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"Quantum Computing with Parafermions", Adrian Hutter, Daniel Loss, 2016

Reviewed August 22, 2023

Citation: Hutter, Adrian, and Daniel Loss. "Quantum computing with parafermions." Physical Review B 93.12 (2016): 125105.

Web: https://arxiv.org/abs/1511.02704

Tags: Abelian-anyons, Universal-scheme

This paper gives a detailed theoretical description of parafermions. Majorana zero modes (Majorana fermions) appear as defects at the end of lines in Z2-gauge-symmetric systems, such as the toric code. Majorana fermions are mathematically the Ising TQFT. Despite lots of theoretical advances, the unfortunate truth is that Majorana fermions are not very computationally powerful. The defects at the end of Zn quantum double models are known as parafermions, and hence generalize Majorana fermions. A good original reference for parafermions at the end of spin chains is this:

> Fendley, Paul. "Parafermionic edge zero modes in Zn-invariant spin chains." Journal of Statistical Mechanics: Theory and Experiment 2012.11 (2012): P11020.

Despite all being based on abelian models, for n>2 there are distinct advantages for quantum computation, which this paper explores.