Electronic interactions can lead to exotic phases with emergent excitations that carry fractional charge and statistics. These fractionalized excitations are called non-Abelian anyons (non-Abelions) if their ground-state degeneracy depends exponentially on their numbers. The non-Abelions are protected against disorder, thermal fluctuation, and interactions; these observations have inspired the use of non-Abelions as building blocks of topological quantum computers. The Fibonacci anyon is capable of performing the entire set of quantum gates needed for a quantum computer through braiding operations. The fractional quantum Hall system has been the focal point of the search for non-Abelions, but the search has thus far been unsuccessful. A fundamental question that arises is whether we can find alternative systems with non-Abelions such as Fibonacci anyons as their local excitations. We provide an affirmative answer to this critical question and explicitly show that the heterostructure of an s -wave superconductor with a fractional quantum Hall state at a 2 / 3 filling fraction yields the Z 3 parafermion state that hosts Fibonacci anyons.

We first show how a fractional quantum Hall state can be viewed as an array of quantum chains coupled through backscattering. Next, we use conformal field theory and study the effect of superconducting pairing on individual chains. We show that when the intrachain electron pairing and backscattering have the same strength, individual chains are critical. We also consider the interchain couplings and demonstrate that they induce a bulk gap; a chiral gapless mode with a parafermion conformal field theory description survives. Thus, the resulting fractional topological superconductor, defined as a fractional quantum Hall system with a filling fraction of 2 / k that has superconducting pairing, is described by Z k parafermion conformal field theory.