Special Issue: Proceedings of the Special Session “Low-Dimensional Topology and Its Relationships with Physics”, held at the AMS/EMS/SPM meeting in Porto, Portugal, June 10–13, 2015; Guest Editors: L. H. Kauffman and P. LopesNo Access

Braiding and Majorana fermions

Louis H. Kauffman

Louis H. Kauffman

Department of Mathematics, Statistics and Computer Science (m/c 249), 851 South Morgan Street, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA

E-mail Address: kauffman@uic.edu

E-mail Address: loukau@gmail.com

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https://doi.org/10.1142/S0218216517430015Cited by:3 (Source: Crossref)

Abstract

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

Keywords:

AMSC: 57M25

References

1. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982). Google Scholar
2. C. W. J. Beenakker, Search for Majorana fermions in superconductors, arXiv:1112.1950. Google Scholar
3. N. E. Bonesteel, L. Hormozi, G. Zikos and S. H. Simon, Braid topologies for quantum computation, Phys. Rev. Lett. 95(14) (2005) 140503, 4 pp. quant-ph/0505665. Crossref, Web of Science, Google Scholar
4. S. H. Simon, N. E. Bonesteel, M. H. Freedman, N. Petrovic and L. Hormozi, Topological quantum computing with only one mobile quasiparticle, Phys. Rev. Lett. 96(7) (2006) 070503, 4 pp., quant-ph/0509175. Crossref, Web of Science, Google Scholar
5. J. L. Brylinski and R. Brylinski, Universal quantum gates, in Mathematics of Quantum Computation, eds. R. Brylinski and G. Chen (Chapman & Hall/CRC Press, Boca Raton, Florida, 2002). Crossref, Google Scholar
6. G. Chen, L. Kauffman and S. Lomonaco (eds.), Mathematics in Quantum Computation and Quantum Technology (Chapman & Hall/CRC, 2007). Crossref, Google Scholar
7. E. Fradkin and P. Fendley, Realizing non-abelian statistics in time-reversal invariant systems, Theory Seminar, Physics Department, UIUC, 4/25/2005. Google Scholar
8. B. Hatfield, Quantum Field Theory of Point Particles and Strings (Perseus Books, Cambridge, Massachusetts, 1991). Google Scholar
9. D. A. Ivanov, Non-abelian statistics of half-quantum vortices in $p$ -wave superconductors, Phys. Rev. Lett. 86 (2001) 268–271. Crossref, Web of Science, Google Scholar
10. L. H. Kauffman, Temperley–Lieb recoupling theory and invariants of three-manifolds, Annals Studies 114 (1994). Google Scholar
11. L. H. Kauffman, Knots and Physics (World Scientific Publishers, 1991), Second Edition (1993), Third Edition (2002), Fourth Edition (2012). Link, Google Scholar
12. L. H. Kauffman, Knot logic and topological quantum computing with Majorana fermions, in Logic and Algebraic Structures in Quantum Computing, eds. Chubb, Eskadarian and Harizanov (Cambridge University Press, 2016), pp. 223–335. Crossref, Google Scholar
13. L. H. Kauffman and S. J. Lomonaco Jr., Quantum entanglement and topological entanglement, New J. Phys. 4 (2002) 73.1–73.18 (http://www.njp.org/). Crossref, Web of Science, Google Scholar
14. L. H. Kauffman, Teleportation topology, quant-ph/0407224 (in the Proc. 2004 Byelorus Conf. Quantum Optics), Opt. Spectrosc. 9 (2005) 227–232. Google Scholar
15. L. H. Kauffman and S. J. Lomonaco Jr., Entanglement criteria — quantum and topological, in Quantum Information and Computation — Spie Proceedings, (21–22 April 2003), eds., Orlando, FL, Donkor, Pinch and Brandt, Vol. 5105, pp. 51–58. Google Scholar
16. L. H. Kauffman and S. J. Lomonaco Jr., Quantum knots, in Quantum Information and Computation II, Proceedings of Spie, ed., Donkor, Pirich and Brandt (12–14 April 2004), pp. 268–284. Google Scholar
