Quantum Enhanced Numerical Homogenization
This work addresses computational challenges in numerical homogenization for non-periodic problems, potentially speeding up simulations in fields like materials science or engineering, though it is incremental as it builds on existing Localized Orthogonal Decomposition methods.
The authors tackled the problem of solving scalar linear partial differential equations with rough coefficients by proposing a quantum-enhanced numerical homogenization method that integrates classical coarse-scale solvers with quantum subroutines for fine-scale corrections, achieving solutions with logarithmic scaling in operations relative to fine-scale resolution.
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.