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Journal articles on the topic 'Dihedral Hidden Subgroup Problem'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Kobayashi, Hirotada, and François Le Gall. "Dihedral Hidden Subgroup Problem: A Survey." IPSJ Digital Courier 1 (2005): 470–77. http://dx.doi.org/10.2197/ipsjdc.1.470.

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2

FENNER, STEPHEN, and YONG ZHANG. "ON THE COMPLEXITY OF THE HIDDEN SUBGROUP PROBLEM." International Journal of Foundations of Computer Science 24, no. 08 (2013): 1221–34. http://dx.doi.org/10.1142/s0129054113500305.

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We study the computational complexity of the HIDDEN SUBGROUP problem, a well-studied problem in quantum computing. First we show that several proposed generalizations or variants of this problem, including HIDDEN COSET, HIDDEN SHIFT, and ORBIT COSET, are all equivalent or reducible to HIDDEN SUBGROUP. Then we study the relationship between the decision version and search version of HIDDEN SUBGROUP over various group classes. We show that the two versions are polynomial-time equivalent over permutation groups, and over dihedral groups given the order of the group is smooth. Finally, we give non

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3

Hayashi, M., A. Kawachi, and H. Kobayashi. "Quantum measurements for hidden subgroup problems with optimal sample complexity." Quantum Information and Computation 8, no. 3&4 (2008): 345–58. http://dx.doi.org/10.26421/qic8.3-4-8.

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One of the central issues in the hidden subgroup problem is to bound the sample complexity, i.e., the number of identical samples of coset states sufficient and necessary to solve the problem. In this paper, we present general bounds for the sample complexity of the identification and decision versions of the hidden subgroup problem. As a consequence of the bounds, we show that the sample complexity for both of the decision and identification versions is $\Theta(\log|\HH|/\log p)$ for a candidate set $\HH$ of hidden subgroups in the case \REVISE{where the candidate nontrivial subgroups} have t

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4

Kuperberg, Greg. "A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem." SIAM Journal on Computing 35, no. 1 (2005): 170–88. http://dx.doi.org/10.1137/s0097539703436345.

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5

Moore, C., and A. Russell. "For distinguishing conjugate Hidden subgroups, the pretty good measurement is as good as it gets." Quantum Information and Computation 7, no. 8 (2007): 752–65. http://dx.doi.org/10.26421/qic7.8-5.

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Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group $D_n$ in the case where the hidden subgroup is chosen uniformly from the $n$ involutions. We show that, for any group and any subgroup $H$, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of $H$. We go on to show that when $H$ forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that t

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6

Li, Fada, Wansu Bao, and Xiangqun Fu. "A quantum algorithm for the dihedral hidden subgroup problem based on lattice basis reduction algorithm." Chinese Science Bulletin 59, no. 21 (2014): 2552–57. http://dx.doi.org/10.1007/s11434-014-0344-0.

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7

Kumar, Gautam, and Hemraj Saini. "Novel Noncommutative Cryptography Scheme Using Extra Special Group." Security and Communication Networks 2017 (2017): 1–21. http://dx.doi.org/10.1155/2017/9036382.

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Noncommutative cryptography (NCC) is truly a fascinating area with great hope of advancing performance and security for high end applications. It provides a high level of safety measures. The basis of this group is established on the hidden subgroup or subfield problem (HSP). The major focus in this manuscript is to establish the cryptographic schemes on the extra special group (ESG). ESG is showing one of the most appropriate noncommutative platforms for the solution of an open problem. The working principle is based on the random polynomials chosen by the communicating parties to secure key

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8

Sdroievski, Nicollas M., Murilo V. G. da Silva, and André L. Vignatti. "The Hidden Subgroup Problem and MKTP." Theoretical Computer Science 795 (November 2019): 204–12. http://dx.doi.org/10.1016/j.tcs.2019.06.012.

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9

Bacon, D. "How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem." Quantum Information and Computation 8, no. 5 (2008): 438–87. http://dx.doi.org/10.26421/qic8.5-6.

