Quantum kernel machine learning of density functionals using a Levy-Lieb pure-state embedding

We illustrate a framework for exact density functional theory using variational quantum circuits based on the constraint-search formulation of Levy and Lieb. Using the Hubbard dimer as a paradigmatic model, we discuss the implementation of explicit density variational energy minimization using a density-constrained variational quantum eigensolver approach. Further, by interpreting the Levy-Lieb mapping from one-body densities to many-electron wavefunctions as a feature embedding into pure states we demonstrate a fidelity based quantum kernel for machine learning observable functionals of the ground-state density. We explore the ability of such a quantum kernel to generalize to unseen data through numerical experiments on the Hubbard dimer.