganized the
sections, comments are welcome
Quantum Lattice Boltzmann is a quantum walk
Numerical methods for the 1-D Dirac equation based on operator splitting and
on the quantum lattice Boltzmann (QLB) schemes are reviewed. It is shown that
these discretizations fall within the class of quantum walks, i.e. discrete
maps for complex fields, whose continuum limit delivers Dirac-like relativistic
quantum wave equations. The correspondence between the quantum walk dynamics
and these numerical schemes is given explicitly, allowing a connection between
quantum computations, numerical analysis and lattice Boltzmann methods. The QLB
method is then extended to the Dirac equation in curved spaces and it is
demonstrated that the quantum walk structure is preserved. Finally, it is
argued that the existence of this link between the discretized Dirac equation
and quantum walks may be employed to simulate relativistic quantum dynamics on
quantum computers.Comment: 18 pages, 3 figure
Simplicial cohomology of band semigroup algebras
We establish simplicial triviality of the convolution algebra ,
where is a band semigroup. This generalizes results of the first author
[Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the
cyclic cohomology of this algebra vanishes in all odd degrees, and is
isomorphic in even degrees to the space of continuous traces on .
Crucial to our approach is the use of the structure semilattice of , and the
associated grading of , together with an inductive normalization procedure
in cyclic cohomology; the latter technique appears to be new, and its
underlying strategy may be applicable to other convolution algebras of
interest.Comment: v1: AMS-LaTeX, 24 pages, 1 figure. v2: some typos corrected; a few
minor adjustments made for clarity; references updated. Accepted June 2011 by
Proc. Royal Soc. Edinburgh Sect.
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