Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS

Abstract: This paper proposes a systematic and explicit quantum circuit framework for solving advection-diffusion equations with boundary conditions, based on the Linear Combination of Hamiltonian Simulations (LCHS) method. By employing the Finite Volume Method (FVM) combined with various flux construction schemes, we elaborate the design of quantum circuits tailored explicitly for Robin boundary conditions (including Dirichlet and Neumann boundary conditions as special cases) and periodic boundary conditions. In contrast to prior works on quantum simulation of advection-diffusion equations, we present a detailed error analysis for the linear combination of unitaries (LCU) induced by the constructed quantum circuits. A comprehensive gate complexity analysis demonstrates the quantum advantages over classical computing in high-dimensional scenarios. We simulate the proposed circuits on a fault-tolerant emulator, and numerical results validate the effectiveness of the proposed framework across homogeneous, inhomogeneous, and high-dimensional cases. The proposed framework is compatible with numerous spatial discretization methods and numerical schemes, extends naturally to other linear PDEs, and establishes a practical foundation for solving large-scale PDE problems on future fault-tolerant quantum computers.