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.

Highlights

  • Topological phases of matter have attracted a significant amount of attention in recent times due to the diversity of emerging physical phenomena heralded by them [1,2,3,4,5]

  • In this paper we develop a Monte Carlo enhanced Solovay-Kitaev algorithm (MCESKA), which inherits its generic structure from the SKA

  • The Fibonacci model is capable of universal quantum computation by only braiding the anyons, whereas the Ising anyon model must be supplemented with an additional unitary operator such as a suitably chosen phase gate

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Summary

INTRODUCTION

Topological phases of matter have attracted a significant amount of attention in recent times due to the diversity of emerging physical phenomena heralded by them [1,2,3,4,5]. Irrespective of the hardware-level implementation details, before any quantum computations can be performed, one must find a mapping between the desired quantum gates and their corresponding braids (or measurement protocols) that must be realized on the anyon hardware. [43] a reinforcement learning technique was deployed in the context of the quantum compilation problem, which appears to have a similar braid word length versus error performance as the MCESKA Since it is fully generic, it will be interesting to see if it too allows accommodating extra functionality such as suppressing the fraction of noisy gates required in hybrid TQC implementations. For the sake of completeness, in Appendix A we provide a brief theoretical background to anyons and TQC and in Appendix B some rudiments of the SU(2)k anyon models are outlined

QUANTUM COMPILATION
A Monte Carlo approach
Mapping the problem onto a ZN spin chain
A thermodynamic picture
A Monte Carlo algorithm
Performance of the Monte Carlo algorithm
The Solovay-Kitaev algorithm
The Shrinking lemma and the Solovay-Kitaev theorem
Implementation of the Solovay-Kitaev algorithm
Monte Carlo enhanced Solovay-Kitaev algorithm
Comparison of compiler algorithms
Fibonacci versus Ising anyon models
Decoherence in nonuniversal anyon models
Noise corrupted braid words in the hybrid Ising anyon model
CONCLUSIONS
F moves and R moves
Universal quantum computation
Findings
The Fibonacci and Ising anyon models

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