Mathematics of topological quantum computing

Rowell, Eric; Wang, Zhenghan

doi:10.1090/bull/1605

Mathematics of topological quantum computing
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by Eric C. Rowell and Zhenghan Wang PDF

Bull. Amer. Math. Soc. 55 (2018), 183-238 Request permission

Abstract:

In topological quantum computing, information is encoded in “knotted” quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of $10^{-10}$ and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics, and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.

References

Scott Aaronson, Quantum computing since Democritus, Cambridge University Press, Cambridge, 2013. MR 3058839, DOI 10.1017/CBO9780511979309
Sven Marian Albrecht, A. Higginbotham, Morten Madsen, Ferdinand Kuemmeth, Thomas Sand Jespersen, Jesper Nygård, Peter Krogstrup, and C. Marcus, Exponential protection of zero modes in Majorana islands, Nature 531 (2016), no. 7593, 206–209., DOI 10.1038/nature17162
Bojko Bakalov and Alexander Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. MR 1797619, DOI 10.1090/ulect/021
Peter Bantay, The Frobenius-Schur indicator in conformal field theory, Phys. Lett. B 394 (1997), no. 1-2, 87–88. MR 1436801, DOI 10.1016/S0370-2693(96)01662-0
Maissam Barkeshli and Michael Freedman, Modular transformations through sequences of topological charge projections, Phys. Rev. B 94 (2016), no. 16, 165108., DOI 10.1103/PhysRevB.94.165108
Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang, Symmetry, defects, and gauging of topological phases (2014), available at arXiv:1410.4540.
Dmitriy Belov and Gregory W Moore, Classification of abelian spin Chern-Simons theories (2005), available at hep-th/0505235.
Parsa Bonderson, Michael Freedman, and Chetan Nayak, Measurement-only topological quantum computation, Phys. Rev. Lett. 101 (2008), no. 1, 010501, 4. MR 2429542, DOI 10.1103/PhysRevLett.101.010501
Parsa Bonderson, Alexei Kitaev, and Kirill Shtengel, Detecting non-abelian statistics in the $\nu = 5/2$ fractional quantum Hall state, Phys. Rev. Lett. 96 (2006), no. 1, 016803., DOI 10.1103/PhysRevLett.96.016803
Parsa Bonderson, Sankar Das Sarma, Michael Freedman, and Chetan Nayak, A blueprint for a topologically fault-tolerant quantum computer (2010), available at arXiv:1003.2856.
Sergey Bravyi and Alexei Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A (3) 71 (2005), no. 2, 022316, 14. MR 2138687, DOI 10.1103/PhysRevA.71.022316
Sergey Bravyi and Alexei Kitaev, Quantum invariants of 3-manifolds and quantum computation, unpublished (2000).
Sergey B. Bravyi and A. Yu. Kitaev, Quantum codes on a lattice with boundary (1998), available at quant-ph/9811052.
Sergey Bravyi, Matthew B. Hastings, and Spyridon Michalakis, Topological quantum order: stability under local perturbations, J. Math. Phys. 51 (2010), no. 9, 093512, 33. MR 2742836, DOI 10.1063/1.3490195
Alain Bruguières, Catégories prémodulaires, modularisations et invariants des variétés de dimension 3, Math. Ann. 316 (2000), no. 2, 215–236 (French, with English summary). MR 1741269, DOI 10.1007/s002080050011
Paul Bruillard, Siu-Hung Ng, Eric C. Rowell, and Zhenghan Wang, Rank-finiteness for modular categories, J. Amer. Math. Soc. 29 (2016), no. 3, 857–881. MR 3486174, DOI 10.1090/jams/842
Paul Bruillard, Siu-Hung Ng, Eric C. Rowell, and Zhenghan Wang, On classification of modular categories by rank, Inter. Math. Res. Not. 2016 (2016), no. 24, 7546, available at http://dx.doi.org/10.1093/imrn/rnw020., DOI 10.1093/imrn/rnw020
Paul Bruillard, Liang Chang, Seung-Moon Hong, Julia Yael Plavnik, Eric C. Rowell, and Michael Yuan Sun, Low-dimensional representations of the three component loop braid group, J. Math. Phys. 56 (2015), no. 11, 111707, 15. MR 3423950, DOI 10.1063/1.4935361
Paul Bruillard, César Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell, and Zhenghan Wang, Fermionic modular categories and the 16-fold way, J. Math. Phys. 58 (2017), no. 4, 041704, 31. MR 3641612, DOI 10.1063/1.4982048
Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, and Mihály Weiner, From vertex operator algebras to conformal nets and back (2015), available at arXiv:1503.01260.
Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen, Symmetry-protected topological orders in interacting bosonic systems, Science 338 (2012), no. 6114, 1604–1606. MR 3025080, DOI 10.1126/science.1227224
Lukasz Cincio and Guifré Vidal, Characterizing topological order by studying the ground states on an infinite cylinder, Phys. Rev. Lett. 110 (2013), no. 6, 067208., DOI 10.1103/PhysRevLett.110.067208
Iris Cong and Zhenghan Wang, Topological quantum computation with gapped boundaries and boundary defects (2017), available at arXiv:1710.07197.
Iris Cong, Meng Cheng, and Zhenghan Wang, Hamiltonian and algebraic theories of gapped boundaries in topological phases of matter, Comm. Math. Phys. 355 (2017), no. 2, 645–689. MR 3681387, DOI 10.1007/s00220-017-2960-4
Iris Cong, Meng Cheng, and Zhenghan Wang, On defects between gapped boundaries in two-dimensional topological phases of matter (2017), available at arXiv:1703.03564.
Iris Cong, Meng Cheng, and Zhenghan Wang, Universal quantum computation with gapped boundaries, Phys. Rev. Lett. (to appear) (2017), available at arXiv.1710.07197.
Iris Cong, Meng Cheng, and Zhenghan Wang, Topological quantum computation with gapped boundaries (2016), available at arXiv:1609.02037.
A. Coste and T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys. Lett. B 323 (1994), no. 3-4, 316–321. MR 1266785, DOI 10.1016/0370-2693(94)91226-2
Xingshan Cui, Higher Categories and Topological Quantum Field Theories, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–University of California, Santa Barbara. MR 3593287
Shawn X. Cui and Zhenghan Wang, Universal quantum computation with metaplectic anyons, J. Math. Phys. 56 (2015), no. 3, 032202, 18. MR 3390917, DOI 10.1063/1.4914941
Shawn X. Cui, Michael H. Freedman, and Zhenghan Wang, Complexity classes as mathematical axioms II, Quantum Topol. 7 (2016), no. 1, 185–201. MR 3459960, DOI 10.4171/QT/75
Shawn X. Cui, Seung-Moon Hong, and Zhenghan Wang, Universal quantum computation with weakly integral anyons, Quantum Inf. Process. 14 (2015), no. 8, 2687–2727. MR 3370675, DOI 10.1007/s11128-015-1016-y
Shawn X. Cui, César Galindo, Julia Yael Plavnik, and Zhenghan Wang, On gauging symmetry of modular categories, Comm. Math. Phys. 348 (2016), no. 3, 1043–1064. MR 3555361, DOI 10.1007/s00220-016-2633-8
Orit Davidovich, Tobias Hagge, and Zhenghan Wang, On arithmetic modular categories (2013), available at arXiv:1305.2229.
Alexei Davydov, Dmitri Nikshych, and Victor Ostrik, On the structure of the Witt group of braided fusion categories, Selecta Math. (N.S.) 19 (2013), no. 1, 237–269. MR 3022755, DOI 10.1007/s00029-012-0093-3
Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. 677 (2013), 135–177. MR 3039775, DOI 10.1515/crelle.2012.014
Colleen Delaney and Zhenghan Wang, Symmetry defects and their application to topological quantum computing, preprint (2017).
