  One of the potential applications of a quantum computer is solving quantum
chemical systems. It is known that one of the fastest ways to obtain somewhat
accurate solutions classically is to use approximations of density functional
theory. We demonstrate a general method for obtaining the exact functional as a
machine learned model from a sufficiently powerful quantum computer. Only
existing assumptions for the current feasibility of solutions on the quantum
computer are used. Several known algorithms including quantum phase estimation,
quantum amplitude estimation, and quantum gradient methods are used to train a
machine learned model. One advantage of this combination of algorithms is that
the quantum wavefunction does not need to be completely re-prepared at each
step, lowering a sizable pre-factor. Using the assumptions for solutions of the
ground-state algorithms on a quantum computer, we demonstrate that finding the
Kohn-Sham potential is not necessarily more difficult than the ground state
density. Once constructed, a classical user can use the resulting machine
learned functional to solve for the ground state of a system self-consistently,
provided the machine learned approximation is accurate enough for the input
system. It is also demonstrated how the classical user can access commonly used
time- and temperature-dependent approximations from the ground state model.
Minor modifications to the algorithm can learn other types of functional
theories including exact time- and temperature-dependence. Several other
algorithms--including quantum machine learning--are demonstrated to be
impractical in the general case for this problem.

Quantum Physics;Other Condensed Matter