Communications in Mathematical Physics

, Volume 227, Issue 3, pp 587–603 | Cite as

Simulation of Topological Field Theories¶by Quantum Computers

  • Michael H. Freedman
  • Alexei Kitaev
  • Zhenghan Wang

Abstract:

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:

1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.

2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michael H. Freedman
    • 1
  • Alexei Kitaev
    • 1
  • Zhenghan Wang
    • 2
  1. 1.Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USAUS
  2. 2.Indiana University, Dept. of Math., Bloomington, IN 47405, USAUS

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