17. S. J. Lomonaco and L. H. Kauffman, Quantum knots and mosaics, J. Quant. Inform. Process. 7(2–3) (2008) 85–115, http://arXiv.org/abs/0805.0339. Crossref, Web of Science, Google Scholar
18. S. J. Lomonaco and L. H. Kauffman, Quantum knots and lattices, or a blueprint for quantum systems that do rope tricks, Quantum information science and its contributions to mathematics, 209276, Proc. Sympos. Appl. Math., Vol. 68 (Amer. Math. Soc., Providence, RI, 2010). Google Scholar
19. S. J. Lomonaco and L. H. Kauffman, Quantizing braids and other mathematical structures: The general quantization procedure, in Quantum Information and Commputation IX — Spie Proceedings, eds. Brandt and Donkor, Pirich, Vol. 8057, (SPIE, April 2011), pp. 805702-1–805702-14. Google Scholar
20. L. H. Kauffman and S. J. Lomonaco, Quantizing knots groups and graphs, in Quantum Information and Commputation IX — Spie Proceeding, eds. Brandt, Donkor and Pirich, Vol. 8057, (SPIE, April 2011), pp. 80570T-1–80570T-15. Google Scholar
21. L. H. Kauffman and S. J. Lomonaco, Braiding operators are universal quantum gates, New J. Phys. 6(134) (2004) 1–39. Google Scholar
22. L. H. Kauffman and S. J. Lomonaco, Jr., $q$ -deformed spin networks, knot polynomials and anyonic topological quantum computation, J. Knot Theory Ramific. 16(3) (2007) 267–332. Link, Web of Science, Google Scholar
23. L. H. Kauffman and S. J. Lomonaco Jr., Spin networks and quantum computation, in Lie Theory and Its Applications in Physics VII, eds. H. D. Doebner and V. K. Dobrev (Heron Press, Sofia, 2008), pp. 225–239. Google Scholar
24. L. H. Kauffman, Quantum computing and the Jones polynomial, math.QA/0105255, in Quantum Computation and Information, S. Lomonaco Jr., (ed.), AMS CONM/305, (2002), pp. 101–137. Crossref, Google Scholar
25. L. H. Kauffman and S. J. Lomonaco Jr., Quantum entanglement and topological entanglement, New J. Phys. 4 (2002) 73.173.18. Crossref, Web of Science, Google Scholar
26. L. H. Kauffman and S. J. Lomonaco Jr., The Fibonacci model and the Temperley–Lieb algebra, Int. J. Modern Phys. B 22(29) (2008) 5065–5080. Link, Web of Science, Google Scholar
27. L. H. Kauffman, Iterants, fermions and Majorana operators, in Unified Field Mechanics — Natural Science Beyond the Veil of Spacetime, eds. R. Amoroso, L. H. Kauffman and P. Rowlands (World Scientific Pub. Co., 2015), pp. 1–32. Link, Google Scholar
28. L. H. Kauffman, Knot logic and topological quantum computing with Majorana fermions, in Logic and Algebraic Structures in Quantum Computing and Information, Lecture Notes in Logic, eds. J. Chubb, Ali Eskandarian and V. Harizanov (Cambridge University Press, 2016), pp. 223–335. Google Scholar
29. L. H. Kauffman and P. Lopes, Eulerian braids (in preparation). Google Scholar
30. V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers and L. P. Kouwenhuven, Signatures of Majorana fermions in hybrid superconductor-semiconductor devices, arXiv:1204.2792. Google Scholar
31. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321(1) (2006) 2111. arXiv.cond-mat/0506438. Crossref, Web of Science, Google Scholar
32. E. Majorana, A symmetric theory of electrons and positrons, I Nuovo Cimento 14 (1937) 171–184. Crossref, Google Scholar
33. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). Google Scholar
34. F. Wilczek, Fractional Statistics and Anyon Superconductivity (World Scientific Publishing Company, 1990). Link, Google Scholar