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

Dissolved organic matter contribution to rain water, throughfall and soil solution chemistry

A method is proposed to determine the acidbase properties of natural water samples containing relatively high amounts of dissolved organic matter. The electroneutrality principle as well as titration data are used to estimate the organic anion concentration in open field precipitation, throughfall and soil solutions, and to develop empirical models based on pH and dissolved organic carbon content. The organic acids dissolved in throughfall have a similar acidic site density but are weaker than those dissolved in soil solution, stream and lake waters. This method is usefull to determine the contribution of organic anions to the charge balance and to the buffering capacity of dissolved organic rich waters with low acid neutralizing capacity. It can be used also to determine the respective contribution of natural organics and anthropogenic minerals to the total acidity of throughfall and rain waters

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