Computational equivalence of the two inequivalent spinor representations of the braidgroup in the Ising topological quantum computer

Abstract

We demonstrate that the two inequivalent spinor representations of the braid group, describing the exchanges of 2n+2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalentfrom the computational point of view, i.e., the sets of topologically protected quantumgates that could be implemented in both cases by braiding exactly coincide. We give theexplicit matrices generating almost all braidings in the spinor representations of the2n+2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us tounderstand better the physical difference between the two inequivalent representations andto propose a process that could determine the type of representation for any concretephysical realization of the Pfaffian quantum computer.