## Simulation of topological field theories by quantum computers

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Venue: | Comm.Math.Phys.227 |

Citations: | 77 - 12 self |

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@ARTICLE{Freedman_simulationof,

author = {Michael H. Freedman and Alexei Kitaev and Zhenghan Wang},

title = {Simulation of topological field theories by quantum computers},

journal = {Comm.Math.Phys.227},

year = {},

pages = {587--603}

}

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### Abstract

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. 1.

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Citation Context ...ivial global features as in the Aharonov-Bohm effect. Before defining the mathematical notions, we would make two comments. First, the converse to the theorem is also true. It has been shown recently =-=[FLW]-=- that a particular UTMF allows efficient simulation of the universal quantum computer. Second, we would like to suggest that the theorem may be viewed as a positive result for computation. Modular fun... |

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Citation Context ...computation is open. ) . H ℓ590 M. H. Freedman, A. Kitaev, Z. Wang 2. Simulating Modular Functors We adopt the axiomatization of [Wa] or [T] to which we refer for details.Also see,Atiyah [At], Segal =-=[Se]-=-, and Witten [Wi]. A surface is a compact oriented 2-manifold with parameterized boundaries. Each boundary component has a label from a finite set L ={1,a,b,c,...} with involution ̂, 1= ̂1. In example... |

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Citation Context ...−2) = W. 2 Lee Mosher has informed us that the existence of the linear bound f (ℓ) (but without control of the constants) follows at least for closed and single punctured surfaces from his two papers =-=[M1]-=- and [M2].Simulation of Topological Field Theories by Quantum Computers 595 α 1 α 2 α 3 α 5 α 4 3 6 j α 5 α 3 4 6 j β 1 β 3 3 6 j β 1 β 5 β 2 β 4 β 3 Fig. 3. Consider the action of braid move about α... |

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Citation Context ...s information can be made to survive the measurement phase of quantum computation is open. ) . H ℓ590 M. H. Freedman, A. Kitaev, Z. Wang 2. Simulating Modular Functors We adopt the axiomatization of =-=[Wa]-=- or [T] to which we refer for details.Also see,Atiyah [At], Segal [Se], and Witten [Wi]. A surface is a compact oriented 2-manifold with parameterized boundaries. Each boundary component has a label f... |

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Citation Context ...[Sh] is nearly exponential time, exp(O(n 1/3 poly(log n))) (an emphiric bound, the proved one is even worse) according to the most refined classical algorithms. The origin of this paper is in thought =-=[Fr]-=- that since ordinary quantum mechanics appears to confer a substantial speed up over classical calculations, that some principle borrowed from the early, string, universe might go still further. Each ... |

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Citation Context ...“exact” model for the operation of the UTMF, faithful simulation as derived above does not depend on a perfectly accurate quantum circuit. Several authors have proved a threshold theorem [Ki,AB], and =-=[KLZ]-=-: If the rate of large errors acting on computational qubits (or qupits) is small enough, the size of ubiquitous error small enough, and both are uncorrelated, then such a computational space may be m... |

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Citation Context ...2 Lee Mosher has informed us that the existence of the linear bound f (ℓ) (but without control of the constants) follows at least for closed and single punctured surfaces from his two papers [M1] and =-=[M2]-=-.Simulation of Topological Field Theories by Quantum Computers 595 α 1 α 2 α 3 α 5 α 4 3 6 j α 5 α 3 4 6 j β 1 β 3 3 6 j β 1 β 5 β 2 β 4 β 3 Fig. 3. Consider the action of braid move about α. This ac... |