Topological quantum computation is considered the best proposal to overcome local errors in the realization of a quantum computer. In this scheme, it is well known that Fibonacci anyons can be used to implement the universal quantum computation, allowing to approximate with their braidings every single-qubit and two-qubit gate at any given accuracy. In this talk we will analyze the key features of these non-abelian anyons and of their braiding group; we will study the main technics used to achieve the compiling of single-qubit gates, focusing our attention on Bonesteel's "brute force" approach and on the quantum hashing through the icosahedral group.
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