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. 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.

Figures (4)

• Figure 1: Diffusion coefficients for the two test cases.
• Figure 2: Relative errors \ref{['eq:err']} for Test Case 1 between the LOD approximations and a reference solution for $\ell=2,3,4$. Results are shown for both the classical LOD and its quantum variant with varying numbers of samples. The classical implementation follows \ref{['algo:2']}, the quantum implementation follows \ref{['algo:4']}. For the quantum variant, results are averaged over six realizations of the full procedure, and the corresponding error bars are shown.
• Figure 3: Relative errors \ref{['eq:err']} for Test Case 2 between the LOD approximations and a reference solution for $\ell=2,3,4$. Results are shown for both the classical LOD and its quantum variant with varying numbers of samples. The classical implementation follows \ref{['algo:2']}, the quantum implementation follows \ref{['algo:4']}. For the quantum variant, results are averaged over six realizations of the full procedure, and the corresponding error bars are shown.
• Figure 4: Extension of the domain $\Omega=[0,1]^2$ with the extended coarse mesh $\mathcal{G}_H^{\ell}$ and $\ell=2$. Additionally the extension of the diffusion coefficient $\mathfrak{A}$ is displayed (inspired from ThesisMohr).

Theorems & Definitions (24)

• Theorem 1: $a$-"orthogonal" decomposition, Theorem 3.5 Altmann21
• Lemma 2: Characterization of the coarse space, Lemma 3.12 Altmann21
• Lemma 3
• proof
• Theorem 4: Theorem 3.19 Altmann21
• Theorem 5: Theorem 3.23 Altmann21
• Remark 6
• Remark 7
• Definition 8: Block encoding
• Lemma 9: Amplitude estimation