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

TL;DR

This paper introduces two quantum algorithms for simulating multiscale linear transport equations, leveraging Schr"odingerization and exponential integrators, achieving superior query complexity and handling multiscale problems efficiently.

Contribution

It presents the first quantum algorithms combining Schr"odingerization with asymptotic-preserving schemes for multiscale linear transport equations.

Findings

Query complexity $\\mathcal{O}(N_vN_x^2\log N_x)$ outperforms existing methods.

Algorithms effectively handle multiscale problems with stiff terms.

First to combine Schr"odingerization with asymptotic-preserving schemes in this context.

Abstract

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.