The Levy-Lieb embedding of density functional theory and its Quantum Kernel: Illustration for the Hubbard Dimer using near-term quantum algorithms

TL;DR

This paper demonstrates a hybrid quantum-classical approach to density functional theory using the Levy-Lieb formulation on a Hubbard dimer, introducing a quantum kernel for learning density-based observables with high accuracy.

Contribution

It implements the Levy-Lieb density embedding and quantum kernel for the Hubbard dimer, showcasing a novel quantum algorithm for density functional theory.

Findings

Successful density variational minimization with a hybrid quantum-classical scheme

Development of a fidelity-based quantum kernel for density embedding

High-accuracy generalization of the kernel in numerical experiments

Abstract

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.