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.