  We show that chemically-accurate potential energy surfaces (PESs) can be
generated from quantum computers by measuring the density along an adiabatic
transition between different molecular geometries. In lieu of using phase
estimation, the energy is evaluated by performing line-integration using the
inverted TDDFT Kohn-Sham potential obtained from the time-varying densities.
The accuracy of this method depends on the validity of the adiabatic evolution
itself and the potential inversion process (which is theoretically exact but
can be numerically unstable), whereas total evolution time is the defining
factor for the precision of phase estimation. We examine the method with a
one-dimensional system of two electrons for both the ground and first triplet
state in first quantization, as well as the ground state of three- and four-
electron systems in second quantization. It is shown that few accurate
measurements can be utilized to obtain chemical accuracy across the full
potential energy curve, with shorter propagation time than may be required
using phase estimation for a similar accuracy. We also show that an accurate
potential energy curve can be calculated by making many imprecise density
measurements (using few shots) along the time evolution and smoothing the
resulting density evolution. We discuss how one can generate full PESs using
either sparse grid representations or machine learning density functionals
where it is known that training the functional using the density (along with
the energy) generates a more transferable functional than only using the
energy. Finally, it is important to note that the method is able to classically
provide a check of its own accuracy by comparing the density resulting from a
time-independent Kohn-Sham calculation using the inverted potential, with the
measured density.
