  Non-Abelian anyons--particles whose exchange noncommutatively transforms a
system's quantum state--are widely sought for the exotic fundamental physics
they harbor as well as for quantum computing applications. There now exist
numerous blueprints for stabilizing the simplest type of non-Abelian anyon,
defects binding Majorana modes, by judiciously interfacing widely available
materials. Following this line of attack, we introduce a device fabricated from
conventional fractional quantum Hall states and s-wave superconductors that
supports exotic non-Abelian anyons that bind `parafermions', which can be
viewed as fractionalized Majorana fermions. We show that these modes can be
experimentally identified (and distinguished from Majoranas) using Josephson
measurements. We also provide a practical recipe for braiding parafermions and
show that they give rise to non-Abelian statistics. Interestingly, braiding in
our setup produces a richer set of topologically protected qubit operations
when compared to the Majorana case. As a byproduct, we establish a new,
experimentally realistic Majorana platform in weakly spin-orbit-coupled
materials such as GaAs.
