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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Received: 4 May 2001 / Accepted: 16 January 2002
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Freedman, M., Kitaev, A. & Wang, Z. Simulation of Topological Field Theories¶by Quantum Computers. Commun. Math. Phys. 227, 587–603 (2002). https://doi.org/10.1007/s002200200635
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DOI: https://doi.org/10.1007/s002200200635