  Fibonacci anyons provide the simplest possible model of non-Abelian fusion
rules: [1] x [1] = [0] + [1]. We propose a conformal field theory construction
of topological quantum registers based on Fibonacci anyons realized as
quasiparticle excitations in the Z_3 parafermion fractional quantum Hall state.
To this end, the results of Ardonne and Schoutens for the correlation function
of n = 4 Fibonacci fields are extended to the case of arbitrary n (and 3 r
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 acting on n-point conformal blocks of
Fibonacci anyons. For low n (up to n = 8), the matrices of braid group
generators are displayed explicitly. A simple recursion formula makes it
possible to extend without efforts the construction to any n. Finally, we
construct N qubit computational spaces in terms of conformal blocks of 2N + 2
Fibonacci anyons.
