  Some theories predict that the filling factor 5/2 fractional quantum Hall
state can exhibit non-Abelian statistics, which makes it a candidate for
fault-tolerant topological quantum computation. Although the non-Abelian
Pfaffian state and its particle-hole conjugate, the anti-Pfaffian state, are
the most plausible wave functions for the 5/2 state, there are a number of
alternatives with either Abelian or non-Abelian statistics. Recent experiments
suggest that the tunneling exponents are more consistent with an Abelian state
rather than a non-Abelian state. Here, we present edge-current-tunneling
experiments in geometrically confined quantum point contacts, which indicate
that Abelian and non-Abelian states compete at filling factor 5/2. Our results
are consistent with a transition from an Abelian state to a non-Abelian state
in a single quantum point contact when the confinement is tuned. Our
observation suggests that there is an intrinsic non-Abelian 5/2 ground state,
but that the appropriate confinement is necessary to maintain it. This
observation is important not only for understanding the physics of the 5/2
state, but also for the design of future topological quantum computation
devices.
