## Quantum algorithms for solvable groups (2001)

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Venue: | In Proceedings of the 33rd ACM Symposium on Theory of Computing |

Citations: | 38 - 1 self |

### BibTeX

@INPROCEEDINGS{Watrous01quantumalgorithms,

author = {John Watrous},

title = {Quantum algorithms for solvable groups},

booktitle = {In Proceedings of the 33rd ACM Symposium on Theory of Computing},

year = {2001},

pages = {60--67}

}

### OpenURL

### Abstract

ABSTRACT In this paper we give a polynomial-time quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomial-time quantum algorithms as well. Our algorithm works in the setting of black-box groups, wherein none of these problems have polynomial-time classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.

### Citations

1368 |
Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ... model, so we will not review this model further except to discuss black-box groups in the context of quantum circuits. The reader not familiar with quantum circuits is referred to Nielsen and Chuang =-=[29-=-]. We also assume the reader is familiar with the basic concepts of group theory (see, for example, Isaacs [23]). Given a group G and elements g; h 2 G we dene the commutator of g and h, denoted [g; h... |

913 |
A Course in Computational Algebraic Number Theory
- Cohen
- 1993
(Show Context)
Citation Context ...Riemann Hypothesis|while no reduction in the other direction is known, the problem of computing class numbers is often considered as a candidate for a problem harder than integer factoring. See Cohen =-=[14]-=- for further information about computing in class groups. 2 proven to be helpful for solving the Graph Isomorphism problem. In this paper we move away from the Hidden Subgroup problem and consider oth... |

878 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...lly all previously identied problems for which quantum algorithms oer exponential speed-up over the best known classical algorithms can be stated as problems regarding abelian groups. In 1994, Shor [3=-=1-=-] presented polynomial time quantum algorithms for integer factoring and computing discrete logarithms, and these algorithms generalize in a natural way to the setting ofsnite groups. Specically, give... |

650 | Quantum theory, the Church-Turing principle, and the universal quantum computer
- Deutsch
- 1985
(Show Context)
Citation Context ...utable function f from G to somesnite set X such that f is constant and distinct on left-cosets of a subgroup H ofsnite index,snd a generating set for H. Mosca and Ekert showed that Deutsch's Problem =-=[15]-=-, Simon's Problem [32], ordersnding and computing discrete logarithms [31],snding hidden linear functions [10], testing self-shift-equivalence of polynomials [19], and the Abelian Stabilizer Problem [... |

457 |
Logical reversibility of computation
- Bennett
(Show Context)
Citation Context ...act that given the gate VG , we maysnd the order of any element g using Shor's algorithm, from which we maysnd the inverse of g. Once we have this, techniques in reversible computation due to Bennett =-=[9]-=- allow for straightforward simulation of UG and U 1 G . Since it is simpler to work directly with the gates UG and U 1 G , however, we will assume that these are the gates made available for a given b... |

356 | On the power of quantum computation
- Simon
- 1994
(Show Context)
Citation Context ... G to somesnite set X such that f is constant and distinct on left-cosets of a subgroup H ofsnite index,snd a generating set for H. Mosca and Ekert showed that Deutsch's Problem [15], Simon's Problem =-=[32]-=-, ordersnding and computing discrete logarithms [31],snding hidden linear functions [10], testing self-shift-equivalence of polynomials [19], and the Abelian Stabilizer Problem [25, 26] can all be sol... |

155 |
Quantum computations: algorithms and error correction
- Kitaev
- 1997
(Show Context)
Citation Context ...lgorithms as taking place in the abelian group generated by g. Shor's algorithms for integer factoring and discrete logarithms were subsequently cast in a dierent group-theoretic framework by Kitaev [=-=25, 26]-=-. This framework involves a problem called the Abelian Stabilizer problem, which may be informally stated as follows. Let k and n be positive integers, and consider some group action of the additive a... |

146 | Quantum measurements and the Abelian stabilizer problem
- Kitaev
- 1995
(Show Context)
Citation Context ...lgorithms as taking place in the abelian group generated by g. Shor's algorithms for integer factoring and discrete logarithms were subsequently cast in a dierent group-theoretic framework by Kitaev [=-=25, 26]-=-. This framework involves a problem called the Abelian Stabilizer problem, which may be informally stated as follows. Let k and n be positive integers, and consider some group action of the additive a... |

