Quantum–Classical Iterative Homogenization Framework for Linear Multiscale Physical Systems

Multiscale physical systems often require resolving interactions between fine-scale fields and large-scale behavior that a single model resolution cannot capture. To address this challenge, a quantum–classical iterative homogenization method is developed that solves linear operator equations while enforcing macroscopic consistency through repeated quantum–classical updates. Each iteration alternates between a quantum linear-operator stage, implemented with structured multi-controlled rotations and quantum Fourier transforms, and a classical projection stage that enforces energy or equilibrium constraints across scales. The same framework is extended to the linear Poisson inversion linking vorticity and velocity in two-dimensional turbulence, isolating the linear subproblem that governs velocity reconstruction. The method has been implemented and benchmarked on a periodic homogenization problem, serving as a linear reference for validating convergence and qubit scalability. This work demonstrates how iterative homogenization principles can be embedded in quantum algorithms, establishing a scalable pathway toward unified quantum–classical simulation of multiscale physical systems.