  The Pfaffian state is an attractive candidate for the observed quantized Hall
plateau at Landau level filling fraction $\nu=5/2$. This is particularly
intriguing because this state has unusual topological properties, including
quasiparticle excitations with non-Abelian braiding statistics. In order to
determine the nature of the $\nu=5/2$ state, one must measure the quasiparticle
braiding statistics. Here, we propose an experiment which can simultaneously
determine the braiding statistics of quasiparticle excitations and, if they
prove to be non-Abelian, produce a topologically-protected qubit on which a
logical NOT operation is performed by quasiparticle braiding. Using the
measured excitation gap at $\nu=5/2$, we estimate the error rate to be
$10^{-30}$ or lower.
