  We propose a quantum inverse iteration algorithm which can be used to
estimate the ground state properties of a programmable quantum device. The
method relies on the inverse power iteration technique, where the sequential
application of the Hamiltonian inverse to an initial state prepares an
approximate groundstate. To apply the inverse Hamiltonian operation, we write
it as a sum of unitary evolution operators using the Fourier approximation
approach. This allows to reformulate the protocol as separate measurements for
the overlap of initial and propagated wavefunction. The algorithm thus
crucially depends on the ability to run Hamiltonian dynamics with an available
quantum device. We benchmark the performance using paradigmatic examples of
quantum chemistry, corresponding to molecular hydrogen and beryllium hydride.
Finally, we show its use for studying the ground state properties of relevant
material science models which can be simulated with existing devices,
considering an example of the Bose-Hubbard atomic simulator.
