Time Complexity Analysis of Quantum Difference Methods for Multiscale Transport Equations

. We investigate time complexities of finite difference methods for solving the multiscale transport equation by quantum algorithms. It is found that the time complexity of classical and quantum treatments of the standard explicit scheme scale is O ( 1 /ε ) , where ε is a small scaling parameter. On the other hand, the complexity of the even-odd parity based asymptotic-preserving (AP) schemes do not depend on ε . This indicates that in quantum computing, AP schemes (and probably other multiscale ones) are of great importance for solving multiscale transport and kinetic equations.