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, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). Our detailed analyses show that for the heat and linear hyperbolic equations the quantum difference methods provide exponential speedup over the classical difference method with respect to the spatial dimension. For the multiscale problem, the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as $O(1/\varepsilon)$, where $\varepsilon$ is the scaling parameter, while the scaling for the 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.