The computational complexity of density functional theory

TL;DR

This paper explores the fundamental computational complexity of density functional theory (DFT), revealing that approximating the universal functional remains NP-complete even with approximate potentials, highlighting intrinsic computational limitations.

Contribution

It formalizes the computational complexity of DFT, extending previous results to show NP-completeness persists with approximate potentials, emphasizing fundamental algorithmic limitations.

Findings

Approximating the universal functional in DFT is NP-complete.

Introducing approximate potentials does not reduce computational complexity.

The complexity results are based on Hamiltonian complexity techniques and perturbative gadgets.

Abstract

Density functional theory is a successful branch of numerical simulations of quantum systems. While the foundations are rigorously defined, the universal functional must be approximated resulting in a `semi'-ab initio approach. The search for improved functionals has resulted in hundreds of functionals and remains an active research area. This chapter is concerned with understanding fundamental limitations of any algorithmic approach to approximating the universal functional. The results based on Hamiltonian complexity presented here are largely based on \cite{Schuch09}. In this chapter, we explain the computational complexity of DFT and any other approach to solving electronic structure Hamiltonians. The proof relies on perturbative gadgets widely used in Hamiltonian complexity and we provide an introduction to these techniques using the Schrieffer-Wolff method. Since the difficulty of this problem has been well appreciated before this formalization, practitioners have turned to a host approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show in DFT, although the introduction of an approximate potential leads to a non-interacting Hamiltonian, it remains, in the worst case, an NP-complete problem.