Effective calculation of the Green's function in the time domain on near-term quantum processors

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 two-, three-, and four-site Hubbard models. This protocol significantly reduces the number of controlled operations when compared to those previously suggested in literature. Such reduction is quite remarkable when considering the two-site Hubbard model, for which we show that it is possible to obtain the exact time propagation of the vertical bar N +/- 1 > states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.