Quantum Computing with Parafermions

TL;DR

This paper explores the computational capabilities of braiding parafermions, showing they can generate the entire single-qudit Clifford group and, for odd d, the full many-qudit Clifford group, based on algebraic relations.

Contribution

It provides a systematic algebraic analysis of parafermion braiding, demonstrating their potential for universal quantum computation without relying on specific physical models.

Findings

Braiding four parafermions can generate the entire single-qudit Clifford group.

For odd d, braiding enables the generation of the full many-qudit Clifford group.

Derived braid group representations match previous physical derivations.

Abstract

$\mathbb{Z}_d$ Parafermions are exotic non-Abelian quasiparticles generalizing Majorana fermions, which correspond to the case $d=2$. In contrast to Majorana fermions, braiding of parafermions with $d>2$ allows to perform an entangling gate. This has spurred interest in parafermions and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a familiy of $2d$ representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a $d$-level qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows to generate the entire single-qudit Clifford group (up to phases), for any $d$. If $d$ is odd, then we show that in fact the entire many-qudit Clifford group can be generated.