145 | Quantum algorithms revisited
- Cleve, Ekert, et al.
- 1998
(Show Context)
Citation Context ...ysis of Shor's algorithm, we will not discuss the analysis in detail and instead refer the reader to Shor [31] and to other sources in which analyses of closely related techniques are given in detail =-=[13, 25]-=-. We assume we are working over a black-box group G with encoding length n, and that a quantum register R has been initialized to state jHi for H some subgroup of G. For given g we are trying tosnd r,... |

103 |
Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix
- Kannan, Bachem
- 1979
(Show Context)
Citation Context ...matrix whose columns are the randomly generated elements of ker(f) ? , we may determine the numbers q 1 ; : : : ; q m in polynomial time by computing the Smith normal form of B (see Kannan and Bachem =-=[24]-=- and Hafner and McCurley [20] for polynomial-time algorithms for computing Smith normal forms). Now it is straightforward to test isomorphism of two given solvable groups G = hg 1 ; : : : ; g k i and ... |

96 | Complexity limitations on quantum computations
- Fortnow, Rogers
- 1990
(Show Context)
Citation Context ...ding membership testing, order verication, and isomorphism testing, are low for the complexity class PP (meaning that an oracle for these problems is useless for PP computations). Fortnow and Rogers [=-=18-=-] proved that any problem in BQP is low for PP, and thus we have obtained an alternate proof that membership testing, order verication, and isomorphism testing for solvable groups are all low for PP. ... |

84 | Local expansion of vertex-transitive graphs and random generation in finite groups
- Babai
- 1991
(Show Context)
Citation Context ...th n and the group operations are performed by a black-box (or group oracle) at unit cost. Black-box groups were introduced by Babai and Szemeredi [7] in 1984 and have since been studied extensively [=-=1, 2, 3, 4, 5, 6]-=-. Any ecient algorithm that works in the context of black-box groups of course remains ecient whenever the group oracle can be replaced by an ecient procedure for computing the group operations. In th... |

84 | An exact quantum polynomial-time algorithm for Simon's problem
- Brassard, Hyer
- 1997
(Show Context)
Citation Context ... that elements of X are unique representations of elements of G), and the group action of Z k on X depends on the group structure of G. Kitaev's approach was further generalized by Brassard and Hyer [=-=11-=-], who formulated the Hidden Subgroup problem. (See also Hyer [22] and Mosca and Ekert [28].) The Hidden Subgroup problem may be informally stated as follows. Given asnitely generated group G and an e... |

75 | On quantum algorithms for noncommutative hidden subgroups
- Ettinger, Høyer
- 1998
(Show Context)
Citation Context .... 1 (See also Cheung and Mosca [12] for further details.) The Hidden Subgroup problem has been considered in the non-abelian case, although with limited success (see, for instance, Ettinger and Hyer [=-=1-=-6], Ettinger, Hyer, and Knill [17], Rotteler and Beth [30], and Hallgren, Russell, and Ta-Shma [21]). No polynomial-time algorithm for the Hidden Subgroup problem is known for any class of non-abelian... |

70 |
On the complexity of matrix group problems
- Babai, Szemeredi
- 1984
(Show Context)
Citation Context ...s are uniquely encoded by strings of some given length n and the group operations are performed by a black-box (or group oracle) at unit cost. Black-box groups were introduced by Babai and Szemeredi [=-=7]-=- in 1984 and have since been studied extensively [1, 2, 3, 4, 5, 6]. Any ecient algorithm that works in the context of black-box groups of course remains ecient whenever the group oracle can be replac... |

66 |
Quantum computation of Fourier transforms over symmetric groups
- BEALS
- 1997
(Show Context)
Citation Context ...lar interest as it relates to the Graph Isomorphism problem; Graph Isomorphism reduces to a special case of the Hidden Subgroup problem in which the groups in question are the symmetric groups. Beals =-=[8]-=- has shown that quantum analogues of Fourier transforms over symmetric groups can be performed in polynomial time, although thus far this has not 1 This is particularly interesting from the standpoint... |

