  Recent experiments have focused attention on the properties of chains of
atoms in which the atoms are either in their ground states or in highly excited
Rydberg states which block similar excitations in their immediate neighbors. As
the low energy Hilbert space of such chains is isomorphic to that of a chain of
Fibonacci anyons, they have been proposed as a platform for topological quantum
computation and for simulating anyon dynamics. We show that generic local
operators in the Rydberg chain correspond to non-local anyonic operators that
do not preserve a topological symmetry of the physical anyons. Consequently, we
argue that Rydberg chains do not yield Fibonacci anyons and quantum computation
with Rydberg atoms is not topologically protected.
