A Quantum Algorithm for the Finite Element Method (Qu-FEM)

The Finite Element Method (FEM) is a cornerstone numerical technique for solving Partial Differential Equations (PDEs) that appear in the engineering sciences, with applications ranging from structural analysis, to multiphysics heat transfer, mass transfer, and fluid mechanics modelling, and to computational chemistry. However, when the dimension of the PDE is large, or when small length scales need to be resolved, the number of degrees of freedom can grow beyond what is feasible with direct numerical simulation on a classical computer. Quantum computers offer a promising route for solving the FEM with high precision in many dimensions. In this talk, we discuss a quantum algorithm that implements the finite element method for elliptic PDEs within a block-encoding framework. We also demonstrate how to incorporate any Neumann or Dirichlet boundary conditions, along with a complexity analysis for our implementation.