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Simulation of topological field theories by quantum computers

Michael H. Freedman·Alexei Kitaev·Zhenghan Wang
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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 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.

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Michael H. Freedman, Alexei Kitaev, Zhenghan Wang. 2000-03-17. Simulation of topological field theories by quantum computers. https://doi.org/10.1007/s002200200635

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