Colleen Delaney, Eric C. Rowell, and Zhenghan Wang, Local unitary representations of the braid group and their applications to quantum computing, Rev. Colombiana Mat. 50 (2016), no. 2, 207–272. MR 3605648, DOI 10.15446/recolma.v50n2.62211
Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325, DOI 10.1007/978-3-540-38955-2
Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699, DOI 10.1007/BF02096988
Chongying Dong, Xingjun Lin, and Siu-Hung Ng, Congruence property in conformal field theory, Algebra Number Theory 9 (2015), no. 9, 2121–2166. MR 3435813, DOI 10.2140/ant.2015.9.2121
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, On braided fusion categories. I, Selecta Math. (N.S.) 16 (2010), no. 1, 1–119. MR 2609644, DOI 10.1007/s00029-010-0017-z
J. Michael Dunn, Lawrence S. Moss, and Zhenghan Wang, Editors’ introduction: the third life of quantum logic: quantum logic inspired by quantum computing, J. Philos. Logic 42 (2013), no. 3, 443–459. MR 3056313, DOI 10.1007/s10992-013-9273-7
Samuel Eilenberg and Saunders Mac Lane, On the groups $H(\Pi ,n)$. II. Methods of computation, Ann. of Math. (2) 60 (1954), 49–139. MR 65162, DOI 10.2307/1969702
Pavel Etingof, Dmitri Nikshych, and Victor Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math. 226 (2011), no. 1, 176–205. MR 2735754, DOI 10.1016/j.aim.2010.06.009
Pavel Etingof, Dmitri Nikshych, and Victor Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010), no. 3, 209–273. With an appendix by Ehud Meir. MR 2677836, DOI 10.4171/QT/6
Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI 10.4007/annals.2005.162.581
Pavel Etingof, Eric Rowell, and Sarah Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008), no. 1, 33–41. MR 2375313, DOI 10.2140/pjm.2008.234.33
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015. MR 3242743, DOI 10.1090/surv/205
David E. Evans and Yasuyuki Kawahigashi, Quantum symmetries on operator algebras, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1642584
Jan-Hendrik Evertse, On sums of $S$-units and linear recurrences, Compositio Math. 53 (1984), no. 2, 225–244. MR 766298
M. Finkelberg, An equivalence of fusion categories, Geom. Funct. Anal. 6 (1996), no. 2, 249–267. MR 1384612, DOI 10.1007/BF02247887
Jens Fjelstad and Jürgen Fuchs, Mapping class group representations from Drinfeld doubles of finite groups (2015), available at arXiv:1506.03263.
Philippe Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Springer Science & Business Media, 2012.
Daniel S. Freed, The cobordism hypothesis, Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 1, 57–92. MR 2994995, DOI 10.1090/S0273-0979-2012-01393-9
Daniel S. Freed and Frank Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no. 3, 435–472. MR 1240583, DOI 10.1007/BF02096860
Michael H. Freedman, Complexity classes as mathematical axioms, Ann. of Math. (2) 170 (2009), no. 2, 995–1002. MR 2552117, DOI 10.4007/annals.2009.170.995
Michael H. Freedman, Quantum computation and the localization of modular functors, Found. Comput. Math. 1 (2001), no. 2, 183–204. MR 1830035, DOI 10.1007/s102080010006
Michael H. Freedman, P/NP, and the quantum field computer, Proc. Natl. Acad. Sci. USA 95 (1998), no. 1, 98–101. MR 1612425, DOI 10.1073/pnas.95.1.98
Michael H. Freedman, Alexei Kitaev, and Zhenghan Wang, Simulation of topological field theories by quantum computers, Comm. Math. Phys. 227 (2002), no. 3, 587–603. MR 1910832, DOI 10.1007/s002200200635
Michael H. Freedman, Michael Larsen, and Zhenghan Wang, A modular functor which is universal for quantum computation, Comm. Math. Phys. 227 (2002), no. 3, 605–622. MR 1910833, DOI 10.1007/s002200200645
Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Topological quantum computation, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 31–38. Mathematical challenges of the 21st century (Los Angeles, CA, 2000). MR 1943131, DOI 10.1090/S0273-0979-02-00964-3
César Galindo, Seung-Moon Hong, and Eric C. Rowell, Generalized and quasi-localizations of braid group representations, Int. Math. Res. Not. IMRN 3 (2013), 693–731. MR 3021796, DOI 10.1093/imrn/rnr269
Michael R. Garey and David S. Johnson, Computers and intractability, A Series of Books in the Mathematical Sciences, W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness. MR 519066
Daniel Gottesman, An introduction to quantum error correction, Quantum computation: a grand mathematical challenge for the twenty-first century and the millennium (Washington, DC, 2000) Proc. Sympos. Appl. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2002, pp. 221–235. MR 1922900, DOI 10.1090/psapm/058/1922900
Paul Gustafson, Finiteness for mapping class group representations from twisted Dijkgraaf-Witten theory (2016), available at arXiv:1610.06069.
