SciELO - Scientific Electronic Library Online

vol.50 issue2Quantum Information and the Representation Theory of the Symmetric GroupSolutions of the hexagon equation for abelian anyons author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand



Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.50 no.2 Bogotá July/Dec. 2016 


Local unitary representations of the braid group and their applications to quantum computing

Colleen Delaney1, Eric C. Rowell2, Zhenghan Wang1, 3

1 University of California Santa Barbara, Santa Barbara, CA, U.S.A.
2 Texas A&M University, College Station, TX, U.S.A.
3 Microsoft Station Q, Santa Barbara, CA, U.S.A.


We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial by a quantum computer and explicit localizations of braid group representations.

Keywords: topological quantum computation, braid group representations, localizations, quantum algebra.

Mathematics Subject Classification: 81P86, 20F36.

Texto completo disponible en PDF


[1] D. Aharonov, V. Jones, and Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, Algorithmica 5 (2009), no. 3, 395-421.         [ Links ]

[2] S. Bigelow, The Burau representation is not faithful for n = 5, Geometry & Topology 3 (1999), 397-404, arXiv:math/9904100v2.         [ Links ]

[3] M. Brannan and B. Collins, Dual bases in Temperley-Lieb algebras, quantum groups, and a question of Jones, (2016), arXiv:1608.03885v2 [math.QA]         [ Links ].

[4] J. Conway and A. Jones, Trigonometric Diophantine equations (on vanishing sums of roots of unity), Acta Arithmetica 30 (1976), no. 3, 229-240.         [ Links ]

[5] M. Epple, Orbits of asteroids, a braid, and the first link invariant, Math. Intelligencer 20 (1998), no. 1, 45-52.         [ Links ]

[6] J. Franko, E. C. Rowell, and Z. Wang, Extraspecial 2-groups and images of braid group representations, J. Knot Theory Ramifications 15 (2006), no. 4, 1-15.         [ Links ]

[7] M. H. Freedman, A. Kitaev, and Z. Wang, Simulation of topological field theories by quantum computers, Comm. Math. Phys. 227 (2002), no. 3, 587-603.         [ Links ]

[8] M. H. Freedman, M. J. Larsen, and Z. Wang, A modular functor which is universal for quantum computation, Comm. Math. Phys. 227 (2002), no. 3, 605-622.         [ Links ]

[9] M.H. Freedman, M. J. Larsen, and Z. Wang, The two-eigenvalue problem and density of Jones representation of braid group, Comm. Math. Phys. 228 (2002), no. 1, 177-199.         [ Links ]

[10] W. Fulton and J.Harris, Representation theory, a first course, Springer, New York, 1991.         [ Links ]

[11] V.F.R. Jones, Braid groups, Hecke algebras and type II1 factors, Geometric methods in operator algebras 123 (1983), 242-273.         [ Links ]

[12] V.F.R. Jones, Hecke-algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335-288.         [ Links ]

[13] S.P. Jordan and P.W. Shor, Estimating Jones polynomials is a complete problem for one clean qubit, Quantum Information and Computation 8 (2008), no. 8, 681-714.         [ Links ]

[14] V. Kliuchnikov, A. Bocharov, and K. M. Svore., Asymptotically optimal topological quantum compiling, Physical Review Letters 112 (2014), no. 140504, 335-288.         [ Links ]

[15] G. Kuperberg, How hard is it to approximate the Jones polynomial?, (2009), arXiv:0908.0512v2 [quant-ph]         [ Links ].

[16] M.J. Larsen and E.C. Rowell, An algebra-level version of a link-polynomial identity of Lickorish, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 623-638.         [ Links ]

[17] M.J. Larsen, E.C. Rowell, and Z. Wang, The N-eigenvalue problem and two applications, Int. Math. Res. Not. 2005 (2005), no. 64, 3987-4018.         [ Links ]

[18] D. Naidu and E.C. Rowell, A finiteness property for braided fusion categories, Algebr. Represent. Theory 15 (2011), no. 5, 837-855.         [ Links ]

[19] E. C. Rowell, R. Stong, and Z. Wang, On classification of modular tensor categories, Comm. Math. Phys. 292 (2009), no. 2, 343-389.         [ Links ]

[20] E. C. Rowell and Z. Wang, Localization of unitary braid representations, Comm. Math. Phys. 311 (2012), no. 3, 595-615, arXiv:1009.0241v2 [math.RT]         [ Links ].

[21] E.C. Rowell, Braid representations from quantum groups of exceptional Lie type, Rev. Un. Mat. Argentina 51 (2010), no. 1, 165-175.         [ Links ]

[22] D.L. Vertigan, On the computational complexity of Tutte, Jones, Homfly and Kauffman invariants, Dissertation, University Of Oxford, 1991.         [ Links ]

[23] Z. Wang, Topological quantum computation, American Mathematical Society, Providence (2008),         [ Links ]

Recibido: julio de 2016 Aceptado: noviembre de 2016

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License