## Quantum computing and the Jones polynomial

Venue: | math.QA/0105255, in Quantum Computation and Information |

Citations: | 10 - 7 self |

### BibTeX

@INPROCEEDINGS{Kauffman_quantumcomputing,

author = {Louis H. Kauffman},

title = {Quantum computing and the Jones polynomial},

booktitle = {math.QA/0105255, in Quantum Computation and Information},

year = {},

pages = {101--137}

}

### OpenURL

### Abstract

This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open. 1

### Citations

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(Show Context)
Citation Context ...ng is just beginning. Acknowledgement. Research on this paper was supported by National Science Foundation Grant DMS 9802859. 2 Dirac Brackets We begin with a discussion of Dirac’s notation, < b|a >, =-=[4]-=-. In this notation < a| and |b > are covectors and vectors respectively. < b|a > is the evaluation of |a > by < b|, hence it is a scalar, and in ordinary quantum mechanics it is a complex number. One ... |

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Citation Context ... This is a topic for further research. 6.3 And Quantum Field Theory Finally, it is important to remark that there is an interpretation of the Jones polynomial in terms of quantum field theory. Witten =-=[26]-=- writes down a functional integral for link invariants in a 3-manifold M: ∫ Z(M, K) = ∫ dAexp[(ik/4π)S(M, A)]tr(Pexp( A)). K Here M denotes a 3-manifold without boundary and A is a gauge field (gauge ... |

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Citation Context .... 7 Summary In relating quantum computing with knot polynomials the key themes are unitarity and measurement. Much is now surely unforseen. For a good survey of quantum computing we recommend [1] and =-=[22]-=- and for another view of topological issues see [6] and [7]. See [25] for an excellent treatment of measurement theory in quantum mechanics and a useage of the Dirac formalism that is in resonance wit... |

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Citation Context ...new road. 7 Summary In relating quantum computing with knot polynomials the key themes are unitarity and measurement. Much is now surely unforseen. For a good survey of quantum computing we recommend =-=[1]-=- and [22] and for another view of topological issues see [6] and [7]. See [25] for an excellent treatment of measurement theory in quantum mechanics and a useage of the Dirac formalism that is in reso... |

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(Show Context)
Citation Context ...on is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open. 1 Introduction This paper is an exploration of issues interrelating the Jones polynomial =-=[10]-=- and quantum computing. In section 2 of the paper we review the formalism of Dirac brackets and some of the quantum physics associated with this formalism. The section ends with a brief description of... |

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(Show Context)
Citation Context ... with braiding remains as remarkable as ever. In order to see how these representations work, it is useful to discuss the combinatorics of these algebras a bit further. The Temperley Lieb algebra TLn =-=[11]-=- is an algebra over a commutative ring k with generators {1, U1, U2, ..., Un−1} and relations U 2 i = δUi, UiUi±1Ui = Ui, UiUj = UjUi, |i − j| > 1, where δ is a chosen element of the ring k. These equ... |

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(Show Context)
Citation Context ...s, one can see that the formalism of Z(S, K) (S 3 denotes the three-dimensional sphere.) yields the Jones polynomial with the basic properties as we have discussed. See Witten’s paper or [26] or [15],=-=[16]-=-. 48The question is: How does the quantum field theory approach to the Jones polynomial relate to quantum computing? One way to discuss this question is to reformulate (topological) quantum field the... |

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Citation Context ...aid group that do produce quantum entanglements corresponding to topological braiding. These phenomena will be the subject of a subsequent paper [20]. We mention one further possibility. In the paper =-=[21]-=- by Lidar and Biham the authors show how to simulate special cases of the Ising model on a quantum computer. Their method is more combinatorial and less algebraic than the approach sketched in this se... |

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Citation Context ...e 3., when translated into algebra, is the famous Yang-Baxter equation. The Yang-Baxter equation occurred for the first time in problems related to exactly solved models in statistical mechanics (See =-=[19]-=-.). All the moves taken together are directly related to the axioms for a quasi-triangular Hopf algebra (aka quantum group). We shall not go into this connection here. There is an intimate connection ... |