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It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using pretty good measurements for obtaining optimal measurements in the hidden subgroup problem. Here we show how to solve the Heisenberg hidden subgroup problem using arguments based instead on the symmetry of certain hidden subgroup states. The symmetry we consider leads naturally to a unitary transform known as the Clebsch-Gordan transform over the Heisenberg g

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10

IVANYOS, GÁBOR, FRÉDÉRIC MAGNIEZ, and MIKLOS SANTHA. "EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM." International Journal of Foundations of Computer Science 14, no. 05 (2003): 723–39. http://dx.doi.org/10.1142/s0129054103001996.

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In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

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11

Jozsa, R. "Quantum factoring, discrete logarithms, and the hidden subgroup problem." Computing in Science & Engineering 3, no. 2 (2001): 34–43. http://dx.doi.org/10.1109/5992.909000.

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12

Grigni, Michelangelo, Leonard J. Schulman, Monica Vazirani, and Umesh Vazirani. "Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem." Combinatorica 24, no. 1 (2004): 137–54. http://dx.doi.org/10.1007/s00493-004-0009-8.

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13

Shakeel, Asif. "An improved query for the hidden subbroup problem." Quantum Information and Computation 14, no. 5&6 (2014): 467–92. http://dx.doi.org/10.26421/qic14.5-6-6.

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The Hidden Subgroup Problem (HSP) is at the forefront of problems in quantum algorithms. In this paper, we introduce a new query, the \textit{character} query, generalizing the well-known phase kickback trick that was first used successfully to efficiently solve Deutsch's problem. An equal superposition query with $\vert 0 \rangle$ in the response register is typically used in the ``standard method" of single-query algorithms for the HSP. The proposed character query improves over this query by maximizing the success probability of subgroup identification under a uniform prior, for the HSP in

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14

Koiran, Pascal, Vincent Nesme, and Natacha Portier. "The quantum query complexity of the abelian hidden subgroup problem." Theoretical Computer Science 380, no. 1-2 (2007): 115–26. http://dx.doi.org/10.1016/j.tcs.2007.02.057.

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15

Hallgren, Sean, Alexander Russell, and Amnon Ta-Shma. "The Hidden Subgroup Problem and Quantum Computation Using Group Representations." SIAM Journal on Computing 32, no. 4 (2003): 916–34. http://dx.doi.org/10.1137/s009753970139450x.

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16

Bae, Eunok, and Soojoon Lee. "Quantum Algorithm for Solving the Continuous Hidden Symmetry Subgroup Problem." IEEE Access 9 (2021): 93248–54. http://dx.doi.org/10.1109/access.2021.3092723.

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17

Ettinger, Mark, Peter Høyer, and Emanuel Knill. "The quantum query complexity of the hidden subgroup problem is polynomial." Information Processing Letters 91, no. 1 (2004): 43–48. http://dx.doi.org/10.1016/j.ipl.2004.01.024.

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18

Ivanyos, Gabor. "Finding hidden Borel subgroups of the general linear group." Quantum Information and Computation 12, no. 7&8 (2012): 661–69. http://dx.doi.org/10.26421/qic12.7-8-10.

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We present a quantum algorithm for solving the hidden subgroup problem in the general linear group over a finite field where the hidden subgroup is promised to be a conjugate of the group of the invertible lower triangular matrices. The complexity of the algorithm is polynomial when size of the base field is not much smaller than the degree.

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19

Childs, A. M., and P. Wocjan. "On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems." Quantum Information and Computation 7, no. 5&6 (2007): 504–21. http://dx.doi.org/10.26421/qic7.5-6-6.

