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## Quantum complexity of testing group commutativity (2005)

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### Other Repositories/Bibliography

Venue: | Proceedings of ICALP’05 |

Citations: | 30 - 5 self |

### Citations

2497 |
Quantum computation and quantum information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ...we will deal with black-box groups we shall shortly describe them in the framework of quantum computing (see also [2] or [5]). For a general introduction to quantum computing the reader might consult =-=[10, 11]-=-. We will work in the quantum circuit model. For a group G of encoding length n, the black-box will be given by two oracles OG and its inverse O −1 , both operating on 2n qubits. For any group element... |

1135 | A fast quantum mechanical algorithm for database search.
- Grover
- 1996
(Show Context)
Citation Context ...Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model. 1 Introduction A direction of research in quantum computation pioneered by Grover =-=[Gro96]-=- around search problems in unstructured, structured, or partially structured databases has recently been infused with new ideas for algorithm design. In contrast to problems based on the Hidden Subgro... |

352 | Quantum lower bounds by polynomials,
- Beals, Buhrman, et al.
- 2001
(Show Context)
Citation Context ...eed can be exponentially smaller than the randomized one. A prominent example is the HSP. On the other hand, for total functions, deterministic and quantum query complexities are polynomially related =-=[3]-=-. In the HSP, the group with its all structure is known to the algorithm designer, and the group operations are generally efficiently computable. In the event that the group is not explicitly known, o... |

288 | Lower bounds for discrete logarithms and related problems
- Shoup
- 1997
(Show Context)
Citation Context ...encodings that did not result from previous queries to the oracle. This family of algorithm are usually called generic algorithms. This notion was introduced to cryptography by Nechaev [18] and Shoup =-=[19]-=-. By suitably randomizing the encoding such as in [20], we can ensure that the probability that the algorithm chances upon a valid pair of encoded group elements is o(1), if this input does not result... |

265 |
Random walks on finite groups and rapidly mixing Markov chains.
- Aldous
- 1983
(Show Context)
Citation Context ...on-negative eigenvalues, λ t ≤ ∆(t) · (minu π(u)) −1 , where λ is the second largest eigenvalue. This bounds the second largest eigenvalue in terms of the total variation distance. 2. (see e.g., Ref. =-=[Ald82]-=-) ∆(t) ≤ 2 exp(−⌊ t τ ⌋). This relates the total variation distance at any time t to the mixing time τ. 3. [Gri78] τ ≤ 2eT . This bounds the mixing time τ in terms of the coupling time T . Combining a... |

211 | Algorithms for random generation and counting: a Markov chain approach - Sinclair - 1993 |

174 | Quantum walk algorithm for element distinctness,
- Ambainis
- 2007
(Show Context)
Citation Context ...for which the approach of Szegedy has no equivalent using other known techniques for constructing quantum algorithms, such as Grover search [Gro96], or the type of quantum walk introduced by Ambainis =-=[Amb04]-=-. Conversely, for Triangle Finding, the approach of Ambainis was more successfully applied. For this problem, Magniez, Szegedy and Santha [MSS05] construct a quantum algorithm that uses recursively tw... |

150 |
Quantum lower bounds for the collision and the element distinctness problems
- Shi
- 2002
(Show Context)
Citation Context ...y an adversary argument. For the quantum case, we reduce UNIQUE COLLISION to UNIQUE SPLIT COLLISION by composing the oracle function with a random permutation. Then Theorem 2 (due to Aaronson and Shi =-=[AS04]-=- and Kutin [Kut05], together with Ambainis [Amb05]) implies the lower bound. Assume that we have an algorithm A for UNIQUE SPLIT COLLISION with constant bounded error γ < 1/4. We run A on oracle F com... |

