Quantum Enhanced Numerical Homogenization

TL;DR

This paper introduces a quantum-enhanced numerical homogenization method for scalar PDEs with rough coefficients, combining classical solvers with quantum subroutines for efficient fine-scale corrections.

Contribution

It presents a novel integration of quantum local problem solvers into classical homogenization, avoiding periodicity reliance and reducing complexity.

Findings

Quantum local solvers achieve accurate solutions with logarithmic complexity.

The method scales efficiently with the smallest length scale in the problem.

Classical simulation demonstrates the potential of the quantum-enhanced approach.

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. Crucially, the approach does not rely on the periodicity of the problem, and the integration of the quantum computation within a coarse model requires only selected measurements of the quantum representative volume elements, overcoming the information bottleneck of quantum interfaces that could eliminate the speed-up. We demonstrate that the local quantum solver can achieve solutions with sufficient accuracy, with a number of operations that scales only logarithmically with the fine-scale resolution, determined by the smallest length scale encoded in the diffusion coefficient. The potential of the approach is illustrated through two-dimensional test cases, using a classical simulation of the local quantum solver.