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We consider an approach to deciding isomorphism of rigid n-vertex graphs (and related isomorphism problems) by solving a nonabelian hidden shift problem on a quantum computer using the standard method. Such an approach is arguably more natural than viewing the problem as a hidden subgroup problem. We prove that the hidden shift approach to rigid graph isomorphism is hard in two senses. First, we prove that \Omega(n) copies of the hidden shift states are necessary to solve the problem (whereas O(n\log n) copies are sufficient). Second, we prove that if one is restricted to single-register measu

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20

Gavinsky, D. "Quantum solution to the hidden subgroup problem for Poly-Near-Hamiltonian groups." Quantum Information and Computation 4, no. 3 (2004): 229–35. http://dx.doi.org/10.26421/qic4.3-8.

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The Hidden Subgroup Problem (HSP) has been widely studied in the context of quantum computing and is known to be efficiently solvable for Abelian groups, yet appears to be difficult for many non-Abelian ones. An efficient algorithm for the HSP over a group \f G\ runs in time polynomial in \f{n\deq\log|G|.} For any subgroup \f H\ of \f G, let \f{N(H)} denote the normalizer of \f H. Let \MG\ denote the intersection of all normalizers in \f G (i.e., \f{\MG=\cap_{H\leq G}N(H)}). \MG\ is always a subgroup of \f G and the index \f{[G:\MG]} can be taken as a measure of ``how non-Abelian'' \f G is (\f

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21

Ivanyos, G. "On solving systems of random linear disequations." Quantum Information and Computation 8, no. 6&7 (2008): 579–94. http://dx.doi.org/10.26421/qic8.6-7-2.

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An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of f

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22

Radhakrishnan, Jaikumar, Martin Rötteler, and Pranab Sen. "Random Measurement Bases, Quantum State Distinction and Applications to the Hidden Subgroup Problem." Algorithmica 55, no. 3 (2008): 490–516. http://dx.doi.org/10.1007/s00453-008-9231-x.

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23

Ivanyos, Gábor, Luc Sanselme, and Miklos Santha. "An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups." Algorithmica 62, no. 1-2 (2010): 480–98. http://dx.doi.org/10.1007/s00453-010-9467-0.

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24

Inui, Y., and F. Le Gall. "Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups." Quantum Information and Computation 7, no. 5&6 (2007): 559–70. http://dx.doi.org/10.26421/qic7.5-6-9.

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In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$, for $p$ and $q$ prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, where $p$ is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this

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25

Decker, T., J. Draisma, and P. Wocjan. "Efficient quantum algorithm for identifying hidden polynomials." Quantum Information and Computation 9, no. 3&4 (2009): 215–30. http://dx.doi.org/10.26421/qic9.3-4-3.

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We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with $d$ elements. The hidden functions of the generalized problem are not restricted to be linear but can also be $m$-variate polynomial functions of total degree $n\geq 2$. The problem of identifying hidden $m$-variate polynomials of degree less or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a constant success probability. In

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26

Gonçalves, Demerson Nunes, Tharso D. Fernandes, and C. M. M. Cosme. "An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups." TEMA (São Carlos) 18, no. 2 (2017): 0215. http://dx.doi.org/10.5540/tema.2017.018.02.0215.

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The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups $\Z_{N}\rtimes\Z_{q^s}$ where $N$ is an integer with a special prime factorization, $q$ prime number and $s$ any positive integer.

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27

Schutzhold, R., and W. G. Unruh. "Hidden symmetry detection on a quantum computer." Quantum Information and Computation 7, no. 1&2 (2007): 83–92. http://dx.doi.org/10.26421/qic7.1-2-4.

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The fastest quantum algorithms (for the solution of classical computational tasks) known so far are basically variations of the hidden subgroup problem with {$f(U[x])=f(x)$}. Following a discussion regarding which tasks might be solved efficiently by quantum computers, it will be demonstrated by means of a simple example, that the detection of more general hidden (two-point) symmetries {$V\{f(x),f(U[x])\}=0$} by a quantum algorithm can also admit an exponential speed-up. E.g., one member of this class of symmetries {$V\{f(x),f(U[x])\}=0$} is discrete self-similarity (or discrete scale invarian

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28

Dinh, Hang, Cristopher Moore, and Alexander Russell. "Limitations of single coset states and quantum algorithms for code equivalence." Quantum Information and Computation 15, no. 3&4 (2015): 260–94. http://dx.doi.org/10.26421/qic15.3-4-4.

