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

Citations: | 11 - 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.

### Citations

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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 ...n the control bit. One can convert a circuit for U into a circuit for controlled-U by replacing each gate G with a circuit for controlled-G. The overhead incurred is thus bounded by a constant factor =-=[23]-=-. Next we’ll show that trace estimation is hard for DQC1. Suppose we are given a classical description of a quantum circuit implementing some unitary transformation U on n qubits. As shown in equation... |

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Citation Context ...if one can be deformed into the other by some sequence of the three Reidemeister moves shown above. 4One of the best known knot invariants is the Jones polynomial, discovered in 1985 by Vaughan Jones=-=[17]-=-. To any oriented knot or link, it associates a Laurent polynomial in the variable t 1/2 . The Jones polynomial has a degree in t which grows at most linearly with the number of crossings in the link.... |

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Citation Context ...e will construct a braid whose Fibonacci representation performs essentially a gate-by-gate simulation of the circuit. To do this, we will use a version of the Solovay-Kitaev theorem which appears in =-=[19]-=-. 10Theorem 1 (Solovay-Kitaev) Suppose matrices U1, . . .,Ur generate a dense subgroup in SU(d). Then, given a desired unitary U ∈ SU(d), and a precision parameter δ > 0, there is an algorithm to fin... |

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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 ...ing the Jones polynomial for trace closures was left open, other than showing that it is contained in BQP. The results of [3, 1, 27] reformulate and generalize the previous results of Freedman et al. =-=[12, 11]-=-, which show that certain approximations of Jones polynomials are BQP-complete. The work of Freedman et al. in turn builds upon Witten’s discovery of a connection between Jones polynomials and topolog... |

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Citation Context ...bility of this problem is highly nontrivial. This was achieved by Haken in 1961[13]. In 1998 it was shown by Hass, Lagarias, and Pippenger that the problem of recognizing the unknot is contained in NP=-=[14]-=-. Starting with the Alexander polynomial, discovered in 1928, a number of knot invariants have been discovered. These are functions which are invariant under Reidemeister moves, thus a knot invariant ... |

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(Show Context)
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 ...t 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.... |

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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 ...rtificial that in the one clean qubit model, there are some pure qubits and some maximally mixed qubits. However, given an arbitrary n-qubit state ρ, one can use the technique of “algorithmic cooling”=-=[23]-=- to unitarily separate this into n − s pure qubits and s maximally mixed qubits, where s is the von Neumann entropy of ρ. Any 2 n × 2 n unitary matrix can be decomposed as a linear combination of n-fo... |

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Citation Context ... construction based on Barrington’s theorem has polynomial overhead and is thus sufficient for our purposes, it seems worth noting that it is possible to achieve better efficiency. As shown by Draper =-=[10]-=-, there exist ancilla-free quantum circuits for performing addition and subtraction, which succeed with high probability and have nearly linear size. Specifically, one can add or subtract a hardcoded ... |

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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 ... braid. 1 One Clean Qubit The one clean qubit model of quantum computation originated as an idealized model of quantum computation on highly mixed initial states, such as appear in NMR implementations=-=[20, 4]-=-. In this model, one is given an initial quantum state consisting of a single qubit in the pure state |0〉, and n qubits in the maximally mixed state. This is described by the density matrix ρ = |0〉 〈0... |

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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 ...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 ...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 ... |

2 |
Bounded-width polynomial-size brancing programs recognize exactly those languages
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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... |

1 | Computation with unitaries and one pure qubit
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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 ... |