  The Green's function plays a crucial role when studying the nature of quantum
many-body systems, especially strongly-correlated systems. Although the
development of quantum computers in the near future may enable us to compute
energy spectra of classically-intractable systems, methods to simulate the
Green's function with near-term quantum algorithms have not been proposed yet.
Here, we propose two methods to calculate the Green's function of a given
Hamiltonian on near-term quantum computers. The first one makes use of a
variational dynamics simulation of quantum systems and computes the dynamics of
the Green's function in real time directly. The second one utilizes the Lehmann
representation of the Green's function and a method which calculates excited
states of the Hamiltonian. Both methods require shallow quantum circuits and
are compatible with near-term quantum computers. We numerically simulated the
Green's function of the Fermi-Hubbard model and demonstrated the validity of
our proposals.
