37 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
Explicit volume-preserving numerical schemes for relativistic trajectories and spin dynamics
A class of explicit numerical schemes is developed to solve for the
relativistic dynamics and spin of particles in electromagnetic fields, using
the Lorentz-BMT equation formulated in the Clifford algebra representation of
Baylis. It is demonstrated that these numerical methods, reminiscent of the
leapfrog and Verlet methods, share a number of important properties: they are
energy-conserving, volume-conserving and second order convergent. These
properties are analysed empirically by benchmarking against known analytical
solutions in constant uniform electrodynamic fields. It is demonstrated that
the numerical error in a constant magnetic field remains bounded for long time
simulations in contrast to the Boris pusher, whose angular error increases
linearly with time. Finally, the intricate spin dynamics of a particle is
investigated in a plane wave field configuration.Comment: 15 pages, 9 figure