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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?

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