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Bulletin of the American Mathematical Society

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Topological quantum computation

Authors: Michael H. Freedman, Alexei Kitaev, Michael J. Larsen and Zhenghan Wang
Journal: Bull. Amer. Math. Soc. 40 (2003), 31-38
MSC (2000): Primary 57R56, 81P68
Published electronically: October 10, 2002
MathSciNet review: 1943131
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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 Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets 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" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about $10^{-4}$) before computation can be stabilized.

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    1S. Bravyi, A. Kitaev, private communications.
  • D. Deutsch, Quantum theory, the Church-Turing 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
  • Richard P. Feynman, Simulating physics with computers, Internat. J. Theoret. Phys. 21 (1981/82), no. 6-7, 467–488. Physics of computation, Part II (Dedham, Mass., 1981). MR 658311, DOI
  • Richard P. Feynman, Quantum mechanical computers, Found. Phys. 16 (1986), no. 6, 507–531. MR 895035, DOI
  • 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 two-eigenvalue 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
  • 12 T. Senthil and M.P.A. Fisher, Fractionalization, topological order, and cuprate superconductivity, cond-mat/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 fault-tolerant quantum computation, Phys. Rev. Lett. A57 (1998), 127-137. 15L. Grover, Quantum Mechanics helps in search for a needle in a haystack, Phys. Rev. Lett., 79 (1997), 325-328.
  • 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
  • 17R. Jozsa, Fidelity for mixed quantum states, Journal of Modern Optics, 41 (1994), no. 12, 2315-2323.
  • Louis H. Kauffman and Sóstenes L. Lins, Temperley-Lieb recoupling theory and invariants of $3$-manifolds, Annals of Mathematics Studies, vol. 134, Princeton University Press, Princeton, NJ, 1994. MR 1280463
  • 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
  • 20A. Kitaev, Fault-tolerant quantum computation by anyons, quant-ph/9707021.
  • Seth Lloyd, Universal quantum simulators, Science 273 (1996), no. 5278, 1073–1078. MR 1407944, DOI
  • 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
  • 23C. Nayak, and K. Shtengel, Microscopic models of two-dimensional magnets with fractionalized excitations, Phys. Rev. B, 64:064422 (2001).
  • Chetan Nayak and Frank Wilczek, $2n$-quasihole states realize $2^{n-1}$-dimensional spinor braiding statistics in paired quantum Hall states, Nuclear Phys. B 479 (1996), no. 3, 529–553. MR 1418835, DOI
  • 25J. Preskill, Fault tolerant quantum computation, quant-ph/9712048. 26N. Read, and E. Rezayi, Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Laudau level, cond-mat/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, 124-134. 28P. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, 2493 (1995). 29P. Shor, Fault-tolerant 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 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
  • 32K. Walker, On Witten’s 3-manifold invariants, preprint, 1991 (available at$\sim$kwalker/math/).
  • Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772

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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
Article copyright: © Copyright 2002 American Mathematical Society