108 |
Quantum Speed-up of Markov Chain Based Algorithms,
- Szegedy
- 2004
(Show Context)
Citation Context ...tic. The linear upper bound on complexity may also be obtained by applying quantum search [Gro96] to locate a pair of generators that do not commute. Using the quantization of random walks by Szegedy =-=[Sze04]-=-, we instead present a sublinear algorithm with time and query complexity in Õ(k2/3 ) (Theorem 3), where the Õ notation means that logarithmic multiplicative factors are omitted. GROUP COMMUTATIVITY b... |

96 |
On the complexity of matrix group problems, I. In
- Babai, Szemeredi
- 1984
(Show Context)
Citation Context ... the group operations are not efficient to implement, it is appropriate to model the group operations by an oracle or a black-box. The notion of black-box groups was introduced by Babai and Szemerédi =-=[4]-=-. In this model, the elements of a group are encoded by words over a finite alphabet, and the group operations are performed by an oracle (the ⋆ For their research support, F.M. thanks the EU 5th fram... |

94 | Quantum algorithms for the triangle problem, in:
- Magniez, Santha, et al.
- 2005
(Show Context)
Citation Context ...6], or the type of quantum walk introduced by Ambainis [Amb04]. Conversely, for Triangle Finding, the approach of Ambainis was more successfully applied. For this problem, Magniez, Szegedy and Santha =-=[MSS05]-=- construct a quantum algorithm that uses recursively two quantum walks à la Ref. [Amb04], while the Szegedy quantization of walks seems to give a less query-efficient algorithm. The problems of GROUP ... |

75 |
Complexity of a determinate algorithm for the discrete logarithm,
- Nechaev
- 1994
(Show Context)
Citation Context ...black-box with encodings that did not result from previous queries to the oracle. This family of algorithm are usually called generic algorithms. This notion was introduced to cryptography by Nechaev =-=[18]-=- and Shoup [19]. By suitably randomizing the encoding such as in [20], we can ensure that the probability that the algorithm chances upon a valid pair of encoded group elements is o(1), if this input ... |

68 | 2002), Quantum lower bound for the collision problem
- Aaronson
(Show Context)
Citation Context ... . , k} to {1, . . . , k}. An oracle decision problem is a boolean function on the set S. For every function F ∈ S, we define the variables xij which are 1 if F (i) = j and 0 otherwise. Definition 1 (=-=[3, 12]-=-). Let Φ : S → {0, 1} be an oracle decision problem. Then the approximation degree of Φ is the lowest degree of real multivariate polynomials P in variables xij, such that |P (x) − Φ(F )| ≤ 1/3, for e... |

48 | Quantum verification of matrix products, in:
- Buhrman, Špalek
- 2006
(Show Context)
Citation Context ...e problems of GROUP COMMUTATIVITY and TRIANGLE FINDING thus give strong evidence that the walks due to Ambainis are not comparable with the ones due to Szegedy. A recent result of Buhrman and ˇSpalek =-=[Bv06]-=- on matrix product verification also relies on the Szegedy quantization for its worst case time complexity. However, for the worst case instances, when there is at most one erroneous entry, the approa... |

45 | Quantum algorithms for solvable groups.
- Watrous
- 2001
(Show Context)
Citation Context ...code the same group element. Mosca [Mos99] showed that one can learn in quantum polynomial time the structure of any black-box abelian group. Such a task is known to be hard classically. Then Watrous =-=[Wat01]-=- pioneered the study of black-box group properties in the quantum context. ∗ A preliminary version of this paper appeared in Proceedings of 32nd International Colloquium on Automata, Languages and Pro... |

44 | Security of signed ElGamal encryption.
- Schnorr, Jakobsson
- 2000
(Show Context)
Citation Context ... the oracle. This family of algorithm are usually called generic algorithms. This notion was introduced to cryptography by Nechaev [18] and Shoup [19]. By suitably randomizing the encoding such as in =-=[20]-=-, we can ensure that the probability that the algorithm chances upon a valid pair of encoded group elements is o(1), if this input does not result from previous queries. If we choose n, the encoding l... |

