  Partial differential equation (PDE) models with multiple temporal/spatial
scales are prevalent in several disciplines such as physics, engineering, and
many others. These models are of great practical importance but notoriously
difficult to solve due to prohibitively small mesh and time step sizes limited
by the scaling parameter and CFL condition. Another challenge in scientific
computing could come from curse-of-dimensionality. In this paper, we aim to
provide a quantum algorithm, based on either direct approximations of the
original PDEs or their homogenized models, for prototypical multiscale problems
in partial differential equations (PDEs), including elliptic, parabolic and
hyperbolic PDEs. To achieve this, we will lift these problems to higher
dimensions and leverage the recently developed Schr\"{o}dingerization based
quantum simulation algorithms to efficiently reduce the computational cost of
the resulting high-dimensional and multiscale problems. We will examine the
error contributions arising from discretization, homogenization, and
relaxation, analyze and compare the complexities of these algorithms in order
to identify the best algorithms in terms of complexities for different
equations in different regimes.

Numerical Analysis;Numerical Analysis