Fibonacci Anyons Versus Majorana Fermions: A Monte Carlo Approach to the Compilation of Braid Circuits in SU(2)_{k} Anyon Models

We study SU(2)_{k} anyon models, assessing their prospects for topological quantum computation. In particular, we compare the Ising (k=2) anyon and Fibonacci (k=3) anyon models, motivated by their potential for future realizations based on Majorana fermion quasiparticles or exotic fractional quantum Hall states, respectively. The quantum computational performance of the different anyon models is quantified at the single-qubit level by the difference between a target unitary operator and its approximation realized by anyon braiding. To facilitate efficient comparisons, we develop a Monte Carlo enhanced Solovay-Kitaev quantum compiler algorithm that finds near-optimal braid words in polynomial time from the exponentially large search tree. Since universal quantum computation cannot be achieved within the Ising anyon model by braiding alone, we introduce an additional elementary phase gate to model a nontopological measurement process, which restores universality of the anyon model at the cost of breaking the full topological protection. We model conventional kinds of decoherence processes algorithmically by introducing a controllable noise term to all nontopological gate operations. We find that, for reasonable levels of decoherence, even the hybrid Ising anyon model retains a significant topological advantage over a conventional, nontopological, quantum computer. Furthermore, we find that only surprisingly short anyon braids are ever required to be compiled due to the gate noise exceeding the intrinsic error of the braid words already for word lengths of the order of 100 elementary braids. We conclude that the future for hybrid topological quantum computation remains promising.