  Geometric phases, generated by cyclic evolutions of quantum systems, offer an
inspiring playground for advancing fundamental physics and technologies, alike.
Intriguingly, the exotic statistics of anyons realised in physical systems can
be interpreted as a topological version of geometric phases. However,
non-Abelian statistics has not yet been demonstrated in the laboratory. Here we
employ an all optical quantum system that simulates the statistical evolution
of Majorana fermions. As a result we experimentally realise non-Abelian Berry
phases with the topological characteristic that they are invariant under
continues deformations of their control parameters. We implement a universal
set of Majorana inspired gates by performing topological and non-topological
evolutions and investigate their resilience against perturbative errors. Our
photonic experiment, while it is not scalable, it suggests the intriguing
possibility of experimentally simulating Majorana statistics with scalable
technologies.
