Quantum Enhanced Numerical Homogenization

Abstract

We propose a numerical homogenization method for scalar linear partial differential equations with rough coefficients that integrates classical coarse-scale solvers with quantum subroutines for fine-scale corrections. Inspired by the Localized Orthogonal Decomposition, we employ quantum local problem solvers to capture fine-scale features efficiently. Unlike periodic homogenization approaches, it does not rely on any periodicity assumption. Moreover, the coupling between quantum computation and the coarse model requires only selected measurements of quantum representative volume elements, thereby mitigating the quantum-interface information bottleneck that could otherwise negate a potential speed-up. We show that the local quantum solver can achieve solutions with the required level of accuracy with an operation count that scales only logarithmically with the fine-scale resolution, as determined by the smallest length scale encoded in the diffusion coefficient. The potential of the approach is illustrated through two-dimensional numerical experiments, using a classical simulation of the local quantum solver.