Fibonacci anyons versus Majorana fermions

TL;DR

This paper compares Fibonacci and Ising anyon models for topological quantum computing, introducing a new compilation algorithm and analyzing their robustness against decoherence, showing promising prospects for hybrid topological quantum computation.

Contribution

The paper develops a Monte Carlo enhanced Solovay-Kitaev algorithm for efficient braid compilation and analyzes the impact of decoherence on the computational advantages of different anyon models.

Findings

Fibonacci anyons can achieve universal quantum computation via braiding.

Hybrid Ising anyons with an additional phase gate retain topological advantages under noise.

Short braids are sufficient for effective quantum gate implementation due to noise levels.

Abstract

We have studied ${\rm SU}(2)_k$ anyon models, assessing their prospects for topological quantum computation. In particular, we have compared 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 single qubit level by the difference between a target unitary operator and its approximation realised by anyon braiding. To facilitate efficient comparisons, we have developed a Monte Carlo enhanced Solovay-Kitaev quantum compiler algorithm that finds 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 have introduced an additional elementary phase gate to model a non-topological 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 non-topological gate operations. We find that for reasonable levels of decoherence, even the hybrid Ising anyon model retains a significant topological advantage over a conventional, non-topological, 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.