This thesis introduces the emerging field of quantum computing, emphasizing its capability to surpass traditional computing by solving complex problems that are beyond the reach of classical computers. Unlike classical systems that operate with bits and logic gates, quantum computing utilizes qubits and quantum gates, exploiting the vast computational space offered by quantum mechanics. A focal point of this study is topological quantum computing, a novel approach designed to overcome the inherent vulnerability of quantum systems to errors, such as decoherence and operational inaccuracies. At the heart of this method lies the use of non-Abelian anyons, with a particular focus on Fibonacci anyons, whose unique topological characteristics and braiding operations present a viable path to fault-tolerant quantum computation. This thesis aims to elucidate how the braiding of Fibonacci anyons can be employed to construct the necessary quantum gates for topological quantum computing. By offering a foundational exploration of quantum computing principles, especially topological quantum computing, and detailing the process for creating quantum gates through braiding of Fibonacci anyons, the work sets the stage for further research and development in this transformative computing paradigm.