  Synthesizing many-body interaction Hamiltonian is a central task in quantum
simulation. However, it is challenging to synthesize interactions including
more than two spins. Borrowing tools from quantum optics, we synthesize
five-body spin-exchange interaction in a superconducting quantum circuit by
simultaneously exciting four independent qubits with time-energy correlated
photon quadruples generated from a qudit. During the dynamic evolution of the
five-body interaction, a Greenberger-Horne-Zeilinger state is generated in a
single step with fidelity estimated to be $0.685$. We compare the influence of
noise on the three-, four- and five-body interaction as a step toward answering
the question on the quantum origin of chiral molecules. We also demonstrate a
many-body Mach-Zehnder interferometer which potentially has a Heisenberg-limit
sensitivity. This study paves a way for quantum simulation involving many-body
interactions and high excited states of quantum circuits.
