QAFE$^2$: Quantum Accelerated Multiscale Finite Element Analysis

TL;DR

QAFE$^2$ introduces a quantum-classical multiscale finite element framework that leverages quantum parallelism to exponentially speed up RVE homogenisation, enabling efficient large-scale simulations.

Contribution

The paper presents a quantum solver for multiscale finite element analysis that achieves polylogarithmic complexity and evaluates all RVE problems simultaneously using quantum superposition.

Findings

Quantum solver attains polylogarithmic complexity for RVE problems.

QAFE$^2$ evaluates all RVE problems at once via quantum superposition.

Numerical experiments confirm accuracy and theoretical speedup.

Abstract

The computational cost of concurrent multiscale finite element methods is dominated by the repeated solution of microscopic representative volume element (RVE) problems at macroscopic quadrature points. In this work, we introduce a quantum-classical framework for multiscale finite element analysis (QAFE$^2$) that leverages quantum parallelism to fundamentally alter the scaling of RVE-based homogenisation. At the single-RVE level, the proposed quantum solver attains polylogarithmic complexity with respect to the microscopic discretisation size, yielding an exponential asymptotic speedup over the best available classical solvers. More importantly, QAFE$^2$ exploits quantum superposition and entanglement to evaluate, in a single quantum execution, the entire ensemble of RVE problems associated with all macroscopic quadrature points. This capability is a form of intrinsic quantum concurrency with no classical analogue. Numerical experiments on one- and two-dimensional model problems with known analytical solutions confirm the accuracy of the proposed formulation and verify the theoretical computational scaling and parallel performance.