  We introduce a hybrid quantum-classical algorithm to compute the Green
function for strongly correlated electrons on noisy intermediate-scale quantum
(NISQ) devices. The technique consists in the construction of a non-orthogonal
excitation basis composed of a set of single-particle excitations generated by
occupation number operators. The excited sectors of the Hamiltonian in this
basis can then be measured on the quantum device and a classical
post-processing procedure yields the Green function in the Lehmann
representation. The technique allow for noise filtering, a useful feature for
NISQ devices. To validate the approach, we carry out a set of
proof-of-principle calculations on the single-band Hubbard model on IBM quantum
hardware. For a 2 site system we find good agreement between the results of
quantum simulations and the exact result for the local spectral function. A
simulation of a 4 site system carried out on classical hardware suggests that
the approach can achieve similar accuracy for larger systems.
