Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling

In this work, we tackle the resolution of partial differential equations (PDEs) on digital quantum computers. Two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. We present a quantum numerical scheme consisting of three steps: quantum state preparation, evolution with diagonal operators, and measurement of observables of interest. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. We provide novel vector-norm analysis to bound the different sources of error. We prove that the number of time-steps required in the evolution can be reduced by a factor $\Theta (16^n)$ for the diffusion equation, and $\Theta (4^n)$ for the convection equation, where $n$ is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis.