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

TL;DR

It is proved 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, an exponential reduction compared to the previously established operator-norm analysis.

Abstract

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 $Θ(16^n)$ for the diffusion equation, and $Θ(4^n)$ for the convection equation, where $n$ is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis.

Figures (1)

• Figure 1: Quantum numerical scheme associated with the resolution of the $d$-dimensional convection or diffusion equations with position- and time- dependent coefficients.