Generating non-Clifford gate operations through exact mapping between Majorana fermions and $\mathbb{Z}_4$ parafermions

Majorana fermions and their generalizations to $\mathbb{Z}_n$ parafermions are considered promising building blocks of fault-tolerant quantum computers for their ability to encode quantum information nonlocally. In such topological quantum computers, highly robust quantum gates are obtained by braiding pairs of these quasi-particles. However, it is well-known that braiding Majorana fermions or parafermions only leads to a Clifford gate, hindering quantum universality. This paper establishes an exact mapping between Majorana fermions to $\mathbb{Z}_4$ parafermions in systems under total parity non-conserving and total parity conserving setting. It is revealed that braiding of Majorana fermions may lead to non-Clifford quantum gates in the 4-dimensional qudit representation spanned by $\mathbb{Z}_4$ parafermions, whilst braiding of $\mathbb{Z}_4$ parafermions may similarly yield non-Clifford quantum gates in the qubit representation spanned by Majorana fermions. This finding suggests that topologically protected universal quantum computing may be possible with Majorana fermions ($\mathbb{Z}_4$ parafermions) by supplementing the usual braiding operations with the braiding of $\mathbb{Z}_4$ parafermions (Majorana fermions) that could be formed out of Majorana fermions ($\mathbb{Z}_4$ parafermions) via the mapping prescribed here. Finally, the paper discusses how braiding of Majorana fermions or $\mathbb{Z}_4$ parafermions could be obtained via a series of parity measurements.