S19 - Computational Fluid Dynamics VII

High-fidelity computational fluid dynamics (CFD) requires a massive number of degrees of freedom, leading to prohibitive memory consumption even on modern supercomputers. Quantum computing is attracting increasing attention as a potential solution to this limitation. However, most existing studies on quantum-assisted flow simulation attempt to directly translate conventional numerical schemes into quantum algorithms. Among these approaches, widely used ones rely on Hamiltonian simulation to represent time evolution. A fundamental limitation of such quantum simulations is that only the final-time state can be efficiently extracted, while intermediate time evolution is not directly accessible. This limitation significantly restricts practical applicability in engineering, as time-resolved information is essential for applications such as aerodynamic design and flow control. To address this issue, we aim to establish a novel quantum computational framework for large-scale unsteady flow simulations that enables access to temporal information beyond final-time states. The proposed approach is not limited to fluid dynamics but is also applicable to a broader class of transport phenomena governed by differential equations.

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.

The Lattice Boltzmann Method (LBM) offers a robust, highly parallelizable kinetic framework for computational fluid dynamics, with strengths in resolving complex transport phenomena in intricate geometries. Yet scaling Direct Numerical Simulation (DNS) to large three-dimensional grids quickly runs into memory bottlenecks, both in capacity and bandwidth, that push current HPC architectures to their limits. To address these bottlenecks, we employ a quantum-inspired approach by embedding the LBM state space (macroscopic fields and distribution functions) in a Tensor Train (TT) decomposition [1]. We exploit correlations in the fluid flow, which lend themselves to low-rank TT structures. With this formulation, we can achieve high compression ratios while maintaining excellent accuracy.