  The constrained-search formulation of Levy and Lieb provides a concrete
mapping from N-representable densities to the space of N-particle wavefunctions
and explicitly defines the universal functional of density functional theory.
We numerically implement the Levy-Lieb procedure for a paradigmatic lattice
system, the Hubbard dimer, using a modified variational quantum eigensolver
approach. We demonstrate density variational minimization using the resulting
hybrid quantum-classical scheme featuring real-time computation of the
Levy-Lieb functional along the search trajectory. We further illustrate a
fidelity based quantum kernel associated with the density to pure-state
embedding implied by the Levy-Lieb procedure and employ the kernel for learning
observable functionals of the density. We study the kernel's ability to
generalize with high accuracy through numerical experiments on the Hubbard
dimer.
