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Last edited by Randl (talk | contribs) 5 months ago. (Update) |
Parafermions is a type of anyons appearing in condensed matter system, specifically fractional quantum Hall effect system coupled to superconductors.[1][2][3] While opening a superconducting gap in 1D semiconductor results in Majorana zero modes, opening a superconducting gap in a 1D wire with fractional excitations results in parafermionic zero modes.
The parafermionic operators satisfy the following algebra:
Generalized Jordan–Wigner transformation
editSimilar to how fermions can be mapped to a two-level (qubit) system via Jordan–Wigner transformation, parafermions can be mapped to a p-level system (qudit) using generalized Jordan–Wigner transformation.
Consider parafermionic chain and define[4]
While braiding of parafermions is not universal, for odd generates full Clifford group,[4] making them more suitable for quantum computing than Majorana zero modes.
References
edit- ^ Lindner, Netanel H.; Berg, Erez; Refael, Gil; Stern, Ady (11 October 2012). "Fractionalizing Majorana Fermions: Non-Abelian Statistics on the Edges of Abelian Quantum Hall States". Physical Review X. 2 (4): 041002. doi:10.1103/PhysRevX.2.041002.
- ^ Alicea, Jason; Fendley, Paul (10 March 2016). "Topological Phases with Parafermions: Theory and Blueprints". Annual Review of Condensed Matter Physics. 7 (1): 119–139. doi:10.1146/annurev-conmatphys-031115-011336.
- ^ Fendley, Paul (28 November 2012). "Parafermionic edge zero modes in Zn-invariant spin chains". Journal of Statistical Mechanics: Theory and Experiment. 2012 (11): P11020. doi:10.1088/1742-5468/2012/11/P11020.
- ^ a b Hutter, Adrian; Loss, Daniel (2 March 2016). "Quantum computing with parafermions". Physical Review B. 93 (12). doi:10.1103/PhysRevB.93.125105.