88 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
- …