  The Green's function has been an indispensable tool to study many-body
systems that remain one of the biggest challenges in modern quantum physics for
decades. The complicated calculation of Green's function impedes the research
of many-body systems. The appearance of the noisy intermediate-scale quantum
devices and quantum-classical hybrid algorithm inspire a new method to
calculate Green's function. Here we design a programmable quantum circuit for
photons with utilizing the polarization and the path degrees of freedom to
construct a highly-precise variational quantum state of a photon, and first
report the experimental realization for calculating the Green's function of the
two-site Fermionic Hubbard model, a prototypical model for strongly-correlated
materials, in photonic systems. We run the variational quantum eigensolver to
obtain the ground state and excited states of the model, and then evaluate the
transition amplitudes among the eigenstates. The experimental results present
the spectral function of Green's function, which agrees well with the exact
results. Our demonstration provides the further possibility of the photonic
system in quantum simulation and applications in solving complicated problems
in many-body systems, biological science, and so on.
