## Explicit Multiregister Measurements for Hidden . . .

Citations: | 7 - 6 self |

### BibTeX

@MISC{Moore_explicitmultiregister,

author = {Cristopher Moore and Alexander Russell},

title = {Explicit Multiregister Measurements for Hidden . . .},

year = {}

}

### OpenURL

### Abstract

We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 |G| registers, and each of the 2^k subsets of the registers contributes some information.

### Citations

877 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...sure the result. For example, in Simon’s problem [31], G = Zn 2 and there is some y such that f(x) = f(x+y) for all x; in this case H = {0, y} and we wish to identify y. In Shor’s factoring algorithm =-=[30]-=- G is the group Z∗ n where n is the number we wish to factor, f(x) = rx mod n for a random r < n, and H is the subgroup of Z∗ n whose index is the multiplicative order of r. In both these algorithms, ... |

373 |
Representation theory: a first course
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Citation Context ...e irreducible representations are those which are not isomorphic to direct sums of representations on lower-dimensional subspaces, and we denote the set of irreducibles by ̂ G. We refer the reader to =-=[9]-=- for an introduction. We denote the set of functions ψ : G → C with ‖ψ‖ 2 = 1, i.e., the Hilbert space of a group-valued register, as C[G]; then the quantum Fourier transform consists of transforming ... |

353 | On the power of quantum computation
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- 1997
(Show Context)
Citation Context ...er the left cosets, with density matrix ρ = 1 ∑ |cH〉 〈cH| . |G| c∈G We then carry out the quantum Fourier transform on |cH〉, or equivalently ρ, and measure the result. For example, in Simon’s problem =-=[31]-=-, G = Zn 2 and there is some y such that f(x) = f(x+y) for all x; in this case H = {0, y} and we wish to identify y. In Shor’s factoring algorithm [30] G is the group Z∗ n where n is the number we wis... |

74 | On Quantum Algorithms for Noncommutative Hidden Subgroups - Ettinger, Hoyer |

66 |
Quantum computation of Fourier transforms over symmetric groups
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- 1997
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Citation Context ... type, or “name,” of an irreducible representation and 1 ≤ i, j ≤ dσ index a row and column (in a chosen basis for V ). This transformation can be carried out efficiently for a wide variety of groups =-=[2, 14, 24]-=-. Several varieties of measurement in the Fourier basis have been proposed. Weak Fourier sampling consists of measuring just the name σ of the irreducible representation. Strong Fourier sampling consi... |

66 | Quantum mechanical algorithms for the nonabelian Hidden Subgroup Problem
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(Show Context)
Citation Context ...d cryptographically important cases of the Shortest Lattice Vector problem [27]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups =-=[8, 10, 13, 15, 17, 25, 28]-=-. However, the basic idea of Fourier sampling can certainly be extended to the nonabelian case. Fourier basis functions are homomorphisms φ : G → C such as the familiar φk(x) = e2πikx/n when G is the ... |

55 | Quantum computation and lattice problems
- Regev
(Show Context)
Citation Context .... Other important motivations include the relationship between the HSP on the dihedral group and hidden shift problems [4] and cryptographically important cases of the Shortest Lattice Vector problem =-=[27]-=-. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups [8, 10, 13, 15, 17, 25, 28]. However, the basic idea of Fourier sampling can ce... |

53 | A subexponential-time quantum algorithm for the dihedral hidden subgroup problem
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(Show Context)
Citation Context ...e-force search through the subgroup lattice of G; however, for groups of interest such as the symmetric groups, this algorithm takes exponential time. For the dihedral groups in particular, Kuperberg =-=[20]-=- devised a subexponential algorithm, which uses 2O(√log n) time and registers, that works by performing entangled measurements on two registers at a time. Regev [27] provided a beautiful kind of worst... |

49 |
Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem
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(Show Context)
Citation Context ...d cryptographically important cases of the Shortest Lattice Vector problem [27]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups =-=[8, 10, 13, 15, 17, 25, 28]-=-. However, the basic idea of Fourier sampling can certainly be extended to the nonabelian case. Fourier basis functions are homomorphisms φ : G → C such as the familiar φk(x) = e2πikx/n when G is the ... |

44 | Polynomial-Time Solution to the Hidden Subgroup Problem for a Class of Non-abelian Groups, Technical report, Quantum Physics e-Print archive
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Citation Context ...d cryptographically important cases of the Shortest Lattice Vector problem [27]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups =-=[8, 10, 13, 15, 17, 25, 28]-=-. However, the basic idea of Fourier sampling can certainly be extended to the nonabelian case. Fourier basis functions are homomorphisms φ : G → C such as the familiar φk(x) = e2πikx/n when G is the ... |

42 | Quantum algorithms for some hidden shift problems
- Dam, Hallgren, et al.
- 2002
(Show Context)
Citation Context ... the trivial subgroup would be sufficient to solve this case of Graph Isomorphism. Other important motivations include the relationship between the HSP on the dihedral group and hidden shift problems =-=[4]-=- and cryptographically important cases of the Shortest Lattice Vector problem [27]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian grou... |

