Journal
COMMUNICATIONS IN MATHEMATICAL PHYSICS
Volume 227, Issue 3, Pages 587-603Publisher
SPRINGER-VERLAG
DOI: 10.1007/s002200200635
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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 equivalent to 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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