Tobias J. Hagge, Graphical calculus for fusion categories and quantum invariants for 3-manifolds, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Indiana University. MR 2712446
F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "parity anomaly", Physical Review Letters 61 (1988), no. 18, 2015., DOI 10.1103/PhysRevLett.61.2015
Andrew Hodges, Alan Turing: the enigma, The centenary edition, Princeton University Press, Princeton, NJ, 2012. With a foreword by Douglas Hofstadter and a new preface by the author. MR 2963548, DOI 10.1515/9781400844975
Seung-Moon Hong, Eric Rowell, and Zhenghan Wang, On exotic modular tensor categories, Commun. Contemp. Math. 10 (2008), no. suppl. 1, 1049–1074. MR 2468378, DOI 10.1142/S0219199708003162
Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture, and modular tensor categories, Proc. Natl. Acad. Sci. USA 102 (2005), no. 15, 5352–5356. MR 2140309, DOI 10.1073/pnas.0409901102
F. Jaeger, D. L. Vertigan, and D. J. A. Welsh, On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 1, 35–53. MR 1049758, DOI 10.1017/S0305004100068936
Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI 10.1006/aima.1993.1055
Zoltán Kádár, Paul Martin, Eric Rowell, and Zhenghan Wang, Local representations of the loop braid group, Glasg. Math. J. 59 (2017), no. 2, 359–378. MR 3628934, DOI 10.1017/S0017089516000215
Torsten Karzig, Christina Knapp, Roman M Lutchyn, Parsa Bonderson, Matthew B Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, and Yuval Oreg et al., Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes, Physical Review B 95 (2017), no. 23, 235305., DOI 10.1103/PhysRevB.95.235305
Yasuyuki Kawahigashi, Roberto Longo, and Michael Müger, Multi-interval subfactors and modularity of representations in conformal field theory, Comm. Math. Phys. 219 (2001), no. 3, 631–669. MR 1838752, DOI 10.1007/PL00005565
Alexander Kirillov Jr, String-net model of Turaev-Viro invariants (2011), available at arXiv:1106.6033.
Alexander Kirillov Jr, On $g$-equivariant modular categories (2004), available at math/0401119.
Alexei Kitaev, Anyons in an exactly solved model and beyond, Ann. Physics 321 (2006), no. 1, 2–111. MR 2200691, DOI 10.1016/j.aop.2005.10.005
A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Physics 303 (2003), no. 1, 2–30. MR 1951039, DOI 10.1016/S0003-4916(02)00018-0
Alexei Kitaev and Liang Kong, Models for gapped boundaries and domain walls, Comm. Math. Phys. 313 (2012), no. 2, 351–373. MR 2942952, DOI 10.1007/s00220-012-1500-5
Vadym Kliuchnikov, Alex Bocharov, and Krysta M Svore, Asymptotically optimal topological quantum compiling, Phys. Rev. Lett. 112 (2014), no. 14, 140504., DOI 10.1103/PhysRevLett.112.140504
Tian Lan, Liang Kong, and Xiao-Gang Wen, Modular extensions of unitary braided fusion categories and $2+1\textrm {D}$ topological/SPT orders with symmetries, Comm. Math. Phys. 351 (2017), no. 2, 709–739. MR 3613518, DOI 10.1007/s00220-016-2748-y
Tian Lan and Xiao-Gang Wen, Hierarchy construction and non-abelian families of generic topological orders (2017), available at arXiv:1701.07820.
Michael J. Larsen, Eric C. Rowell, and Zhenghan Wang, The $N$-eigenvalue problem and two applications, Int. Math. Res. Not. 64 (2005), 3987–4018. MR 2206918, DOI 10.1155/IMRN.2005.3987
Michael A Levin and Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Physical Review B 71 (2005), no. 4, 045110., DOI 10.1103/PhysRevB.71.045110
Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
RM Lutchyn, EPAM Bakkers, LP Kouwenhoven, P Krogstrup, CM Marcus, and Y Oreg, Realizing Majorana zero modes in superconductor-semiconductor heterostructures (2017), available at arXiv:1707.04899.
Michaël Mignard and Peter Schauenburg, Modular categories are not determined by their modular data (2017), available at arXiv:1708.02796.
Gregory Moore and Nicholas Read, Nonabelions in the fractional quantum Hall effect, Nuclear Phys. B 360 (1991), no. 2-3, 362–396. MR 1118790, DOI 10.1016/0550-3213(91)90407-O
Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. MR 1002038, DOI 10.1007/BF01238857
Scott Morrison and Noah Snyder, Non-cyclotomic fusion categories, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4713–4733. MR 2922607, DOI 10.1090/S0002-9947-2012-05498-5
Vincent Mourik, Kun Zuo, Sergey M Frolov, SR Plissard, EPAM Bakkers, and Leo P Kouwenhoven, Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices, Science 336 (2012), no. 6084, 1003–1007., DOI 10.1126/science.1222360
Michael Mueger, Modular categories (2012), available at arXiv:1201.6593.
