Time complexity analysis of quantum difference methods for the multiscale transport equations

We investigate time complexities of finite difference methods for solving the multiscale transport equation with quantum algorithms. We find that the time complexities of both the classical treatment and quantum treatment for a standard explicit scheme scale as $\mathcal{O}(1/\varepsilon)$, where $\varepsilon$ is the small scaling parameter, while the complexities for the even-odd parity based Asymptotic-Preserving (AP) scheme do not depend on $\varepsilon$. This indicates that it is still of great importance to use AP (and probably other efficient multiscale) schemes for multiscale problems in quantum computing when solving multiscale transport or kinetic equations.