Systematically Generated Two-Qubit Braids for Fibonacci Anyons

We show how two-qubit Fibonacci anyon braids can be generated using a simple iterative procedure which, in contrast to previous methods, does not require brute force search [1]. Our construction is closely related to that of [2], but with the new feature that it can be used for three-anyon qubits as well as four-anyon qubits. The iterative procedure we use, which was introduced by Reichardt [3], generates sequences of three-anyon weaves that asymptotically conserve the total charge of two of the three anyons, without control over the corresponding phase factors. The resulting two-qubit gates are independent of these factors and their length grows as log 1/$\epsilon$, where $\epsilon$ is the error, which is asymptotically better than the Solovay-Kitaev method.\\ \ \ [1] C. Carnahan, D. Zeuch, and N. E. Bonesteel, arXiv:1511.00719v1 (2015).\\ \ [2] H. Xu and X. Wan, Phys. Rev. A \textbf{78}, 042325 (2008).\\ \ [3] B. W. Reichardt, Quantum Information & Computation \textbf{12}, 876 (2012).