The finite element method (FEM) is one of the most widely used numerical methods for solving partial differential equations, in large part because of its ability to accommodate complex geometries, high-order approximations, and general boundary conditions. Existing quantum algorithms for PDEs, however, have largely focused on the linear-system solve once a discretized operator is already available. In this work, we develop Qu-FEM, a framework for implementing the surrounding finite-element machinery on a fault-tolerant quantum computer.

The central challenge is finite-element assembly. A direct quantum analogue of classical element-by-element assembly introduces a linear combination whose size grows with the number of elements. We instead introduce the unit of interaction and the related local-to-global node-number indicator matrix, which use the mesh connectivity to group equivalent local interactions across the mesh and construct block-encodings of global finite-element arrays. For spatially varying coefficients, we extend this construction by performing Gauss–Legendre quadrature directly on the quantum computer, using quantum eigenvalue transformation to evaluate coefficient functions over all elements simultaneously.

We further show how Dirichlet boundary conditions can be imposed through a Lagrange-multiplier formulation without modifying the block-encodings produced during assembly. Finally, we discuss the output problem associated with quantum linear-system algorithms: recovering the physical scale of the finite-element solution and extracting low-dimensional quantities such as fluxes and flow rates rather than reconstructing the complete solution field. Together, these constructions provide a modular route from finite-element discretization and assembly to constrained quantum linear solves, while highlighting preconditioning, general mesh connectivity, and observable extraction as central remaining challenges.