  We propose an improved quantum algorithm to calculate the Green's function
through real-time propagation, and use it to compute the retarded Green's
function for the 2-, 3- and 4-site Hubbard models. This novel protocol
significantly reduces the number of controlled operations when compared to
those previously suggested in literature. Such reduction is quite remarkable
when considering the 2-site Hubbard model, for which we show that it is
possible to obtain the exact time propagation of the $|N\pm 1\rangle$ states by
exponentiating one single Pauli component of the Hamiltonian, allowing us to
perform the calculations on an actual superconducting quantum processor.
