  The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle
e^{-itH}\rangle$, where $t$ is the time and where the expectation value is
taken on a given initial quantum state. This function gives access to the
different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders
$K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on
quantum computers using one extra ancillary qubit with a set of measurement for
each value of the time $t$. The low cost in terms of qubits renders it very
attractive in the near term period where the number of qubits is limited.
Assuming that the generating function can be precisely computed using quantum
devices, we show how the information content of this function can be used a
posteriori on classical computers to solve quantum many-body problems. Several
methods of classical post-processing are illustrated with the aim to predict
approximate ground or excited state energies and/or approximate long-time
evolutions. This post-processing can be achieved using methods based on the
Krylov space and/or on the $t$-expansion approach that is closely related to
the imaginary time evolution. Hybrid quantum-classical calculations are
illustrated in many-body interacting systems using the pairing and
Fermi-Hubbard models.