42 |
From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
- Bacon, Childs, et al.
- 2005
(Show Context)
Citation Context ...d cryptographically important cases of the Shortest Lattice Vector problem [30]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups =-=[3, 10, 12, 15, 18, 20, 28, 31]-=-. However, the basic idea of Fourier sampling can certainly be extended to the nonabelian case. Fourier basis functions are homomorphisms φ : G → C such as the familiar φk(x) = e 2πikx/n when G is the... |

38 | Normal subgroup reconstruction and quantum computation using group representations
- Hallgren, Russell, et al.
(Show Context)
Citation Context ...d cryptographically important cases of the Shortest Lattice Vector problem [30]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups =-=[3, 10, 12, 15, 18, 20, 28, 31]-=-. However, the basic idea of Fourier sampling can certainly be extended to the nonabelian case. Fourier basis functions are homomorphisms φ : G → C such as the familiar φk(x) = e 2πikx/n when G is the... |

37 | An Improved Quantum Fourier Transform Algorithm and Applications, FOCS 2000, (also available at http://www.cs.caltech.edu/~hallgren
- Hallgren, Hales
(Show Context)
Citation Context ... < n, and H is the subgroup of Z∗ n whose index is the multiplicative order of r. (However, since |Z∗ n| is unknown, we actually perform the Fourier transform over Zq for some q = O(n2 ); see [33] or =-=[13, 14]-=-.) In both these algorithms, G is abelian, and it is not hard to see that for any abelian group a polynomial number1 of experiments of this type allow us to determine H. In essence, each experiment yi... |

36 | Efficient quantum transforms
- Høyer
- 1997
(Show Context)
Citation Context ... type, or “name,” of an irreducible representation and 1 ≤ i, j ≤ dσ index a row and column (in a chosen basis for V ). This transformation can be carried out efficiently for a wide variety of groups =-=[2, 14, 24]-=-. Several varieties of measurement in the Fourier basis have been proposed. Weak Fourier sampling consists of measuring just the name σ of the irreducible representation. Strong Fourier sampling consi... |

34 | Hidden subgroup states are almost orthogonal
- Ettinger, Høyer, et al.
- 1999
(Show Context)
Citation Context ...6] showed that the optimal measurement in the dihedral group is already entangled in the two-register case. In one sense we already know that such a measurement can succeed. Ettinger, Høyer and Knill =-=[6]-=- showed that the density matrices ρ become nearly orthogonal for distinct subgroups for some k = O(log |G|). As a consequence, a measurement exists which determines the hidden subgroup with high proba... |

29 | Optimal measurements for the dihedral hidden subgroup problem
- Bacon, Childs, et al.
- 2006
(Show Context)
Citation Context ...orst-case to average-case quantum reduction, by showing that the HSP for the dihedral group Dn can be reduced to uniformly random instances of the Subset Sum problem on Zn. Bacon, Childs, and van Dam =-=[1]-=- deepened this connection by determining the optimal multiregister measurement for the dihedral group, and showing that it consists of the so-called pretty good measurement (PGM); they used this to sh... |

29 |
The power of basis selection in Fourier sampling: Hidden subgroup problems in affine groups
- Moore, Rockmore, et al.
(Show Context)
Citation Context ...d cryptographically important cases of the Shortest Lattice Vector problem [30]. So far, explicit polynomial-time quantum algorithms for the HSP are known only for a few families of nonabelian groups =-=[3, 10, 12, 15, 18, 20, 28, 31]-=-. However, the basic idea of Fourier sampling can certainly be extended to the nonabelian case. Fourier basis functions are homomorphisms φ : G → C such as the familiar φk(x) = e 2πikx/n when G is the... |

28 |
The quantum query complexity of the hidden subgroup problem is polynomial
- Ettinger, Høyer, et al.
(Show Context)
Citation Context ...t the density matrices ρ become nearly orthogonal for distinct subgroups for some k = O(log |G|). As a consequence, a measurement exists which determines the hidden subgroup with high probability. In =-=[7]-=- they make this result somewhat more constructive by giving an algorithm which solves the HSP by performing a brute-force search through the subgroup lattice of G; however, for groups of interest such... |

27 | Generic quantum Fourier transforms
- Moore, Rockmore, et al.
(Show Context)
Citation Context ... type, or “name,” of an irreducible representation and 1 ≤ i, j ≤ dσ index a row and column (in a chosen basis for V ). This transformation can be carried out efficiently for a wide variety of groups =-=[2, 14, 24]-=-. Several varieties of measurement in the Fourier basis have been proposed. Weak Fourier sampling consists of measuring just the name σ of the irreducible representation. Strong Fourier sampling consi... |

