By taking advantage of the superposition of quantum states, the hope is that algorithms not available to a classical computer can be run efficiently on a quantum computer. By doing so, larger and more exact solutions for quantum chemistry models may be possible. The question of which algorithm to use is the subject of recent research; however, it has already been shown that many classical algorithms will not be efficient on the quantum computer.

One method that I propose to use is a direct application of Lanczos' method [1]. The quantum computer is a natural vehicle to surpass the limits of this method on classical computers and has several advantages in terms of other quantum algorithms. Solving the problem in this way also gives access to the continued fraction representation of the Green's function. I will also discuss extensions using excitations [2] and how the same techniques can be applied to obtain exact density functional theory with a minimum of measurements of the wavefunction [3].

[1] T.E. Baker, "Lanczos recursion on a quantum computer for the Green’s function and ground state" {\it Phys. Rev. A} {\bf 103}, 032404 (2021) [arXiv: 2008.05593] [2] T.E. Baker, "Block Lanczos method for excited states on a quantum computer" (2021) [arxiv: 2109.14114] [3] T.E. Baker and D. Poulin, "Density functionals and Kohn-Sham potentials with minimal wavefunction preparations on a quantum computer" {\it Phys. Rev. Research} {\bf 2}, 043238 (2020) [arXiv: 2008.05592]

The author is grateful to the US-UK Fulbright Commission for financial support under the Fulbright U.S. Scholarship programme as hosted by the University of York. This research was undertaken in part thanks to funding from the Bureau of Education and Cultural Affairs from the United States Department of State.