  Quantum algorithms are touted as a way around some classically intractable
problems such as the simulation of quantum mechanics. At the end of all quantum
algorithms is a quantum measurement whereby classical data is extracted and
utilized. In fact, many of the modern hybrid-classical approaches are
essentially quantum measurements of states with short quantum circuit
descriptions. Here, we compare and examine three methods of extracting the
time-dependent one-particle probability density from a quantum simulation:
direct $Z$-measurement, Bayesian phase estimation and harmonic inversion. We
have tested these methods in the context of the potential inversion problem of
time-dependent density functional theory. Our test results suggest that direct
measurement is the preferable method. We also highlight areas where the other
two methods may be useful and report on tests using Rigetti's quantum virtual
device. This study provides a starting point for imminent applications of
quantum computing.
