Communications in Mathematical Physics

, Volume 227, Issue 3, pp 587-603

First online:

Simulation of Topological Field Theories¶by Quantum Computers

  • Michael H. FreedmanAffiliated withMicrosoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA
  • , Alexei KitaevAffiliated withMicrosoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA
  • , Zhenghan WangAffiliated withIndiana University, Dept. of Math., Bloomington, IN 47405, USA

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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.