A prospective application of quantum computing is solving quantum chemistry problems, however, obtaining exact solutions is difficult due to the lack of general method of obtaining such solutions. Typically the solution lies in finding the ground state energy, even though the energy is not descriptive enough to fully characterize all desired properties of a system. In order to find such properties, many measurements of the wavefunction are required. These measurements are expensive, because the wavefunction cannot be copied and must often be re-prepared before a second measurement is performed. Finding the full wavefunction would require exponentially many measurements, so one option would be to encode many solutions into one measurement by using a machine learned (ML) model. Training of the ML model requires finding exact quantities at several different external potentials. Besides that, one can use density functional theory (DFT), to replace the wavefunction with the one-body density, n(r), which has fewer variables. When DFT is used, instead of the Hamiltonian, the universal functional, F[n], must be found. The quantities required for the classical user to find self-consistent solutions are the exact functional (determining the energy) and the functional derivative. So, in addition to finding F[n], one also must find some other quantity such as the density, n(r), or the Kohn-Sham(KS) potential, vs(r). With these components, one can fully characterize a quantum ground state and solve for other measurable quantities.

The authors propose an algorithm that finds the ML model for F[n] on the quantum computer if a ground-state wavefunction is available. The algorithm leaves the wavefunction largely undisturbed so it can be used as the starting state for another system, greatly reducing the pre-factor required to solve other systems through a quantum counting algorithm to extract descriptive quantities such as the density. Most of this algorithm is kept entirely on the quantum computer to motivate future improvements for speed, but the counting algorithm does allow for information to be output classically. Moreover, the authors demonstrate that the exact Kohn-Sham potential can be solved in a faster way with a gradient evaluated on a cost function for the Kohn-Sham system.

The novelty of the proposed algorithm suggested in this work is the limitation of the number of measurements and re-preparations of the wavefunction especially in the case of time-dependent quantities. Since there is no algorithm for the general case that is exponentially better than the proposed algorithm, limiting the number of measurements and re-preparations of the wavefunction is as best as one can achieve. The proposed algorithm is a combination of several known algorithms including quantum phase estimation, quantum amplitude estimation, and quantum gradient methods that are iteratively used to train a machine learned model.