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