  Topological quantum computation (TQC) is one of the most striking
architectures that can realize fault-tolerant quantum computers. In TQC, the
logical space and the quantum gates are topologically protected, i.e., robust
against local disturbances. The topological protection, however, requires
rather complicated lattice models and hard-to-manipulate dynamics; even the
simplest system that can realize universal TQC--the Fibonacci anyon
system--lacks a physical realization, let alone braiding the non-Abelian
anyons. Here, we propose a disk model that can realize the Fibonacci anyon
system, and construct the topologically protected logical spaces with the
Fibonacci anyons. Via braiding the Fibonacci anyons, we can implement universal
quantum gates on the logical space. Our proposal is platform-independent. As a
demonstration, we implement a topological Hadamard gate on a logical qubit
through a sequence of $15$ braiding operations of three Fibonacci anyons with
merely $2$ nuclear spin qubits. The gate fidelity reaches 97.18% by randomized
benchmarking. We further prove by experiment that the logical space and
Hadamard gate are topologically protected: local disturbances due to thermal
fluctuations result in a global phase only. Our work is a proof of principle of
TQC and paves the way towards fault-tolerant quantum computation.
