  One of the central problems in quantum mechanics is to determine the ground
state properties of a system of electrons interacting via the Coulomb
potential. Since its introduction by Hohenberg, Kohn, and Sham, Density
Functional Theory (DFT) has become the most widely used and successful method
for simulating systems of interacting electrons, making their original work one
of the most cited in physics. In this letter, we show that the field of
computational complexity imposes fundamental limitations on DFT, as an
efficient description of the associated universal functional would allow to
solve any problem in the class QMA (the quantum version of NP) and thus
particularly any problem in NP in polynomial time. This 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, while given the
universal functional it can be computed efficiently using DFT. This provides a
clear illustration how the field of quantum computing is useful even if quantum
computers would never be built.