Michael Müger, Galois extensions of braided tensor categories and braided crossed $G$-categories, J. Algebra 277 (2004), no. 1, 256–281. MR 2059630, DOI 10.1016/j.jalgebra.2004.02.026
M. Müger, On the structure of modular categories, Proc. London Math. Soc. 87 (2003), no. 2, 291–308., DOI 10.1112/S0024611503014187
Michael Müger, From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors, J. Pure Appl. Algebra 180 (2003), no. 1-2, 159–219. MR 1966525, DOI 10.1016/S0022-4049(02)00248-7
Deepak Naidu and Eric C. Rowell, A finiteness property for braided fusion categories, Algebr. Represent. Theory 14 (2011), no. 5, 837–855. MR 2832261, DOI 10.1007/s10468-010-9219-5
Sonia Natale, The core of a weakly group-theoretical braided fusion category, 2017.
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Modern Phys. 80 (2008), no. 3, 1083–1159. MR 2443722, DOI 10.1103/RevModPhys.80.1083
Siu-Hung Ng and Peter Schauenburg, Congruence subgroups and generalized Frobenius-Schur indicators, Comm. Math. Phys. 300 (2010), no. 1, 1–46. MR 2725181, DOI 10.1007/s00220-010-1096-6
Siu-Hung Ng and Peter Schauenburg, Frobenius-Schur indicators and exponents of spherical categories, Adv. Math. 211 (2007), no. 1, 34–71. MR 2313527, DOI 10.1016/j.aim.2006.07.017
Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000. MR 1796805
R. Walter Ogburn and John Preskill, Topological quantum computation, Quantum computing and quantum communications (Palm Springs, CA, 1998) Lecture Notes in Comput. Sci., vol. 1509, Springer, Berlin, 1999, pp. 341–356. MR 1750535, DOI 10.1007/3-540-49208-9_{3}1
Victor Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), no. 2, 177–206. MR 1976459, DOI 10.1007/s00031-003-0515-6
Jiannis K. Pachos, Introduction to topological quantum computation, Cambridge University Press, Cambridge, 2012. MR 3157248, DOI 10.1017/CBO9780511792908
John Preskill, Lecture notes for Physics 219: Quantum computation, Caltech Lecture Notes (1999).
N. Read, Conformal invariance of chiral edge theories, Physical Review B 79 (2009), no. 24, 245304., DOI 10.1103/PhysRevB.79.245304
Nicholas Read and E Rezayi, Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level, Physical Review B 59 (1999), no. 12, 8084., DOI 10.1103/PhysRevB.59.8084
Eric C. Rowell, An invitation to the mathematics of topological quantum computation, Journal of Physics: Conference series, 2016, pp. 012012.
Eric C. Rowell, Braid representations from quantum groups of exceptional Lie type, Rev. Un. Mat. Argentina 51 (2010), no. 1, 165–175. MR 2681262
Eric C. Rowell, Two paradigms for topological quantum computation, Advances in quantum computation, Contemp. Math., vol. 482, Amer. Math. Soc., Providence, RI, 2009, pp. 165–177. MR 2568418, DOI 10.1090/conm/482/09418
Eric C. Rowell, From quantum groups to unitary modular tensor categories, Representations of algebraic groups, quantum groups, and Lie algebras, Contemp. Math., vol. 413, Amer. Math. Soc., Providence, RI, 2006, pp. 215–230. MR 2263097, DOI 10.1090/conm/413/07848
Eric C. Rowell and Zhenghan Wang, Degeneracy and non-Abelian statistics, Phys. Rev. A 93 (2016), no. 3, 030102., DOI 10.1103/PhysRevA.93.030102
Eric C. Rowell and Zhenghan Wang, Localization of unitary braid group representations, Comm. Math. Phys. 311 (2012), no. 3, 595–615. MR 2909757, DOI 10.1007/s00220-011-1386-7
Eric C. Rowell and Hans Wenzl, $\textrm {SO}(N)_2$ braid group representations are Gaussian, Quantum Topol. 8 (2017), no. 1, 1–33. MR 3630280, DOI 10.4171/QT/85
Eric Rowell, Richard Stong, and Zhenghan Wang, On classification of modular tensor categories, Comm. Math. Phys. 292 (2009), no. 2, 343–389. MR 2544735, DOI 10.1007/s00220-009-0908-z
Sankar Das Sarma, Michael Freedman, and Chetan Nayak, Majorana zero modes and topological quantum computation (2015), available at arXiv:1501.02813.
Modjtaba Shokrian Zini and Zhenghan Wang, Conformal field theories as scaling limit of anyonic chains (2017), available at arXiv:1706.08497.
