# Do we know anything about the computational complexity of the exchange-correlation functional?

Density functional theory is based on the Hohenberg-Kohn (HK) theorems and aims to compute the ground-state many-body wavefunction of a physical material and/or molecules.

To put it simply, the HK theorems show that there is a unique one-to-one mapping between a many-body Hamiltonian $$\mathcal{H}$$ (like you'd encounter in quantum chemistry or in the solid-state), and the electron density $$\rho(\mathbf{r})$$. This relationship is quite surprising, especially since you reduce the information of the wavefunction $$\psi$$ with $$3N$$ variables to a function of just $$3$$ variables.

The catch here is that extracting information from $$\rho(\mathbf{r})$$ is quite difficult, and you can only really do it exactly if you know the exchange-correlation functional term $$\frac{\delta E_{xc}[\rho]}{\delta \rho(\mathbf{r})}$$, but no one knows what this functional looks like exactly.

My question is the following:

Do we know anything about the computational complexity of evaluating the exchange-correlation functional in either the classical or quantum computing cases?