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Citation: Hutter, Adrian, and Daniel Loss. "Quantum computing with parafermions." Physical Review B 93.12 (2016): 125105.
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.
This gives a key point where abelian models can't be all clumped together. Parafermions, despite being qudit variants of Majorana zero modes, are strictly more computationally powerful - they can perform entangling gates. In particular, for any n>2 Parafermions can generate the full Clifford group by braiding alone.