66 | Quantum mechanical algorithms for the nonabelian hidden subgroup problem
- Grigni, Schulman, et al.
(Show Context)
Citation Context ...den Subgroup Problem has been considered in the non-abelian case, although with limited success (see Ettinger and Hyer [17], Ettinger, Hyer, and Knill [18], Grigni, Schulman, Vazirani, and Vazirani [2=-=0-=-], Hallgren, Russell, and Ta-Shma [23], Ivanyos, Magniez, and Santha [26], and Rotteler and Beth [35]). Quantum polynomial-time algorithms forsnding non-abelian hidden subgroups are known for only lim... |

63 | Succinct quantum proofs for properties of finite groups
- Watrous
- 2000
(Show Context)
Citation Context ...hat g \Gamma 1 i hjgi 2 hh1; : : : ; hli for each i and j (as well as hh1; : : : ; hli ^ hg1; : : : ; gki). See Babai [3] for more examples of problems reducing to order computation. In another paper =-=[38]-=- we have shown that there exist succinct quantum certificates for various group-theoretic properties, including the property that a given integer divides the order of a group (i.e., given an integer d... |

61 | Quantum algorithms and the Fourier transform
- Jozsa
- 1998
(Show Context)
Citation Context ...he group G in question corresponds to the set X and the group action of Z k on X is determined by the group structure of G. Kitaev's approach was further generalized by Brassard and Hyer [12], Jozsa [=-=27-=-], and Mosca and Ekert [33], who formulated the Hidden Subgroup Problem. (See also Hyer [24].) The Hidden Subgroup Problem may be informally stated as follows. Given asnitely generated group G and an ... |

60 | Quantum cryptanalysis of hidden linear functions (ex- tended abstract
- Boneh, Lipton
- 1995
(Show Context)
Citation Context ...H ofsnite index,snd a generating set for H. Mosca and Ekert showed that Deutsch's Problem [15], Simon's Problem [32], ordersnding and computing discrete logarithms [31],snding hidden linear functions =-=[10]-=-, testing self-shift-equivalence of polynomials [19], and the Abelian Stabilizer Problem [25, 26] can all be solved in polynomial time within the framework of the Hidden Subgroup problem. In the black... |

60 | The hidden subgroup problem and eigenvalue estimation on a quantum computer, arXive e-print quant-ph/9903071
- Mosca, Ekert
- 1999
(Show Context)
Citation Context ... k on X depends on the group structure of G. Kitaev's approach was further generalized by Brassard and Hyer [11], who formulated the Hidden Subgroup problem. (See also Hyer [22] and Mosca and Ekert [2=-=8]-=-.) The Hidden Subgroup problem may be informally stated as follows. Given asnitely generated group G and an eciently computable function f from G to somesnite set X such that f is constant and distinc... |

59 |
Asymptotically fast triangularization of matrices over rings
- Hafner, McCurley
- 1991
(Show Context)
Citation Context ...randomly generated elements of ker(f) ? , we may determine the numbers q 1 ; : : : ; q m in polynomial time by computing the Smith normal form of B (see Kannan and Bachem [24] and Hafner and McCurley =-=[20]-=- for polynomial-time algorithms for computing Smith normal forms). Now it is straightforward to test isomorphism of two given solvable groups G = hg 1 ; : : : ; g k i and H = hh 1 ; : : : ; h l i as f... |

44 | Polynomial-time solution to the Hidden Subgroup Problem for a class of non-abelian groups. Technical report, Quantum Physics e-Print archive
- Rötteler, Beth
- 1998
(Show Context)
Citation Context ...The Hidden Subgroup problem has been considered in the non-abelian case, although with limited success (see, for instance, Ettinger and Hyer [16], Ettinger, Hyer, and Knill [17], Rotteler and Beth [30=-=]-=-, and Hallgren, Russell, and Ta-Shma [21]). No polynomial-time algorithm for the Hidden Subgroup problem is known for any class of non-abelian groups except for a special class of groups based on wrea... |

40 | A polynomial-time theory of black-box groups
- Babai, Beals
(Show Context)
Citation Context ...th n and the group operations are performed by a black-box (or group oracle) at unit cost. Black-box groups were introduced by Babai and Szemeredi [7] in 1984 and have since been studied extensively [=-=1, 2, 3, 4, 5, 6]-=-. Any ecient algorithm that works in the context of black-box groups of course remains ecient whenever the group oracle can be replaced by an ecient procedure for computing the group operations. In th... |

