Computational complexity of interacting electrons and fundamental limitations of density functional theory

Abstract

One of the central problems in quantum mechanics is to determine the ground-state properties of a system of electrons interacting through the Coulomb potential. Since its introduction^1,2, density functional theory has become the most widely used and successful method for simulating systems of interacting electrons. Here, we show that the field of computational complexity imposes fundamental limitations on density functional theory. In particular, if the associated ‘universal functional’ could be found efficiently, this would imply that any problem in the computational complexity class Quantum Merlin Arthur could be solved efficiently. Quantum Merlin Arthur is the quantum version of the class NP and thus any problem in NP could be solved in polynomial time. This is considered highly unlikely. Our result 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, but, given the universal functional, it can be computed efficiently using density functional theory. This work illustrates how the field of quantum computing could be useful even if quantum computers were never built.