nLab topological quantum computation with anyons -- references

Topological quantum computation with anyons

The idea of topological quantum computation via the Chern-Simons theory of anyons (e.g. in the quantum Hall effect) is due to:

• Alexei Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2-30 (arXiv:quant-ph/9707021, doi:10.1016/S0003-4916(02)00018-0)

• Michael Freedman, Alexei Kitaev, Michael Larsen, Zhenghan Wang, Topological quantum computation, Bull. Amer. Math. Soc. 40 (2003), 31-38 (arXiv:quant-ph/0101025, doi:10.1090/S0273-0979-02-00964-3, pdf)

• Michael Freedman, Michael Larsen, Zhenghan Wang, A modular functor which is universal for quantum computation, Communications in Mathematical Physics. 2002, Vol 227, Num 3, pp 605-622 (arXiv:quant-ph/0001108)

  (specifically via su(2)-anyons)

• D. Melnikov, A. Mironov, S. Mironov, A. Morozov, An. Morozov, Towards topological quantum computer, Nucl. Phys. B926 (2018) 491-508 (arXiv:1703.00431, doi:10.1016/j.nuclphysb.2017.11.016)

Textbook accounts:

• Zhenghan Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS 2010 (ISBN-13: 978-0-8218-4930-9, pdf)

• Jiannis K. Pachos, Introduction to Topological Quantum Computation, Cambridge University Press (2012) $[$doi:10.1017/CBO9780511792908$]$

• Tudor D. Stanescu, Part IV of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)

• Steven H. Simon, Topological Quantum, 2021 $[$pdf, webpage$]$

Review:

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) $[$arXiv:0707.1888, doi:10.1103/RevModPhys.80.1083$]$

• Ady Stern, Netanel H. Lindner, Topological Quantum Computation – From Basic Concepts to First Experiments, Science 339 6124 (2013) 1179-1184 (doi:10.1126/science.1231473)

• Eric C. Rowell, An Invitation to the Mathematics of Topological Quantum Computation, J. Phys.: Conf. Ser. 698 (2016) 012012 (doi:10.1088/1742-6596/698/1/012012)

• Ville Lahtinen, Jiannis K. Pachos, A Short Introduction to Topological Quantum Computation, SciPost Phys. 3 021 (2017) (arXiv:1705.04103)

• Eric C. Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)

• Bernard Field, Tapio Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 3 (2018) 045004 (arXiv:1802.06176, doi:10.1088/2058-9565/aacad2)

Focus on abelian anyons:

• Jiannis K. Pachos, Quantum computation with abelian anyons on the honeycomb lattice, International Journal of Quantum Information 4 6 (2006) 947-954 (arXiv:quant-ph/0511273)

• James Robin Wootton, Dissecting Topological Quantum Computation, 2010 (pdf, pdf)

  “non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models.”

• Wade Bloomquist, Zhenghan Wang, On Topological Quantum Computing With Mapping Class Group Representations, J. Phys. A: Math. Theor. 52 (2019) 015301 (arXiv:1805.04622, doi:10.1088/1751-8121/aaeea1)

• Seth Lloyd, Quantum computation with abelian anyons, Quantum Information Processing 1 1/2 (2002) (arXiv:quant-ph/0004010, doi:10.1023/A:1019649101654)

Realization in experiment:

• Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,

  Experimental Quantum Computations on a Topologically Encoded Qubit, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 (arXiv:1403.5426, doi:10.1126/science.1253742)

  (for quantum error correction)

Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):

• Ivan Marin, On the representation theory of braid groups, Annales mathématiques Blaise Pascal, 20 2 (2013) 193-260 (arXiv:math/0502118, dml:275607)

Review:

• Camilo Arias Abad, Introduction to representations of braid groups, Rev. colomb. mat. vol.49 no.1 (2015) (arXiv:1404.0724, doi:10.15446/recolma.v49n1.54160)

• Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 (pdf, pdf)

in relation to modular tensor categories:

• Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

Braid representations seen inside the topological K-theory of the braid group‘s classifying space:

• Alejandro Adem, Daniel C. Cohen, Frederick R. Cohen, On representations and K-theory of the braid groups, Math. Ann. 326 (2003) 515-542 (arXiv:math/0110138, doi:10.1007/s00208-003-0435-8)

• Frederick R. Cohen, Section 3 of: On braid groups, homotopy groups, and modular forms, in: J.M. Bryden (ed.), Advances in Topological Quantum Field Theory, Kluwer 2004, 275–288 (pdf)

See also:

• R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary $q$ and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

As quantum gates for topological quantum computation with anyons:

• Louis H. Kauffman, Samuel J. Lomonaco, Braiding Operators are Universal Quantum Gates, New Journal of Physics, Volume 6, January 2004 (arXiv:quant-ph/0401090, doi:10.1088/1367-2630/6/1/134)

• Samuel J. Lomonaco, Louis Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (2006) (arXiv:quant-ph/0605004)

  (braid group representation serving as a topological quantum gate to compute the Jones polynomial)

• Louis H. Kauffman, Samuel J. Lomonaco, Topological quantum computing and $SU(2)$ braid group representations, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) (doi:10.1117/12.778068, rg:228451452)

• C.-L. Ho, A.I. Solomon, C.-H.Oh, Quantum entanglement, unitary braid representation and Temperley-Lieb algebra, EPL 92 (2010) 30002 (arXiv:1011.6229)

• Louis H. Kauffman, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (arXiv:1710.04650, doi:10.1142/S0217751X18300235)

Introduction and review:

• Colleen Delaney, Eric C. Rowell, Zhenghan Wang, Local unitary representations of the braid group and their applications to quantum computing, Revista Colombiana de Matemáticas(2017), 50 (2):211 (arXiv:1604.06429, doi:10.15446/recolma.v50n2.62211)

Approximating all topological quantum gates by just the weaves among all braids:

• Steven H. Simon, Nick E. Bonesteel, Michael H. Freedman, N. Petrovic, Layla Hormozi, Topological Quantum Computing with Only One Mobile Quasiparticle, Phys. Rev. Lett. 96 (2006) 070503 (arXiv:quant-ph/0509175, doi:10.1103/PhysRevLett.96.070503)

• Layla Hormozi, Georgios Zikos, Nick E. Bonesteel, Steven H. Simon, Topological quantum compiling, Phys. Rev. B 75, 165310 (doi:10.1103/PhysRevB.75.165310, arXiv:quant-ph/0610111)

• Mohamed Taha Rouabah, Compiling single-qubit braiding gate for Fibonacci anyons topological quantum computation (arXiv:2008.03542)

• Simon 2021, Sec. 11.4.1