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Quantum computers can break the RSA, El Gamal, and elliptic curve public-key cryptosystems, as they can efficiently factor integers and extract discrete logarithms. The power of such quantum attacks lies in \emph{quantum Fourier sampling}, an algorithmic paradigm based on generating and measuring coset states. %This motivates the investigation of the power or limitations of quantum Fourier sampling, especially in attacking candidates for ``post-quantum'' cryptosystems -- classical cryptosystems that can be implemented with today's computers but will remain secure even in the presence of quantu

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29

Montanaro, A. "Quantum algorithms for shifted subset problems." Quantum Information and Computation 9, no. 5&6 (2009): 500–512. http://dx.doi.org/10.26421/qic9.5-6-10.

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We consider a recently proposed generalisation of the abelian hidden subgroup problem: the {\em shifted subset problem}. The problem is to determine a subset $S$ of some abelian group, given access to quantum states of the form $\ket{S+x}$, for some unknown shift $x$. We give quantum algorithms to find Hamming spheres and other subsets of the boolean cube $\{0,1\}^n$. The algorithms have time complexity polynomial in $n$ and give rise to exponential separations from classical computation.

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30

Denney, DA, C. Moore, and A. Russell. "Finding conjugate stabilizer subgroups in $\PSL$ and related groups." Quantum Information and Computation 10, no. 3&4 (2010): 282–91. http://dx.doi.org/10.26421/qic10.3-4-8.

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We reduce a case of the hidden subgroup problem (HSP) in $\SL$, $\PSL$, and $\PGL$, three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group $\AGL$. These groups act on projective space in an ``almost'' 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive resu

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31

Elgharabawy, Ayman, Mukesh Prasad, and Chin-Teng Lin. "Subgroup Preference Neural Network." Sensors 21, no. 18 (2021): 6104. http://dx.doi.org/10.3390/s21186104.

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Subgroup label ranking aims to rank groups of labels using a single ranking model, is a new problem faced in preference learning. This paper introduces the Subgroup Preference Neural Network (SGPNN) that combines multiple networks have different activation function, learning rate, and output layer into one artificial neural network (ANN) to discover the hidden relation between the subgroups’ multi-labels. The SGPNN is a feedforward (FF), partially connected network that has a single middle layer and uses stairstep (SS) multi-valued activation function to enhance the prediction’s probability an

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32

Chi, Dong Pyo, Jeong San Kim, and Soojoon Lee. "Quantum algorithms for the hidden subgroup problem on some semi-direct product groups by reduction to Abelian cases." Physics Letters A 359, no. 2 (2006): 114–16. http://dx.doi.org/10.1016/j.physleta.2006.06.014.

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33

Heinenken, Hermann. "Fitting classes of certain metanilpotent groups." Glasgow Mathematical Journal 36, no. 2 (1994): 185–95. http://dx.doi.org/10.1017/s001708950003072x.

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There are two families of group classes that are of particular interest for clearing up the structure of finite soluble groups: Saturated formations and Fitting classes. In both cases there is a unique conjugacy class of subgroups which are maximal as members of the respective class combined with the property of being suitably mapped by homomorphisms (in the case of saturated formations) or intersecting suitably with normal subgroups (when considering Fitting classes). While it does not seem too difficult, however, to determine the smallest saturated formation containing a given group, the sam

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34

Bub, J. "Quantum computaton from a quantum logical perspective." Quantum Information and Computation 7, no. 4 (2007): 281–96. http://dx.doi.org/10.26421/qic7.4-1.

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It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be s

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35

Van den Nest, Maarten. "Efficient classical simulations of quantum Fourier transforms and Normalizer circuits over Abelian groups." Quantum Information and Computation 13, no. 11&12 (2013): 1007–37. http://dx.doi.org/10.26421/qic13.11-12-7.

