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

TL;DR

This paper analyzes the time complexities of classical and quantum finite difference methods for multiscale transport equations, highlighting the efficiency of asymptotic-preserving schemes in quantum computing.

Contribution

It provides a comparative analysis of classical and quantum time complexities for multiscale transport equations, emphasizing the importance of AP schemes.

Findings

Classical and quantum explicit schemes scale as O(1/ε).

AP schemes have complexities independent of ε.

AP schemes are crucial for efficient quantum solutions to multiscale problems.

Abstract

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.