Braiding Fibonacci anyons
Ludmil K. Hadjiivanov, Lachezar S. Georgiev · Journal of High Energy Physics · 2024
A bstract 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 ℤ 3 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 = 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 $$ \mathcal{B} $$ 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 $$ \mathcal{N} $$ N qubit computational spaces in terms of conformal blocks of $$ 2\mathcal{N} $$ 2 N + 2 Fibonacci anyons.