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

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.

arXiv comment9 pages, 6 figures

References in corpus (22)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• A variational eigenvalue solver on a quantum processor
• Supervised learning with quantum enhanced feature spaces
• Quantum machine learning in feature Hilbert spaces
• The Variational Quantum Eigensolver: a review of methods and best practices
• Power of data in quantum machine learning
• Variational Quantum Computation of Excited States
• Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation
• Training variational quantum algorithms is NP-hard
• Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory
• Pure density functional for strong correlations and the thermodynamic limit from machine learning
• The Hubbard Dimer: A density functional case study of a many-body problem
• Deep learning the Hohenberg-Kohn maps of Density Functional Theory
• Penalty methods for variational quantum eigensolver
• Machine learning density functional theory for the Hubbard model
• Linear response time-dependent density functional theory of the Hubbard dimer
• Quantum mechanics in metric space: wave functions and their densities
• Density functional theory and quantum computation
• Degenerate ground states and nonunique potentials: breakdown and restoration of density functionals
• Variational calculation of many-body wave functions and energies from density-functional theory
• The entanglement of few-particle systems when using the local-density approximation
• Density functionals and Kohn-Sham potentials with minimal wavefunction preparations on a quantum computer

Cited by in corpus (2)

• Seven Useful Questions in Density Functional Theory
• Geometrical Perspective on Spin-Lattice Density-Functional Theory