Minimal Quantum Circuits for Simulating Fibonacci Anyons

The Fibonacci topological order is the prime candidate for the realization of universal fault-tolerant topological quantum computation. Here 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 minimal designs can be used as realistic blueprints to demonstrate the properties of Fibonacci anyons on many modern quantum architectures.