Simulation of topological field theories by quantum computers

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 filed 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. Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399 Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399 (On leave from Landau Institute for Theoretical Physics (Moscow) Indiana University, Dept. of Math., Bloomington, IN 47405