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 [doi:10.1016/S0003-4916(02)00018-0, arXiv:quant-ph/9707021]

• 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 [doi:10.1090/S0273-0979-02-00964-3, arXiv:quant-ph/0101025, 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 [doi:10.1007/s002200200645, arXiv:quant-ph/0001108]

  (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) [doi:10.1103/RevModPhys.80.1083, arXiv:0707.1888]

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

  (via R-matrix-solutions to the quantum Yang-Baxter equation)

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) [doi:10.1103/PhysRevA.67.022315, arXiv:quant-ph/0206128]

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

Proposals more specifically for topological quantum gates implemented in fractional quantum Hall systems:

• D. V. Averin, V.J. Goldman: Quantum computation with quasiparticles of the fractional quantum Hall effect, Solid State Communications 121 1 (2001) 25-28 [doi:10.1016/S0038-1098(01)00447-1, arXiv:cond-mat/0110193]

• Sankar Das Sarma, Michael Freedman, Chetan Nayak: Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State, Phys. Rev. Lett. 94 166802 (2005) [doi:10.1103/PhysRevLett.94.166802, arXiv:cond-mat/0412343]

• Sergey Bravyi: Universal Quantum Computation with the $\nu = 5/2$ Fractional Quantum Hall State, Phys. Rev. A 73 042313 (2006) [doi:10.1103/PhysRevA.73.042313, arXiv:quant-ph/0511178]

Monographs:

• 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, Oxford University Press (2023) [ISBN:9780198886723, 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)

• John Preskill: Topological Quantum Computation, Chapter 9 in: Quantum Computation, lecture notes (since 2004) [pdf, pdf]

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

• 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, spire:2748124]

• 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, arXiv:1601.05288]

• 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) [doi: 10.21468/SciPostPhys.3.3.021, 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 [doi:10.1088/2058-9565/aacad2, arXiv:1802.06176]

• 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, spire:2141848]

• Fabian Hassler: Topological quantum computing [arXiv:2410.13547]

• Hisham Sati, Sachin Valera: Topological Quantum Computing, Encyclopedia of Mathematical Physics 2nd ed 4 (2025) 325-345 [doi:10.1016/B978-0-323-95703-8.00262-7]

• Jason Alicea: What does it mean to create a topological qubit?, Quantum Frontiers blog (March 2025) [webpage]

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 [doi:10.1142/S0219749906002328, arXiv:quant-ph/0511273]

• James Robin Wootton, Dissecting Topological Quantum Computation, PhD thesis, Leeds (2010) [ethesis:1163, 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. […] Though universal non-Abelian models are admittedly the holy grail of topological quantum computation, and rightly so, this thesis has shown that Abelian models are just as useful as non-universal non-Abelian models. […] Abelian models are a computationally powerful, fault-tolerant and experimentally realistic prospect for quantum computation.”

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

• James R. Wootton, Jiannis K. Pachos: Universal Quantum Computation with Abelian Anyon Models, Electronic Notes in Theoretical Computer Science 270 2 (2011) 209-218 [doi:10.1016/j.entcs.2011.01.032, arXiv:0904.4373]

see also:

• Menelaos Zikidis: Abelian Anyons and Fractional Quantum Hall Effect, Seminar notes (2017) [pdf, pdf]

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

• Yichen Hu, Biao Lian: Chiral Sachdev-Ye model: Integrability and chaos of anyons in $1+1d$, Phys. Rev. B 105 (2022) 125109 [doi:10.1103/PhysRevB.105.125109]

Realization in experiment (so far via quantum simulation of anyons on non-topological quantum hardware, cf. FF24, Fig 5, as in “topological codes” for quantum error correction):

on superconducting qbits:

• T. Andersen et al.: Non-Abelian braiding of graph vertices in a superconducting processor, Nature 618 (2023) 264–269 [arXiv:2210.10255, doi:10.1038/s41586-023-05954-4]

on trapped-ion quantum hardware:

• 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)

• Mohsin Iqbal, Nathanan Tantivasadakarn: Topological Order from Measurements and Feed-Forward on a Trapped Ion Quantum Computer, Nature Communications Physics 7 (2024) 205 [doi:10.1038/s42005-024-01698-3, arXiv:2302.01917]

• Mohsin Iqbal, Nathanan Tantivasadakarn, R. Verresen et al., Figure 5 in : Non-Abelian topological order and anyons on a trapped-ion processor, Nature 626 (2024) 505–511 [doi:10.1038/s41586-023-06934-4]

  Nature research briefing: Topological matter created on a quantum chip produces quasiparticles with computing power [doi:10.1038/d41586-023-04126-8]

• Michael Foss-Feig, Guido Pagano, Andrew C. Potter, Norman Y. Yao: Progress in Trapped-Ion Quantum Simulation, Annual Reviews of Condensed Matter Physics (2024) [arXiv:2409.02990 doi:10.1146/annurev-conmatphys-032822-045619]

Discussion of anyon braid gates via homotopy type theory:

• David Jaz Myers, Hisham Sati, Urs Schreiber: Topological Quantum Gates in Homotopy Type Theory, Comm. Math. Phys. 405 172 (2024) [arXiv:2303.02382, doi:10.1007/s00220-024-05020-8]

• David Jaz Myers: Topological Quantum Gates, talk at Running HoTT 2024, CQTS@NYUAD (April 2024) [video:kt]

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. 49 1 (2015) [doi:10.15446/recolma.v49n1.54160, arXiv:1404.0724]

• 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, J. High Energ. Phys. 2022 15 (2022) [doi:10.1007/JHEP09(2022)015, arXiv:2112.07195]

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 6 (2004) [doi:10.1088/1367-2630/6/1/134, arXiv:quant-ph/0401090]

  (via R-matrix solutions to the quantum Yang-Baxter equation)

• Yong Zhang, Louis H. Kauffman, Mo-Lin Ge: Yang–Baxterizations, Universal Quantum Gates and Hamiltonians, Quantum Inf Process 4 (2005) 159–197 [doi:10.1007/s11128-005-7655-7]

  (via R-matrix solutions to the quantum Yang-Baxter equation)

• 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 [doi:10.1209/0295-5075/92/30002, arXiv:1011.6229]

• Rebecca Chen: Generalized Yang-Baxter Equations and Braiding Quantum Gates, Journal of Knot Theory and Its Ramifications 21 09 (2012) 1250087 [arXiv:1108.5215, doi:10.1142/S0218216512500873]

• Louis H. Kauffman, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics A 33 23 (2018) 1830023 [doi:10.1142/S0217751X18300235, arXiv:1710.04650]

• 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) [doi:10.1103/PhysRevLett.95.140503, arXiv:quant-ph/0505065]

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

• 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) [doi:10.1103/PhysRevLett.112.140504, arXiv:1310.4150, talk recording: doi:10.48660/13100129]

• Joren W. Brunekreef, Topological Quantum Computation and Quantum Compilation, Utrecht (2014) [hdl: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:10.1103/PhysRevLett.125.170501, arXiv:2004.04743]

• 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) [doi:10.1103/PRXQuantum.2.010334, arXiv:2008.10790]

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

  (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 2023, Sec. 11.4.1