Quantum-assisted hλ-adaptive finite element method

Quantum computing is a rapidly advancing field, driven by the potential advantages derived from the unique properties of quantum entanglement. In particular, the exponential speedup of certain carefully designed algorithms, compared to their classical counterparts, promises to significantly enhance the numerical solution of a wide range of problems. This paper investigates the integration of quantum computing with the finite element method, focusing on singularly perturbed advection-diffusion-reaction problems. We introduce a novel finite element scheme that combines classical and quantum algorithms. In this approach, the primary mesh adaptation loop is managed by a classical computer, while a specific stabilization procedure is executed on a quantum computer. This procedure leverages the Harrow-Hassidim-Lloyd algorithm in conjunction with the swap test to estimate the value of a linear functional, which constitutes a substantial portion of the computational workload. We demonstrate that this hybrid approach effectively eliminates parasitic oscillations in the finite element approximation, even at the early stages of the adaptation process. This leads to a significant improvement in the quality of intermediate finite element solutions. As a result, our scheme offers more efficient feedback with reduced computational costs for researchers using the method to investigate physical phenomena. To support the scheme, we prove special explicit a posteriori error estimates. Possible benefits of the proposed finite element scheme are analyzed using the numerical comparison with the typical adaptive scheme.