Quantum Carleman linearisation efficiency in nonlinear fluid dynamics

TL;DR

This paper investigates the potential of quantum computing to efficiently solve nonlinear fluid dynamics equations by connecting Carleman linearisation parameters with physical fluid flow scales.

Contribution

It introduces a framework linking Carleman linearisation truncation efficiency to physical parameters like the Kolmogorov scale in quantum fluid dynamics simulations.

Findings

Establishes a connection between numerical and physical parameters for quantum CFD.

Provides a formalism for vector field simulation across spatial dimensions.

Analyzes regimes where quantum algorithms outperform classical methods.

Abstract

Computational fluid dynamics (CFD) is a specialised branch of fluid mechanics that utilises numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing, which has the potential to resolve nonlinear differential equations more efficiently than classical computers. Here, we try to answer the question of which regimes of nonlinear partial differential equations (PDEs) for fluid dynamics can have an efficient quantum algorithm. We propose a connection between the numerical parameter, $R$, that guarantees efficiency in the truncation of the Carleman linearisation, and the physical parameters that describe the fluid flow. This link can be made thanks to the Kolmogorov scale, which determines the minimum size of the grid needed to properly resolve the energy cascade induced by the nonlinear term. Additionally, we introduce the formalism for vector field simulation in different spatial dimensions, providing the discretisation of the operators and the boundary conditions.