Peter W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52 (1995), no. 4, R2493., DOI 10.1103/PhysRevA.52.R2493
Peter W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994) IEEE Comput. Soc. Press, Los Alamitos, CA, 1994, pp. 124–134. MR 1489242, DOI 10.1109/SFCS.1994.365700
Ady Stern and Bertrand I Halperin, Proposed experiments to probe the non-abelian $\nu = 5/2$ quantum Hall state, Phys. Rev. Lett. 96 (2006), no. 1, 016802., DOI 10.1103/PhysRevLett.96.016802
James E. Tener, Geometric realization of algebraic conformal field theories (2016), available at arXiv:1611.01176.
James E. Tener and Zhenghan Wang, On classification of extremal non-holomorphic conformal field theories, J. Phys. A 50 (2017), no. 11, 115204, 22. MR 3622581, DOI 10.1088/1751-8121/aa59cd
V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673, DOI 10.1515/9783110883275
Vladimir G. Turaev, Modular categories and $3$-manifold invariants, Internat. J. Modern Phys. B 6 (1992), no. 11-12, 1807–1824. Topological and quantum group methods in field theory and condensed matter physics. MR 1186845, DOI 10.1142/S0217979292000876
Vladimir Turaev and Hans Wenzl, Semisimple and modular categories from link invariants, Math. Ann. 309 (1997), no. 3, 411–461. MR 1474200, DOI 10.1007/s002080050120
A. M. Turing, Computing machinery and intelligence, Mind 59 (1950), 433–460. MR 37064, DOI 10.1093/mind/LIX.236.433
A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction, Proc. London Math. Soc. (2) 43 (1937), no. 7, 544–546. MR 1575661, DOI 10.1112/plms/s2-43.6.544
Dirk Llewellyn Vertigan, On the computational complexity of Tutte, Jones, homfly and Kauffman invariants., PhD Thesis, University of Oxford, 1991.
Kevin Walker, On Witten’s 3-manifold invariants, preprint (1991).
Kevin Walker and Zhenghan Wang, $(3+1)$-TQFTs and topological insulators, Front. Phys. 7 (2012), no. 2, 150–159., DOI 10.1007/s11467-011-0194-z
Zhenghan Wang, Beyond anyons (2017), available at arXiv:1710.00464.
Zhenghan Wang, Quantum computing: a quantum group approach, Symmetries and groups in contemporary physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 11, World Sci. Publ., Hackensack, NJ, 2013, pp. 41–50. MR 3221449, DOI 10.1142/9789814518550_{0}009
Zhenghan Wang, Topological quantum computation, CBMS Regional Conference Series in Mathematics, vol. 112, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2010. MR 2640343, DOI 10.1090/cbms/112
Zhenghan Wang, Topologization of electron liquids with Chern-Simons theory and quantum computation, Differential geometry and physics, Nankai Tracts Math., vol. 10, World Sci. Publ., Hackensack, NJ, 2006, pp. 106–120. MR 2322390
D. J. A. Welsh, Complexity: knots, colourings and counting, London Mathematical Society Lecture Note Series, vol. 186, Cambridge University Press, Cambridge, 1993. MR 1245272, DOI 10.1017/CBO9780511752506
Xiao-Gang Wen, Zoo of quantum-topological phases of matter (2016), available at arXiv:1610.03911.
Xiao-Gang Wen, Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons, Oxford University Press on Demand, 2004.
Xiao-Gang Wen, Theory of the edge states in fractional quantum Hall effects, Internat. J. Modern Phys. B 6 (1992), no. 10, 1711–1762. MR 1170236, DOI 10.1142/S0217979292000840
X. G. Wen, Nonabelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991), no. 6, 802–805. MR 1089602, DOI 10.1103/PhysRevLett.66.802
Frank Wilczek, Fractional statistics and anyon superconductivity, Vol. 5, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990., DOI 10.1142/0961
Robert L Willett, Loren N Pfeiffer, and KW West, Measurement of filling factor $5/2$ quasiparticle interference with observation of charge $e/4$ and $e/2$ period oscillations, Proc. Natl. Acad. Sci. USA 106 (2009), no. 22, 8853–8858., DOI 10.1073/pnas.0812599106
Dominic J. Williamson and Zhenghan Wang, Hamiltonian models for topological phases of matter in three spatial dimensions, Ann. Physics 377 (2017), 311–344. MR 3606141, DOI 10.1016/j.aop.2016.12.018
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399., DOI 10.1007/BF01217730