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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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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), 31-38 Request permission


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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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:
  • MathSciNet review: 1943131