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Simulation of topological field theories by quantum computers. (English) Zbl 1014.81006
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, \(H\equiv 0\). 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.

MSC:
81P68 Quantum computation
81T45 Topological field theories in quantum mechanics
57R56 Topological quantum field theories (aspects of differential topology)
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