Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, hyperbolic equation and multiscale system with quantum algorithms (hence referred to as "quantum methods"). For equations we study impact explicit implicit discretizations on advantages over classical method. problem, find complexity both treatment scheme scales $\mathcal{O}(1/\varepsilon)$, where $\varepsilon$ is scaling parameter, while Asymptotic-Preserving (AP) schemes does not depend $\varepsilon$. This indicates that it still great importance develop AP problems in computing.