On the NP-completeness of the Hartree-Fock method for translationally invariant systems

TL;DR

This paper demonstrates that the Hartree-Fock method, widely used in quantum chemistry, is NP-complete in worst-case scenarios, and explores specific instances where the problem remains computationally hard.

Contribution

The authors identify and construct new NP-complete variants of the Hartree-Fock problem, including translationally invariant cases and embeddings of spin glasses.

Findings

Translationally invariant Hartree-Fock solutions can be trivial while broken symmetry solutions are NP-complete.

Instances of spin glasses can be embedded into translationally invariant Hartree-Fock problems.

The work delineates boundaries of computational feasibility for the Hartree-Fock method.

Abstract

The self-consistent field method utilized for solving the Hartree-Fock (HF) problem and the closely related Kohn-Sham problem, is typically thought of as one of the cheapest methods available to quantum chemists. This intuition has been developed from the numerous applications of the self-consistent field method to a large variety of molecular systems. However, as characterized by its worst-case behavior, the HF problem is NP-complete. In this work, we map out boundaries of the NP-completeness by investigating restricted instances of HF. We have constructed two new NP-complete variants of the problem. The first is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions are trivial, but whose broken symmetry solutions are NP-complete. Second, we demonstrate how to embed instances of spin glasses into translationally invariant Hartree-Fock instances and provide a numerical example. These findings are the first steps towards understanding in which cases the self-consistent field method is computationally feasible and when it is not.