A quantum algorithm for the dihedral hidden subgroup problem based on lattice basis reduction algorithm

2014 ◽  
Vol 59 (21) ◽  
pp. 2552-2557 ◽  
Author(s):  
Fada Li ◽  
Wansu Bao ◽  
Xiangqun Fu
Keyword(s):  
Quantum Algorithm ◽  
Reduction Algorithm ◽  
Lattice Basis Reduction ◽  
Hidden Subgroup Problem ◽  
Basis Reduction ◽  
Subgroup Problem ◽  
Lattice Basis Reduction Algorithm

scholarly journals Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups

2007 ◽  
Vol 7 (5&6) ◽  
pp. 559-570
Author(s):  
Y. Inui ◽  
F. Le Gall
Keyword(s):  
Direct Product ◽  
Polynomial Time ◽  
Efficient Algorithm ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Black Box ◽  
Hidden Subgroup Problem ◽  
Time Quantum ◽  
Subgroup Problem

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 result and present an efficient algorithm solving the HSP over the groups $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.

Hidden symmetry detection on a quantum computer

2007 ◽  
Vol 7 (1&2) ◽  
pp. 83-92
Author(s):  
R. Schutzhold ◽  
W.G. Unruh
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Self Similarity ◽  
Hidden Subgroup Problem ◽  
Hidden Symmetry ◽  
Speed Up ◽  
Subgroup Problem ◽  
Discrete Scale

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 invariance).

EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM

2003 ◽  
Vol 14 (05) ◽  
pp. 723-739 ◽  
Author(s):  
GÁBOR IVANYOS ◽  
FRÉDÉRIC MAGNIEZ ◽  
MIKLOS SANTHA
Keyword(s):  
Commutator Subgroup ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Solvable Groups ◽  
Normal Subgroups ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem ◽  
Small Index ◽  
Hidden Subgroups

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.

scholarly journals An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups

2017 ◽  
Vol 18 (2) ◽  
pp. 0215 ◽  
Author(s):  
Demerson Nunes Gonçalves ◽  
Tharso D Fernandes ◽  
C M M Cosme
Keyword(s):  
Positive Integer ◽  
Quantum Computation ◽  
Prime Number ◽  
Abelian Groups ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Prime Factorization ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem

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.

scholarly journals How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem

2008 ◽  
Vol 8 (5) ◽  
pp. 438-487
Author(s):  
D. Bacon
Keyword(s):  
Finite Groups ◽  
Algorithm Design ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Quantum Computers ◽  
Hidden Subgroup Problem ◽  
Unitary Transform ◽  
Subgroup Problem ◽  
Good Measurement ◽  
Optimal Measurements

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 group. This gives a new representation theoretic explanation for the pretty good measurement derived algorithm for efficiently solving the Heisenberg hidden subgroup problem and provides evidence that Clebsch-Gordan transforms over finite groups are a new primitive in quantum algorithm design.

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