Towards Hybrid Quantum-Classical Computing for Large-Scale CFD: Tackling Scalability Challenges of the Variational Quantum Linear Solver

Quantum computing (QC) represents an emerging frontier with significant potential for transformative advancements in Computational Fluid Dynamics (CFD). This potential has been explicitly recognized by the CFD Vision 2030 report, emphasizing the strategic importance of exploring QC for future breakthroughs. Among quantum algorithms relevant to CFD, the Variational Quantum Linear Solver (VQLS) is particularly promising due to its suitability for solving linear and Hamiltonian-based problems, characteristic of fundamental CFD equations such as the pressure-Poisson equation. However, the practical implementation of VQLS for large-scale CFD problems faces critical scalability challenges, most notably the exponential increase in the number of quantum circuit evaluations and the barren plateau phenomenon, which severely hampers trainability and algorithm convergence. In this study, two novel strategies are introduced to address these challenges. First, we utilize a block encoding technique based on efficient oracles, thereby significantly reducing quantum circuit complexity and qubit overhead while accessing system matrix. Second, we incorporate a quantum-inspired evolutionary optimization (QIEO) approach, demonstrating enhanced training efficiency and effectively mitigating the barren plateau issue. These improvements are integrated within a Hybrid Quantum-Classical Method (HQCM) linear solver framework and demonstrated through solving a 2D steady-state heat equation, representative of the pressure-Poisson equations common in incompressible CFD simulations. The proposed methodologies yield promising results, showcasing improved circuit efficiency, enhanced stability during optimization, and accurate numerical solutions. This work lays a foundational step toward realizing practical, scalable quantum-assisted CFD solvers and sets clear pathways for future research aimed at further algorithmic and architectural enhancements.