  The realization of Majorana zero modes is in the centre of intense
theoretical and experimental investigations. Unfortunately, their exchange that
can reveal their exotic statistics needs manipulations that are still beyond
our experimental capabilities. Here we take an alternative approach. Through
the Jordan-Wigner transformation, the Kitaev's chain supporting two Majorana
zero modes is mapped to the spin-1/2 chain. We experimentally simulated the
spin system and its evolution with a photonic quantum simulator. This allows us
to probe the geometric phase, which corresponds to the exchange of two Majorana
zero modes positioned at the ends of a three-site chain. Finally, we
demonstrate the immunity of quantum information encoded in the Majorana zero
modes against local errors through the simulator. Our photonic simulator opens
the way for the efficient realization and manipulation of Majorana zero modes
in complex architectures.
