## Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups (1998)

Citations: | 44 - 0 self |

### BibTeX

@TECHREPORT{Rötteler98polynomial-timesolution,

author = {Martin Rötteler and Thomas Beth},

title = {Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups},

institution = {},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer. 1 supported by DFG grant GRK 209/3-98 1

### Citations

860 | Algorithms for quantum computation: discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ... We adopt the definition of the hidden subgroup problem given in [4]. The history of the hidden subgroup problem parallels the history of quantum computing since the algorithms of Simon [14] and Shor =-=[13]-=- can be formulated in the language of hidden subgroups (see e.g. [9] for this reduction) for certain abelian groups. In the paper [3] an exact quantum algorithm (running in polynomial time in the numb... |

482 |
The representation theory of the symmetric group
- James, Kerber
- 1981
(Show Context)
Citation Context ...rmore f takes different values on different cosets. The problem is to find generators for U. 3 Wreath Products In this section we recall the definition of wreath products in general (see also [6] and =-=[8]-=-) and define the family of groups for which we will solve the hidden subgroup problem. Definition 3.1 Let G be a group and H ⊆ Sn be a subgroup of the symmetric group on n letters. The wreath product ... |

367 | On the Power of Quantum Computation
- Simon
- 1997
(Show Context)
Citation Context ...tum computation compared to classical computation becomes apparent in this problem. The first occurrence of a hidden subgroup problem for an abelian group appeared has been implicitly in Simon’s work =-=[14]-=-. In fact he solved the hidden subgroup problem for the group Z n 2 under the promise that the hidden subgroup is of order 2. The group theoretical interpretation of this algorithm has been formulated... |

215 | Elementary gates for quantum computation
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ...nsform for Z2n 2 and therefore a tensor product of 2n Hadamard matrices. The circuits for the case of Wn are shown in figure 3. Quantum circuits in general are built from certain gate primitives (see =-=[1]-=-) and it is clear that the complexity cost for this circuit is linear in the number of qubits, since the conditional gate representing the evaluation at the transversal � t∈T Φ(t) can be realized with... |

75 | On quantum algorithms for noncommutative hidden subgroups
- Ettinger, Hoyer
(Show Context)
Citation Context ...orithm over any classical algorithm, even probabilistic ones. Recently the question has been raised as to whether the hidden subgroup problem could also be solved for non-abelian groups. In the paper =-=[4]-=- the problem is addressed for the dihedral groups DN. The authors have found an interesting way to circumvent the application of the Fourier transform for the dihedral groups and instead use the Fouri... |

60 | Quantum Algorithms and the Fourier Transform
- Jozsa
- 1998
(Show Context)
Citation Context ... subgroup problem for the group Z n 2 under the promise that the hidden subgroup is of order 2. The group theoretical interpretation of this algorithm has been formulated by several authors (see [3], =-=[9]-=-, [10]). In the paper [3] it is shown that subgroups of arbitrary order can be found and furthermore that this can be done by an exact quantum polynomial time algorithm. Thus there is an exponential s... |

59 | The hidden subgroup problem and eigenvalue estimation on a quantum computer
- Mosca, Ekert
- 1998
(Show Context)
Citation Context ...roup problem for the group Z n 2 under the promise that the hidden subgroup is of order 2. The group theoretical interpretation of this algorithm has been formulated by several authors (see [3], [9], =-=[10]-=-). In the paper [3] it is shown that subgroups of arbitrary order can be found and furthermore that this can be done by an exact quantum polynomial time algorithm. Thus there is an exponential speed-u... |

36 | Efficient quantum transforms
- Høyer
(Show Context)
Citation Context ...Fourier transforms on a quantum computer described in [12] can be applied directly in case of wreath products A ≀ Z2 where A is an arbitrary abelian 2-group (for efficient quantum transforms see also =-=[5]-=-). 9sThe recursion of the algorithm follows the chain A ≀ Z2 ⊲ A × A ⊲ E, where the second composition factor is the base group. We first want to determine the irreducible representations of G := A ≀ ... |

30 | Fast quantum Fourier transforms for a class of non-Abelian groups
- Püschel, Rötteler, et al.
- 1999
(Show Context)
Citation Context ...rier transform for the groups Wn effectively on a quantum computer. We want to do this in brief since the general recursive method to obtain fast Fourier transforms on a quantum computer described in =-=[12]-=- can be applied directly in case of wreath products A ≀ Z2 where A is an arbitrary abelian 2-group (for efficient quantum transforms see also [5]). 9sThe recursion of the algorithm follows the chain A... |

15 |
Basic Algebra II
- Jacobson
- 1980
(Show Context)
Citation Context ... . . . , χk} the set of irreducible representations of A recall that the irreducible representations of G ∗ are given by the set {χi ⊗ χj : i, j = 1, . . ., k} of pairwise tensor products (see, e.g., =-=[7]-=- section 5.6). Since G ∗ ⊳ G the group G operates on the representations of G ∗ via inner conjugation. Because G is a semidirect product of G ∗ with Z2 we can write each element g ∈ G as g = (a1, a2; ... |

11 |
Konstruktive Darstellungstheorie und Algorithmengenerierung
- Püschel
- 1998
(Show Context)
Citation Context ...nce the recursive formula � DFTG∗ · Φ(t) · DFTZ2 t∈T provides a Fourier transform for G. Here Φ(t) denotes the extension (as a whole) of the regular representation of G∗ to a representation of G (see =-=[11]-=-, [2], [12]). In case of Wn the transform DFTG∗ is the Fourier transform for Z2n 2 and therefore a tensor product of 2n Hadamard matrices. The circuits for the case of Wn are shown in figure 3. Quantu... |

9 |
An exact polynomial-time algorithm for Simon’s problem
- Brassard, Høyer
- 1997
(Show Context)
Citation Context ...idden subgroup problem for the group Z n 2 under the promise that the hidden subgroup is of order 2. The group theoretical interpretation of this algorithm has been formulated by several authors (see =-=[3]-=-, [9], [10]). In the paper [3] it is shown that subgroups of arbitrary order can be found and furthermore that this can be done by an exact quantum polynomial time algorithm. Thus there is an exponent... |

6 |
Methoden der Schnellen Fouriertransformation
- Beth
- 1984
(Show Context)
Citation Context ... i.e., only the factor group G/G ∗ = Z2 operates via permutation of the tensor factors. The operation of τ is to map χ1 ⊗ χ2 ↦→ χ2 ⊗ χ1. Therefore it is easy to determine the inertia groups (see [6], =-=[2]-=- for definitions) Tρ of a representation ρ of G ∗ . We have to consider two cases: a) ρ = χi⊗χi. Then Tρ = G since permutation of the factors leaves ρ invariant. b) ρ = χi ⊗ χj, i �= j. Here we have T... |

3 |
Endliche Gruppen, volume I
- Huppert
- 1983
(Show Context)
Citation Context ...) Furthermore f takes different values on different cosets. The problem is to find generators for U. 3 Wreath Products In this section we recall the definition of wreath products in general (see also =-=[6]-=- and [8]) and define the family of groups for which we will solve the hidden subgroup problem. Definition 3.1 Let G be a group and H ⊆ Sn be a subgroup of the symmetric group on n letters. The wreath ... |