Topological quantum computation is an implementation of a quantum computer in a way that radically reduces decoherence. Topological qubits are encoded in the topological evolution of two-dimensional quasi-particles called anyons and universal set of quantum gates can be constructed by braiding these anyons yielding to a topologically protected circuit model. In the present study we remind the basics of this emerging quantum computation scheme and illustrate how a topological qubit built with three Fibonacci anyons might be adopted to achieve leakage free braiding gate by exchanging the anyons composing it. A single-qubit braiding gate that approximates the Hadamard quantum gate to a certain accuracy is numerically implemented using a brute force search method. The algorithms utilized for that purpose are explained and the numerical programs are publicly shared for reproduction and further use.