38 | Normal subgroup reconstruction and quantum computation using group representations
- Hallgren, Russell, et al.
- 2000
(Show Context)
Citation Context ...idered in the non-abelian case, although with limited success (see, for instance, Ettinger and Hyer [16], Ettinger, Hyer, and Knill [17], Rotteler and Beth [30], and Hallgren, Russell, and Ta-Shma [21=-=-=-]). No polynomial-time algorithm for the Hidden Subgroup problem is known for any class of non-abelian groups except for a special class of groups based on wreath products considered by Rotteler and B... |

37 |
Fast Monte Carlo algorithms for permutation groups
- Babai, Cooperman, et al.
- 1995
(Show Context)
Citation Context ...th n and the group operations are performed by a black-box (or group oracle) at unit cost. Black-box groups were introduced by Babai and Szemeredi [7] in 1984 and have since been studied extensively [=-=1, 2, 3, 4, 5, 6]-=-. Any ecient algorithm that works in the context of black-box groups of course remains ecient whenever the group oracle can be replaced by an ecient procedure for computing the group operations. In th... |

34 | Hidden subgroup states are almost orthogonal
- Ettinger, Høyer, et al.
- 1999
(Show Context)
Citation Context ...2] for further details.) The Hidden Subgroup problem has been considered in the non-abelian case, although with limited success (see, for instance, Ettinger and Hyer [16], Ettinger, Hyer, and Knill [1=-=7-=-], Rotteler and Beth [30], and Hallgren, Russell, and Ta-Shma [21]). No polynomial-time algorithm for the Hidden Subgroup problem is known for any class of non-abelian groups except for a special clas... |

32 |
Randomization in group algorithms: Conceptual questions
- Babai
- 1997
(Show Context)
Citation Context |

30 |
Quantum Computer Algorithms
- Mosca
- 1999
(Show Context)
Citation Context ...thin the framework of the Hidden Subgroup problem. In the black-box group setting, the Hidden Subgroup problem can be solved in quantum polynomial time whenever G is abelian, as demonstrated by Mosca =-=[27]-=-. Mosca also proved that several other interesting group-theoretic problems regarding abelian black-box groups can be reduced to the Hidden Subgroup problem, and thus can be computed in quantum polyno... |

30 |
Las Vegas algorithms for matrix groups
- Beals, Babai
(Show Context)
Citation Context ...th n and the group operations are performed by a black-box (or group oracle) at unit cost. Black-box groups were introduced by Babai and Szemeredi [7] in 1984 and have since been studied extensively [=-=1, 2, 3, 4, 5, 6, 9]-=-. Any ecient algorithm that works in the context of black-box groups of course remains ecient whenever the group oracle can be replaced by an ecient procedure for computing the group operations. In th... |

28 | Computing in solvable matrix groups
- Luks
- 1992
(Show Context)
Citation Context ...ng the order of G = hg1 ; : : : ; gk i for given elements g1 ; : : : ; gk 2 GL(n; F) under the assumption that G is solvable. The most ecient classical algorithm known for this problem is due to Luks =-=[31]-=-, and runs in time polynomial in the input size plus the largest prime other than p dividing jGj. Our quantum algorithm solves this problem in polynomial time without dependence on the primes dividing... |

20 |
Decomposing finite Abelian groups
- Cheung, Mosca
(Show Context)
Citation Context ...ox group, one can find the order of the group, and in fact one can decompose the group into a direct product of cyclic subgroups of prime power order, in polynomial time. 1 (See also Cheung and Mosca =-=[13]-=- for further details.) The Hidden Subgroup Problem has been considered in the non-abelian case, although with limited success (see Ettinger and Ho/yer [17], Ettinger, Ho/yer, and Knill [18], Grigni, S... |

19 |
Bounded round interactive proofs in finite groups
- Babai
- 1992
(Show Context)
Citation Context ...th n and the group operations are performed by a black-box (or group oracle) at unit cost. Black-box groups were introduced by Babai and Szemer'edi [7] in 1984 and have since been studied extensively =-=[1, 2, 3, 4, 5, 6, 9]-=-. Any efficient algorithm that works in the context of black-box groups of course remains efficient whenever the group oracle can be replaced by an efficient procedure for computing the group operatio... |

