Topological quantum computation
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 by Michael H. Freedman, Alexei Kitaev, Michael J. Larsen and Zhenghan Wang PDF
 Bull. Amer. Math. Soc. 40 (2003), 3138 Request permission
Abstract:
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2Dmagnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like $e^{\alpha \ell }$, where $\ell$ is a length scale, and $\alpha$ is some positive constant. In contrast, the “presumptive" qubitmodel of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about $10^{4}$) before computation can be stabilized.References

1S. Bravyi, A. Kitaev, private communications.
 D. Deutsch, Quantum theory, the ChurchTuring principle and the universal quantum computer, Proc. Roy. Soc. London Ser. A 400 (1985), no. 1818, 97–117. MR 801665
 D. Deutsch, Quantum computational networks, Proc. Roy. Soc. London Ser. A 425 (1989), no. 1868, 73–90. MR 1019288
 Torbjörn Einarsson, Fractional statistics on a torus, Phys. Rev. Lett. 64 (1990), no. 17, 1995–1998. MR 1048107, DOI 10.1103/PhysRevLett.64.1995
 Richard P. Feynman, Simulating physics with computers, Internat. J. Theoret. Phys. 21 (1981/82), no. 67, 467–488. Physics of computation, Part II (Dedham, Mass., 1981). MR 658311, DOI 10.1007/BF02650179
 Richard P. Feynman, Quantum mechanical computers, Found. Phys. 16 (1986), no. 6, 507–531. MR 895035, DOI 10.1007/BF01886518 7M. Freedman, A. Kitaev, and Z. Wang, Simulation of topological field theories by quantum computers, Comm. Math. Phys. 227 (2002), no. 3, 587–603. 8M. Freedman, M. Larsen, and Z. Wang, A modular functor which is universal for quantum computation, Comm. Math. Phys. 227 (2002), no. 3, 605–622. 9M. Freedman, M. Larsen, and Z. Wang, The twoeigenvalue problem and density of Jones representation of braid groups, Comm. Math. Phys. 228 (2002), no. 1, 177–199. 10M.H. Freedman, Quantum computation and the localization of modular functors, Found. Comput. Math. 1 (2001), no. 2, 183–204.
 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 12 T. Senthil and M.P.A. Fisher, Fractionalization, topological order, and cuprate superconductivity, condmat/0008082. 13S. Girvin, The quantum Hall effect: novel excitations and broken symmetries, in Topological aspects of low dimensional systems, Edited by A. Comtet, T. Jolicoeur, S. Ouvry, and F. David EDP Sci., Les Ulis, 1999. 14D. Gottesman, Theory of faulttolerant quantum computation, Phys. Rev. Lett. A57 (1998), 127137. 15L. Grover, Quantum Mechanics helps in search for a needle in a haystack, Phys. Rev. Lett., 79 (1997), 325328.
 V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403 17R. Jozsa, Fidelity for mixed quantum states, Journal of Modern Optics, 41 (1994), no. 12, 23152323.
 Louis H. Kauffman and Sóstenes L. Lins, TemperleyLieb recoupling theory and invariants of $3$manifolds, Annals of Mathematics Studies, vol. 134, Princeton University Press, Princeton, NJ, 1994. MR 1280463, DOI 10.1515/9781400882533
 A. Yu. Kitaev, Quantum computations: algorithms and error correction, Uspekhi Mat. Nauk 52 (1997), no. 6(318), 53–112 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 6, 1191–1249. MR 1611329, DOI 10.1070/RM1997v052n06ABEH002155 20A. Kitaev, Faulttolerant quantum computation by anyons, quantph/9707021.
 Seth Lloyd, Universal quantum simulators, Science 273 (1996), no. 5278, 1073–1078. MR 1407944, DOI 10.1126/science.273.5278.1073
 Gregory Moore and Nicholas Read, Nonabelions in the fractional quantum Hall effect, Nuclear Phys. B 360 (1991), no. 23, 362–396. MR 1118790, DOI 10.1016/05503213(91)90407O 23C. Nayak, and K. Shtengel, Microscopic models of twodimensional magnets with fractionalized excitations, Phys. Rev. B, 64:064422 (2001).
 Chetan Nayak and Frank Wilczek, $2n$quasihole states realize $2^{n1}$dimensional spinor braiding statistics in paired quantum Hall states, Nuclear Phys. B 479 (1996), no. 3, 529–553. MR 1418835, DOI 10.1016/05503213(96)004300 25J. Preskill, Fault tolerant quantum computation, quantph/9712048. 26N. Read, and E. Rezayi, Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Laudau level, condmat/9809384. 27P. Shor, Algorithms for quantum computation, discrete logarithms and factoring, Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, 124134. 28P. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, 2493 (1995). 29P. Shor, Faulttolerant quantum computation, Proc. 37th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1996. 30R. Solvay, private communication.
 V. G. Turaev, Quantum invariants of knots and 3manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673, DOI 10.1515/9783110883275 32K. Walker, On Witten’s 3manifold invariants, preprint, 1991 (available at http://www.xmission.com/$\sim$kwalker/math/).
 Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772, DOI 10.1007/BF01217730
Additional Information
 Michael H. Freedman
 Affiliation: (M. H. Freedman) Microsoft Research, One Microsoft Way, Redmond, Washington 98052
 Alexei Kitaev
 Affiliation: (A. Kitaev) On leave from L.D. Landau Institute for Theoretical Physics, Kosygina St. 2, Moscow, 117940, Russia
 Address at time of publication: Microsoft Research, One Microsoft Way, Redmond, Washington 98052
 Michael J. Larsen
 Affiliation: (M. J. Larsen and Z. Wang) Indiana University, Department of Mathematics, Bloomington, Indiana 47405
 MR Author ID: 293592
 Received by editor(s): November 16, 2000
 Received by editor(s) in revised form: February 21, 2002
 Published electronically: October 10, 2002
 © Copyright 2002 American Mathematical Society
 Journal: Bull. Amer. Math. Soc. 40 (2003), 3138
 MSC (2000): Primary 57R56, 81P68
 DOI: https://doi.org/10.1090/S0273097902009643
 MathSciNet review: 1943131