  The Fibonacci topological order is the prime candidate for the realization of
universal topological quantum computation. We devise minimal quantum circuits
to demonstrate the non-Abelian nature of the doubled Fibonacci topological
order, as realized in the Levin-Wen string net model. Our circuits effectively
initialize the ground state, create excitations, twist and braid them, all in
the smallest lattices possible. We further design methods to determine the
fusion amplitudes and braiding phases of multiple excitations by carrying out a
single qubit measurement. We show that the fusion channels of the doubled
Fibonacci model can be detected using only three qubits, twisting phases can be
measured using five, and braiding can be demonstrated using nine qubits. These
designs provide the simplest possible settings for demonstrating the properties
of Fibonacci anyons and can be used as realistic blueprints for implementation
on many modern quantum architectures.
