We present a quantum algorithm for the simulation of the linear advection-diffusion equation based on block encodings of high order finite-difference operators and the quantum singular value transform. Our complexity analysis shows that the higher order methods significantly reduce the number of gates and qubits required to reach a given accuracy. The theoretical results are supported by numerical simulations of one- and two-dimensional benchmarks.
@article{arxiv.2512.22163,
title = {A quantum advection-diffusion solver using the quantum singular value transform},
author = {Gard Olav Helle and Tommaso Benacchio and Anna Bomme Ousager and Jørgen Ellegaard Andersen},
journal= {arXiv preprint arXiv:2512.22163},
year = {2026}
}