## Estimating Jones polynomials is a complete problem for one clean qubit (2007)

Citations: | 10 - 4 self |

### BibTeX

@MISC{Shor07estimatingjones,

author = {Peter W. Shor and Stephen P. Jordan},

title = {Estimating Jones polynomials is a complete problem for one clean qubit },

year = {2007}

}

### OpenURL

### Abstract

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model[12, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems [20]. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid.

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Citation Context ... problems, that is, problems which admit yes/no answers. This is mathematically convenient, and the implications for the complexity of nondecision problems are usually straightforward to obtain (c.f. =-=[22]-=-). The one clean qubit complexity class consists of the decision problems which can be solved in polynomial time by a one clean qubit machine with correctness probability of at least 2/3. The experime... |

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Citation Context ...that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model=-=[12, 3, 1]-=-. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard... |

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Citation Context ...that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model=-=[13, 3, 1]-=-. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard... |

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Citation Context ...ynomial of the plat or trace closure of a braid at t = e i2π/k can be computed on a quantum computer in time which scales polynomially in the number of strands and crossings in the braid and in k. In =-=[1, 27]-=-, it is shown that for plat closures, this problem is BQP-complete. The complexity of approximating the Jones polynomial for trace closures was left open, other than showing that it is contained in BQ... |

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Citation Context ...it computer. The others are estimating the Pauli decomposition of the unitary matrix corresponding to a polynomial-size quantum circuit 2 , [21, 27], estimating quadratically signed weight enumerators=-=[22]-=-, and estimating average fidelity decay of quantum maps[25, 26]. 7 Acknowledgements The authors thank David Vogan, Pavel Etingof, Raymond Laflamme, Pawel Wocjan, Sergei Bravyi, Wim van Dam, Aram Harro... |

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Citation Context ... and σi+1σiσi+1 = σi+1σiσi+1 for all i. The group operation corresponds to concatenation of braids. 3 Fibonacci Representation The Fibonacci representation ρ (n) of the braid group Bn is described in =-=[18]-=- in the context of Temperley-Lieb F recoupling theory. This is a mathematical framework, which in this case describes two species of idealized “particles” denoted by p and ∗. We will not delve into th... |

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Citation Context ...modulo 2 n by performing quantum Fourier transform, followed by O(n 2 ) controlledrotations, followed by an inverse quantum Fourier transform. Furthermore, using approximate quantum Fourier transforms=-=[6]-=-, [10] describes an approximate version of the circuit, which, for any value of parameter m, uses a total of only O(mn log n) gates to produce an output having an inner product with |x + a mod 2 n 〉 o... |

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Citation Context ...large unitary matrices can be estimated to this precision on a classical computer in polynomial time. Thus it seems unlikely that DQC1 is contained in P. (For more detailed analysis of this point see =-=[9]-=-.) However, it also appears unlikely that DQC1 contains all of P. By applying Barrington’s theorem[7], it has been shown that DQC1 contains NC1, the class of problems solvable by logarithmic depth cla... |

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Citation Context ...ion of the unitary matrix corresponding to a polynomial-size quantum circuit 2 , [21, 27], estimating quadratically signed weight enumerators[22], and estimating average fidelity decay of quantum maps=-=[25, 26]-=-. 7 Acknowledgements The authors thank David Vogan, Pavel Etingof, Raymond Laflamme, Pawel Wocjan, Sergei Bravyi, Wim van Dam, Aram Harrow, and Daniel Nagaj for useful discussions, and an anonymous re... |

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Citation Context ...action, the comparison is complete. Alternatively, one could use the linear size quantum comparison circuit devised by Takahashi and Kunihiro, which uses n uninitialized ancillas but no clean ancillas=-=[28]-=-. Unfortunately, most crossings in a given braid will not be acting on the leftmost strand. However, we can reduce the problem of extracting a general symbol to the problem of extracting the leftmost ... |

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Citation Context ... Thus it seems unlikely that DQC1 is contained in P. (For more detailed analysis of this point see [9].) However, it also appears unlikely that DQC1 contains all of P. By applying Barrington’s theorem=-=[7]-=-, it has been shown that DQC1 contains NC1, the class of problems solvable by logarithmic depth classical circuits [4]. These relationships are summarized in figure 3. 3BQP P NC1 DQC1 Figure 3: Summa... |

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Citation Context ...tum circuit to polynomial accuracy is a DQC1-complete problem. Estimating the normalized trace of a quantum circuit is a special case of this, and it is also DQC1-complete. This point is discussed in =-=[27]-=-. To make our presentation self-contained, we will sketch here a proof that trace estimation is DQC1-complete. Technically, we should consider the decision problem of determining whether the trace is ... |