Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory

TL;DR

This paper demonstrates that fundamental computational complexity limits the effectiveness of Density Functional Theory (DFT) in quantum mechanics, linking its computational difficulty to the hardness of solving certain quantum problems even with quantum computers.

Contribution

It establishes a theoretical connection between the computational complexity of quantum problems and the limitations of DFT, showing that an efficient universal functional would imply solving QMA-hard problems.

Findings

Finding the ground state energy of the Hubbard model is QMA-hard.

Efficiently computing the universal functional would solve NP problems in polynomial time.

Quantum computing insights reveal fundamental limits of DFT.

Abstract

One of the central problems in quantum mechanics is to determine the ground state properties of a system of electrons interacting via the Coulomb potential. Since its introduction by Hohenberg, Kohn, and Sham, Density Functional Theory (DFT) has become the most widely used and successful method for simulating systems of interacting electrons, making their original work one of the most cited in physics. In this letter, we show that the field of computational complexity imposes fundamental limitations on DFT, as an efficient description of the associated universal functional would allow to solve any problem in the class QMA (the quantum version of NP) and thus particularly any problem in NP in polynomial time. This follows from the fact that finding the ground state energy of the Hubbard model in an external magnetic field is a hard problem even for a quantum computer, while given the universal functional it can be computed efficiently using DFT. This provides a clear illustration how the field of quantum computing is useful even if quantum computers would never be built.