10 | Solvable black-box group problems are low for PP
- Arvind, Vinodchandran
- 1997
(Show Context)
Citation Context |

8 | Testing the shift–equivalence of polynomials using quantum mechanics
- Grigoriev
- 1996
(Show Context)
Citation Context ...d Ekert showed that Deutsch's Problem [15], Simon's Problem [32], ordersnding and computing discrete logarithms [31],snding hidden linear functions [10], testing self-shift-equivalence of polynomials =-=[19]-=-, and the Abelian Stabilizer Problem [25, 26] can all be solved in polynomial time within the framework of the Hidden Subgroup problem. In the black-box group setting, the Hidden Subgroup problem can ... |

8 |
Algebra: a Graduate Course. Brooks/Cole
- Isaacs
- 1994
(Show Context)
Citation Context ...circuits. The reader not familiar with quantum circuits is referred to Nielsen and Chuang [29]. We also assume the reader is familiar with the basic concepts of group theory (see, for example, Isaacs =-=[2-=-3]). Given a group G and elements g; h 2 G we dene the commutator of g and h, denoted [g; h], as [g; h] = g 1 h 1 gh, and for any two subgroups H;K G we write [H; K] to denote the subgroup of G gener... |

7 |
Bounded round interactive proofs in groups
- Babai
- 1992
(Show Context)
Citation Context |

5 |
Quantum Algorithms
- Høyer
- 2000
(Show Context)
Citation Context ... by the group structure of G. Kitaev's approach was further generalized by Brassard and Ho/yer [12], Jozsa [27], and Mosca and Ekert [33], who formulated the Hidden Subgroup Problem. (See also Ho/yer =-=[24]-=-.) The Hidden Subgroup Problem may be informally stated as follows. Given a finitely generated group G and an efficiently computable function f from G to some finitesset X such that f is constant and ... |

2 |
Efficient algorithms for some instances of the non-Abelian hidden subgroup problem
- Ivanyos, Magniez, et al.
- 2001
(Show Context)
Citation Context ...gh with limited success (see Ettinger and Hyer [17], Ettinger, Hyer, and Knill [18], Grigni, Schulman, Vazirani, and Vazirani [20], Hallgren, Russell, and Ta-Shma [23], Ivanyos, Magniez, and Santha [2=-=6-=-], and Rotteler and Beth [35]). Quantum polynomial-time algorithms forsnding non-abelian hidden subgroups are known for only limited classes ofsnite groups|most notably, the recent paper of Ivanyos, M... |

1 |
Decomposing Abelian groups
- Cheung, Mosca
- 2000
(Show Context)
Citation Context ...-box group, one cansnd the order of the group, and in fact one can decompose the group into a direct product of cyclic subgroups of prime power order, in polynomial time. 1 (See also Cheung and Mosca =-=[1-=-2] for further details.) The Hidden Subgroup problem has been considered in the non-abelian case, although with limited success (see, for instance, Ettinger and Hyer [16], Ettinger, Hyer, and Knill [1... |

1 |
Quantum Algorithms
- Hyer
- 2000
(Show Context)
Citation Context ...and the group action of Z k on X depends on the group structure of G. Kitaev's approach was further generalized by Brassard and Hyer [11], who formulated the Hidden Subgroup problem. (See also Hyer [2=-=2]-=- and Mosca and Ekert [28].) The Hidden Subgroup problem may be informally stated as follows. Given asnitely generated group G and an eciently computable function f from G to somesnite set X such that ... |

1 |
Succinct quantum proofs for properties of groups
- Watrous
- 2000
(Show Context)
Citation Context ...i h j g i 2 hh 1 ; : : : ; h l i for each i and j (as well as hh 1 ; : : : ; h l i hg 1 ; : : : ; g k i). See Babai [3] for more examples of problems reducing to order computation. In another paper [=-=33-=-] we have shown that there exist succinct quantum certicates for various group-theoretic properties, including the property that a given integer divides the order of a group (i.e., given an integer d ... |