Nonlinear partial differential equations (PDEs) are fundamental descriptions of the physical world and are key to science and engineering. Examples include compressible Navier-Stokes equations for fluids and reaction-diffusion equations for population dynamics and chemical reactions. However, for quantum algorithms, nonlinearities inherent in these PDEs present a significant challenge. A known and well-studied technique to overcome these difficulties, is to use Carleman linearization to embed a finite-dimensional nonlinear system, resulting from spatial discretization of a PDE, into an infinite-dimensional linear system, thereby enabling solving on quantum computers after truncation [1].
Understanding the convergence properties of Carleman linearization, as applied to nonlinear ODEs that result from spatial discretization of PDEs, is an active area of research. Nevertheless, the set of known convergence results are commonly applied to the analysis of finite rectilinear meshes [2]. However, the implication on convergence of Carleman linearization in the context of using high-order curvilinear meshes has not been well studied. Curvilinear meshes have important uses for practical problems with complex geometry which are commonly found in industries such as aerospace. The aim of this work is to understand, based on current Carleman theory, what classes of problems can be solved on quantum computers, in the context of reaction-diffusion equations.
