The Computational Complexity of Density Functional Theory

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 [20]. 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 [20], we show in DFT, although the introduction of an approximate potential leads to a noninteracting Hamiltonian, it remains, in the worst case, an NP-complete problem.

Electronic Structure in a Fixed Basis is QMA-complete

“…However, while such a functional exists, it may not be computationally efficient. Indeed, computing it has been shown to be QMA-hard, also under Turing reductions [23]. For both N -representability and DFT, it remains an open question whether they remain hard when the inputs (2-RDMs and electron densities, respectively) are restricted to the ground states of electronic structure Hamiltonians.…”

Theoretical and computational studies on nanofriction: A review

“…Despite its ability to give finely configured data, DFT is not commonly used in studying nanoscale friction due to the complexity and large time involved in studying interactions between large surfaces. 58 In addition, while DFT most accurately simulates the electronic interactions and friction caused due to it, it may not be used to directly simulate mechanical forces and interactions between atoms, and thus, methods such as MD or MC are required to accurately simulate the process of mechanical wear of a material due to friction. 52 So, DFT, in the context of studying friction, is mostly used to verify theories and models that have been developed using other computational methods and to understand the atomistic mechanisms of the processes being studied.…”

Solver for the electronic V-representation problem of time-dependent density functional theory

“…Despite the numerous approximations to the universal functional, computational complexity arguments [6,7] showed that obtaining the numerically exact functional is intractable even with quantum computation. It may be possible to obtain a functional using a particular form of the Kohn-Sham orbitals.…”