Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2024 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Topological quantum computation
HTML articles powered by AMS MathViewer

by Michael H. Freedman, Alexei Kitaev, Michael J. Larsen and Zhenghan Wang PDF
Bull. Amer. Math. Soc. 40 (2003), 31-38 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 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.
References
    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 10.1103/PhysRevLett.64.1995
  • 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 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 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 10.1073/pnas.95.1.98
    • 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 10.2307/1971403
    • 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, 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, Fault-tolerant quantum computation by anyons, quant-ph/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. 2-3, 362–396. MR 1118790, DOI 10.1016/0550-3213(91)90407-O
    • 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 10.1016/0550-3213(96)00430-0
    • 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, DOI 10.1515/9783110883275
    • 32K. Walker, On Witten’s 3-manifold 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
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 57R56, 81P68
  • Retrieve articles in all journals with MSC (2000): 57R56, 81P68
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), 31-38
  • MSC (2000): Primary 57R56, 81P68
  • DOI: https://doi.org/10.1090/S0273-0979-02-00964-3
  • MathSciNet review: 1943131