Systematic Construction of Braids for Topological Quantum Computation

In topological quantum computation, quantum gates are carried out by braiding worldlines of non-Abelian anyons in 2+1 dimensional space-time. The simplest such anyons for which braiding is universal for quantum computation are Fibonacci anyons. Reichardt [1] has shown how to construct nontrivial braids for three Fibonacci anyons which yield $2 \times 2$ unitary operations whose off-diagonal matrix elements (in the appropriate basis) can be made arbitrarily small through a simple and efficient iterative procedure. A great advantage of this construction is that it does not require either brute force search or the Solovay-Kitaev method. There is, however, a downside---the phases of the diagonal matrix elements cannot be directly controlled. Despite this, we show that the resulting braids can be used to construct leakage-free entangling two-qubit gates for qubits encoded using four Fibonacci anyons each. We give two explicit constructions---one based on the ``functional braid" approach of Hu and Wan [2], and another based on the ``effective qubit" approach of Hormozi et al. [3]. \newline [1] B.W. Reichardt, Quant. Inf. and Comp. {\bf 12}, 876 (2012). \newline [2] H. Xu and X. Wan, PRA \textbf{78}, 042325 (2008). \newline [3] L. Hormozi et al., PRL {\bf 103}, 160501 (2009).