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

Calculation of generating function in many-body systems with quantum computers: technical challenges and use in hybrid quantum-classical methods