Braiding Fibonacci anyons

Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ₃ parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group $\mathcal{B}$ \( \mathcal{B} \) _n acting on n-point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n. Finally, we construct $\mathcal{N}$ \( \mathcal{N} \) qubit computational spaces in terms of conformal blocks of $2\mathcal{N}$ \( 2\mathcal{N} \) + 2 Fibonacci anyons.