  Photonic quantum simulators are promising candidates for providing insight
into other small- to medium-sized quantum systems. The available photonic
quantum technology is reaching the state where significant advantages arise for
the quantum simulation of interesting questions in Heisenberg spin systems.
Here we experimentally simulate such spin systems with single photons and
linear optics. The effective Heisenberg-type interactions among individual
single photons are realized by quantum interference at the tunable direction
coupler followed by the measurement process. The effective interactions are
characterized by comparing the entanglement dynamics using pairwise concurrence
of a four-photon quantum system. We further show that photonic quantum
simulations of generalized Heisenberg interactions on a four-site square
lattice and a six-site checkerboard lattice are in reach of current technology.