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The quantum Fourier transform (QFT) is an important ingredient in various quantum algorithms which achieve superpolynomial speed-ups over classical computers. In this paper we study under which conditions the QFT can be simulated efficiently classically. We introduce a class of quantum circuits, called \emph{normalizer circuits}: a normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. In our main result we prove that all normalizer circuits have polynom

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36

Berkovich, Ya G., G. A. Freiman, and Cheryl E. Praeger. "Small squaring and cubing properties for finite groups." Bulletin of the Australian Mathematical Society 44, no. 3 (1991): 429–50. http://dx.doi.org/10.1017/s0004972700029932.

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A group G is said to have the small squaring property on k-sets if |K2| < k2 for all k-element subsets K of G, and is said to have the small cubing property on k-sets if |K3| < k3 for all k-element subsets K. It is shown that a finite nonabelian group with the small squaring property on 3-sets is either a 2-group or is of the form TP with T a normal abelian odd order subgroup and P a nontrivial 2-group such that Q = Cp(T) has index 2 in P and P inverts T. Moreover either P is abelian and Q is elementary abelian, or Q is abelian and each element of P − Q inverts Q. Conversely each group o

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37

Biasse, Jean-François, та Fang Song. "On the quantum attacks against schemes relying on the hardness of finding a short generator of an ideal in ℚ(𝜁2𝑠 )". Journal of Mathematical Cryptology 13, № 3-4 (2019): 151–68. http://dx.doi.org/10.1515/jmc-2015-0046.

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Abstract A family of ring-based cryptosystems, including the multilinear maps of Garg, Gentry and Halevi [Candidate multilinear maps from ideal lattices, Advances in Cryptology—EUROCRYPT 2013, Lecture Notes in Comput. Sci. 7881, Springer, Heidelberg 2013, 1–17] and the fully homomorphic encryption scheme of Smart and Vercauteren [Fully homomorphic encryption with relatively small key and ciphertext sizes, Public Key Cryptography—PKC 2010, Lecture Notes in Comput. Sci. 6056, Springer, Berlin 2010, 420–443], are based on the hardness of finding a short generator of a principal ideal (short-PIP)

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38

Childs, Andrew M., and Gábor Ivanyos. "Quantum computation of discrete logarithms in semigroups." Journal of Mathematical Cryptology 8, no. 4 (2014). http://dx.doi.org/10.1515/jmc-2013-0038.

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AbstractWe describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and the discrete logarithm problem as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks. In contrast, we show that some generalizations of the discrete logarithm problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete logarithm problem in semigroups to the dihedral hidden subgroup problem, and we show that the c

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39

Bacon, Dave, and Thomas Decker. "Optimal single-copy measurement for the hidden-subgroup problem." Physical Review A 77, no. 3 (2008). http://dx.doi.org/10.1103/physreva.77.032335.

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40

Panduranga Rao, M. V. "Solving a hidden subgroup problem using the adiabatic quantum-computing paradigm." Physical Review A 67, no. 5 (2003). http://dx.doi.org/10.1103/physreva.67.052306.

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41

Panduranga Rao, M. V. "Erratum: Solving a hidden subgroup problem using the adiabatic quantum-computing paradigm [Phys. Rev. A67, 052306 (2003)]." Physical Review A 73, no. 1 (2006). http://dx.doi.org/10.1103/physreva.73.019902.

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42

Bucicovschi, Orest, Daniel Copeland, David Meyer, and James Pommersheim. "Single-query quantum algorithms for symmetric problems." Quantum Information and Computation, January 2016, 19–38. http://dx.doi.org/10.26421/qic16.1-2-2.

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Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered n-tuple of group elements is to be determined by querying elements of the tuple. For any finite group G, we give an algorithm to find the product of two elements of G with a single quantum query with probability 2/|G|. This generalizes Deutsch’s Algorithm from Z2 to an ar

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