nLab topological quantum computation with anyons -- references

Topological quantum computation with anyons

The idea of topological quantum computation via a Chern-Simons theory with anyon braiding defects 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, P/NP, and the quantum field computer, Proc. Nat. Acad. Sci. 95 1 (1998) 98-101 $[$doi:10.1073/pnas.95.1.9$]$

• 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. 227 3 (2002) 605-622 $[$arXiv:quant-ph/0001108, doi:10.1007/s002200200645$]$

  (specifically via su(2)-anyons)

• 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$]$

• Dmitry Melnikov, Andrei Mironov, Sergey Mironov, Alexei Morozov, Andrey Morozov, Towards topological quantum computer, Nucl. Phys. B926 (2018) 491-508 (arXiv:1703.00431, doi:10.1016/j.nuclphysb.2017.11.016)

and via a Dijkgraaf-Witten theory (like Chern-Simons theory but with discrete gauge group):

• R. Walter Ogburn, John Preskill, Topological Quantum Computation, in: Quantum Computing and Quantum Communications, Lecture Notes in Computer Science 1509, Springer (1998) $[$doi:10.1007/3-540-49208-9_31$]$

• Carlos Mochon, Anyons from non-solvable finite groups are sufficient for universal quantum computation, Phys. Rev. A 67 022315 (2003) $[$arXiv:quant-ph/0206128, doi:10.1103/PhysRevA.67.022315$]$

• Carlos Mochon, Anyon computers with smaller groups, Phys. Rev. A 69 032306 (2004) $[$arXiv:quant-ph/0306063, doi:10.1103/PhysRevA.69.032306$]$

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:

• Louis Kauffman, Quantum Topology and Quantum Computing, in: Samuel J. Lomonaco (ed.), Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium, Proceedings of Symposia in Applied Mathematics 58, AMS (2002) [pdf, doi:10.1090/psapm/058]

  (in relation to quantum topology)

• Gavin K. Brennen, Jiannis K. Pachos, Why should anyone care about computing with anyons?, Proc. R. Soc. A 464 (2008) 1-24 $[$arXiv:0704.2241, doi:10.1098/rspa.2007.0026$]$

• 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)

• Ananda Roy, David P. DiVincenzo, Topological Quantum Computing, Lecture notes of the 48th IFF Spring School (2017) $[$arXiv:1701.05052$]$

• Ville Lahtinen, Jiannis K. Pachos, A Short Introduction to Topological Quantum Computation, SciPost Phys. 3 021 (2017) $[$arXiv:1705.04103, doi:10.21468/SciPostPhys.3.3.021$]$

• 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$]$

• Muhammad Ilyas, Quantum Field Theories, Topological Materials, and Topological Quantum Computing $[$arXiv:2208.09707$]$

• Eric C. Rowell, Braids, Motions and Topological Quantum Computing $[$arXiv:2208.11762$]$

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)

Simulation of Ising anyons in a lattice of ordinary superconducting qbits:

• T. Andersen et al. Observation of non-Abelian exchange statistics on a superconducting processor $[$arXiv:2210.10255$]$

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:

• Chen Ning Yang, M. L. Ge (eds.). Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) $[$doi:10.1142/0796$]$

  (focus on quantum Yang-Baxter equation)

• 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 from the monodromy of the Knizhnik-Zamolodchikov connection on bundles of conformal blocks over configuration spaces of points:

• Ivan Todorov, Ludmil Hadjiivanov, Monodromy Representations of the Braid Group, Phys. Atom. Nucl. 64 (2001) 2059-2068; Yad.Fiz. 64 (2001) 2149-2158 $[$arXiv:hep-th/0012099, doi:10.1134/1.1432899$]$

• Ivan Marin, Sur les représentations de Krammer génériques, Annales de l’Institut Fourier, 57 6 (2007) 1883-1925 $[$numdam:AIF_2007__57_6_1883_0$]$

and understood in terms of anyon statistics:

• Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations, Journal of High Energy Physics 2022 15 (2022) $[$arXiv:2112.07195, doi:10.1007/JHEP09(2022)015$]$

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)

• David Lovitz, Universal Braiding Quantum Gates [arXiv:2304.00710]

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)

• Eric C. Rowell, Braids, Motions and Topological Quantum Computing $[$arXiv:2208.11762$]$

Realization of Fibonacci anyons on quasicrystal-states:

• Marcelo Amaral, David Chester, Fang Fang, Klee Irwin, Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing, Symmetry 14 9 (2022) 1780 $[$arXiv:2207.08928, doi:10.3390/sym14091780$]$

Realization on supersymmetric spin chains:

• Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains $[$arXiv:2209.03822$]$

See also:

• Yuanjie Ren, Peter Shor, Topological quantum computation assisted by phase transitions [arXiv:2311.00103]

Compilation to braid gate circuits

On approximating (cf. the Solovay-Kitaev theorem) given quantum gates by (i.e. compiling them to) cicuits of anyon braid gates (generally considered for su(2)-anyons and here mostly for universal Fibonacci anyons, to some extent also for non-universal Majorana anyons):

• Nicholas E. Bonesteel, Layla Hormozi, Georgios Zikos, Steven H. Simon, Braid Topologies for Quantum Computation, Phys. Rev. Lett. 95 140503 (2005) $[$arXiv:quant-ph/0505065, doi:10.1103/PhysRevLett.95.140503$]$

• Layla Hormozi, Georgios Zikos, Nicholas E. Bonesteel, Steven H. Simon, Topological Quantum Compiling, Phys. Rev. B 75 165310 (2007) $[$arXiv:quant-ph/0610111, doi:10.1103/PhysRevB.75.165310$]$

• Layla Hormozi, Nicholas E. Bonesteel, Steven H. Simon, Topological Quantum Computing with Read-Rezayi States, Phys. Rev. Lett. 103 160501 (2009) $[$doi:10.1103/PhysRevLett.103.160501, arXiv:0903.2239$]$

• M. Baraban, Nicholas E. Bonesteel, Steven H. Simon, Resources required for topological quantum factoring, Phys. Rev. A 81 062317 (2010) $[$doi:10.1103/PhysRevA.81.062317, arXiv:1002.0537$]$

  (focus on compiling Shor's algorithm)

• Vadym Kliuchnikov, Alex Bocharov, Krysta M. Svore, Asymptotically Optimal Topological Quantum Compiling, Phys. Rev. Lett. 112 140504 (2014) $[$arXiv:1310.4150, journal: doi:10.1103/PhysRevLett.112.140504, talk recording: doi:10.48660/13100129$]$

• Joren W. Brunekreef, Topological Quantum Computation and Quantum Compilation, Utrecht (2014) $[$studenttheses.uu.nl:20.500.12932/17738$]$

• Yuan-Hang Zhang, Pei-Lin Zheng, Yi Zhang, Dong-Ling Deng, Topological Quantum Compiling with Reinforcement Learning, Phys. Rev. Lett. 125 170501 (2020) $[$doi:2004.04743, doi:10.1103/PhysRevLett.125.170501$]$

• Emil Génetay-Johansen, Tapio Simula, Section IV of: Fibonacci anyons versus Majorana fermions – A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models, PRX Quantum 2 010334 (2021) $[$arXiv:2008.10790, doi:10.1103/PRXQuantum.2.010334$]$

• Cheng-Qian Xu, D. L. Zhou, Quantum teleportation using Ising anyons, Phys. Rev. A 106 012413 (2022) $[$arXiv:2201.11923, doi:10.1103/PhysRevA.106.012413$]$

  (focus on implemening the quantum teleportation-protocol with Ising anyons)

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