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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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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

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

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