26 | The symmetric group defies strong fourier sampling
- Moore, Russell, et al.
(Show Context)
Citation Context ... will not succeed in solving the Hidden Subgroup Problem in the cases we care most about—in particular, the case relevant to Graph Isomorphism [13, 10, 19]. Most recently, Moore, Russell and Schulman =-=[22]-=- showed that strong Fourier sampling requires an exponential number of experiments to distinguish the order-2 subgroups H defined above from each other or from the trivial subgroup. However, there is ... |

24 | Quantum factoring, discrete logarithms, and the hidden subgroup problem
- Jozsa
- 2001
(Show Context)
Citation Context ...d provide enormous benefits. In particular, solving the HSP for the symmetric group Sn would provide an efficient quantum algorithm for the Graph Automorphism and Graph Isomorphism problems (see e.g. =-=[18]-=- for a review). Let G1, G2 be two rigid, connected graphs of size n, and let H ⊂ S2n be the automorphism group of their disjoint union. If G1 ∼ = G2, then H = {1, m} is of order 2, consisting of the i... |

23 |
Amnon Ta-Shma. Normal subgroup reconstruction and quantum computation using group representations
- Hallgren, Russell
(Show Context)
Citation Context |

20 |
The hidden subgroup problem and permutation group theory
- Kempe, Shalev
- 2005
(Show Context)
Citation Context ...gative results have shown that these types of measurement will not succeed in solving the Hidden Subgroup Problem in the cases we care most about—in particular, the case relevant to Graph Isomorphism =-=[13, 10, 19]-=-. Most recently, Moore, Russell and Schulman [22] showed that strong Fourier sampling requires an exponential number of experiments to distinguish the order-2 subgroups H defined above from each other... |

19 |
Quantum complexity theory (preliminary abstract
- Bernstein, Vazirani
- 1993
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Citation Context ...own efficient algorithm for this problem—and, indeed, almost every quantum algorithm that provides an exponential speedup over the best known classical algorithm—uses the approach of Fourier sampling =-=[3]-=-. By preparing a uniform superposition over the elements of G, querying the function f, and then measuring the value of f, we obtain a uniform superposition over one of the (left) cosets of H, |cH〉 = ... |

19 | Quantum Fourier sampling simplified
- Hales, Hallgren
- 1999
(Show Context)
Citation Context ... < n, and H is the subgroup of Z∗ n whose index is the multiplicative order of r. (However, since |Z∗ n| is unknown, we actually perform the Fourier transform over Zq for some q = O(n2 ); see [33] or =-=[13, 14]-=-.) In both these algorithms, G is abelian, and it is not hard to see that for any abelian group a polynomial number1 of experiments of this type allow us to determine H. In essence, each experiment yi... |

14 |
Gábor Ivanyos, Frédéric Magniez, Miklos Santha, and Pranab Sen. Hidden translation and orbit coset in quantum computing
- Friedl
- 2003
(Show Context)
Citation Context |

13 | Shor’s Algorithm is Optimal
- Ip
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Citation Context ...dent measurements; rather, it is an entangled measurement, in which we measure vectors in C[Gk ] along a basis whose basis vectors are not tensor products of k basis vectors in C[G]. For instance, Ip =-=[16]-=- showed that the optimal measurement in the dihedral group is already entangled in the two-register case. In one sense we already know that such a measurement can succeed. Ettinger, Høyer and Knill [6... |

11 | For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets”, quant-ph/0501177
- Moore, Russell
(Show Context)
Citation Context ... showing that it consists of the so-called pretty good measurement (PGM); they used this to show a sharp threshold at k = log2 n for the number of registers needed to solve the HSP. Moore and Russell =-=[21]-=- generalized their results to some extent, showing that the PGM is optimal for arbitrary groups G in the single-register case whenever we wish to distinguish the conjugates of some subgroup H from eac... |

10 | Linear Representations of Finite Groups. Number 42 - Serre - 1977 |

5 | On the power of random bases - Radhakrishnan, Rotteler, et al. |

5 |
Subexponentialtime algorithms for hidden subgroup problems over product groups
- Alagić, Moore, et al.
- 2006
(Show Context)
Citation Context ...then perform a measurement inside σ0, and if we ever observe the sign representation π we know that the hidden subgroup is trivial. Using a somewhat different type of sieve, Alagic, Moore and Russell =-=[1]-=- recently obtained a subexponentialtime algorithm for the HSP in groups of the form Gn for finite G. Even though these groups have a simple structure, they are similar to Sn in that most of their irre... |

2 |
and Aner Shalev, The hidden subgroup problem and permutation group theory
- Kempe
- 2004
(Show Context)
Citation Context ...gative results have shown that these types of measurement will not succeed in solving the Hidden Subgroup Problem in the cases we care most about—in particular, the case relevant to Graph Isomorphism =-=[15, 12, 22]-=-. In particular, Moore, Russell and Schulman [25] showed that strong Fourier sampling fails, in the sense that we need an exponential number of experiments on single coset states to distinguish the or... |