Quantum simulation of multiscale linear transport equations via Schr\"odingerization and exponential integrators

In this paper, we present two Hamiltonian simulation algorithms for multiscale linear transport equations, combining the Schr\"odingerization method [S. Jin, N. Liu and Y. Yu, Phys. Rev. Lett, 133 (2024), 230602][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603] and exponential integrator while incorporating incoming boundary conditions. These two algorithms each have advantages in terms of design easiness and scalability, and the query complexity of both algorithms, $\mathcal{O}(N_vN_x^2\log N_x)$, outperforms existing quantum and classical algorithms for solving this equation. In terms of the theoretical framework, these are the first quantum Hamiltonian simulation algorithms for multiscale linear transport equation to combine the Schr\"odingerization method with an effective asymptotic-preserving schemes, which are efficient for handling multiscale problems with stiff terms.