39 | A quantum lower bound for the collision problem
- Kutin
- 2005
(Show Context)
Citation Context ...gument. For the quantum case, we reduce UNIQUE COLLISION to UNIQUE SPLIT COLLISION by composing the oracle function with a random permutation. Then Theorem 2 (due to Aaronson and Shi [AS04] and Kutin =-=[Kut05]-=-, together with Ambainis [Amb05]) implies the lower bound. Assume that we have an algorithm A for UNIQUE SPLIT COLLISION with constant bounded error γ < 1/4. We run A on oracle F composed with a rando... |

36 |
Quantum Computer Algorithms
- Mosca
- 1999
(Show Context)
Citation Context ... structured, or partially structured databases has recently been infused with new ideas for algorithm design. In contrast to problems based on the Hidden Subgroup Problem (HSP) (see for instance Ref. =-=[Mos99]-=-), the speed up for these search problems is often only polynomial. Usually in search problems, the access to the input is done via an oracle. This leads to the notion of query complexity which measur... |

30 |
Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range
- Ambainis
(Show Context)
Citation Context ... reduce UNIQUE COLLISION to UNIQUE SPLIT COLLISION by composing the oracle function with a random permutation. Then Theorem 2 (due to Aaronson and Shi [AS04] and Kutin [Kut05], together with Ambainis =-=[Amb05]-=-) implies the lower bound. Assume that we have an algorithm A for UNIQUE SPLIT COLLISION with constant bounded error γ < 1/4. We run A on oracle F composed with a random permutation on the domain. If ... |

19 |
Quantum lower bounds for collision and element distinctness with small range
- Ambainis
(Show Context)
Citation Context ...omized reduction from the problem Collision which detects between a bijection and a two-to-one function. However, the reduction is still valid for the special case we consider. As noticed by Ambainis =-=[14]-=-, this reduction also implies the lower bound on the approximation degree. 3 A quantum algorithm for Group Commutativity We are given a black-box group G with generators g1, . . . , gk. The problem is... |

10 |
Classical and quantum computation, volume 47
- Kitaev, Vyalyi
- 2002
(Show Context)
Citation Context ...we will deal with black-box groups we shall shortly describe them in the framework of quantum computing (see also [2] or [5]). For a general introduction to quantum computing the reader might consult =-=[10, 11]-=-. We will work in the quantum circuit model. For a group G of encoding length n, the black-box will be given by two oracles OG and its inverse O −1 , both operating on 2n qubits. For any group element... |

8 |
Coupling methods for Markov processes
- Griffeath
- 1978
(Show Context)
Citation Context ...econd largest eigenvalue in terms of the total variation distance. 2. (see e.g., Ref. [Ald82]) ∆(t) ≤ 2 exp(−⌊ t τ ⌋). This relates the total variation distance at any time t to the mixing time τ. 3. =-=[Gri78]-=- τ ≤ 2eT . This bounds the mixing time τ in terms of the coupling time T . Combining all three relations, we may deduce the following relationship between the spectral gap of a Markov chain and coupli... |

7 |
Testing commutativity of a group and the power of randomization. Electronic version at http://www-math.mit. edu/ pak/research.html
- PAK
- 2000
(Show Context)
Citation Context ...his context, we study the problem of testing the commutativity of a black-box group (GROUP COMMUTATIVITY) given by its generators. The classical complexity of this problem was first considered by Pak =-=[Pak00]-=-. The straightforward algorithm for the problem has complexity O(k2 ), where k is the number of generators, since it suffices to check if every pair of generators commute. Pak presented a surprising r... |

1 | BS84] [BˇS04 - Babai, Szemerédi - 2001 |

1 | Quantum verification of matrix products. Technical Report quant-ph/0409035, arXiv archive - Buhrman, Spalek - 2004 |

1 | On the complexity of matrix group problems I - unknown authors - 1984 |