scholarly journals From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

2005 ◽  
Author(s):  
D. Bacon ◽  
A.M. Childs ◽  
W. van Dam
Keyword(s):  
Semidirect Product ◽  
Quantum Algorithms ◽  
Hidden Subgroup Problem ◽  
Optimal Measurement ◽  
Subgroup Problem ◽  
Semidirect Product Groups

An improved query for the hidden subbroup problem

2014 ◽  
Vol 14 (5&6) ◽  
pp. 467-492
Author(s):  
Asif Shakeel
Keyword(s):  
Abelian Group ◽  
Standard Method ◽  
Finite Abelian Group ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Hidden Subgroup Problem ◽  
Subgroup Identification ◽  
Subgroup Problem ◽  
Query Algorithms

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 which the oracle functions take values in a finite abelian group. We apply our results to the case when the subgroups are drawn from a set of conjugate subgroups and obtain a success probability greater than that found by Moore and Russell.

scholarly journals For distinguishing conjugate Hidden subgroups, the pretty good measurement is as good as it gets

2007 ◽  
Vol 7 (8) ◽  
pp. 752-765
Author(s):  
C. Moore ◽  
A. Russell
Keyword(s):  
Dihedral Group ◽  
Hidden Subgroup Problem ◽  
Parent Group ◽  
Optimal Measurement ◽  
Subgroup Problem ◽  
Good Measurement ◽  
Hidden Subgroups

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 the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest.

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.

Efficient classical simulations of quantum Fourier transforms and Normalizer circuits over Abelian groups

2013 ◽  
Vol 13 (11&12) ◽  
pp. 1007-1037
Author(s):  
Maarten Van den Nest
Keyword(s):  
Fourier Transforms ◽  
Finite Abelian Group ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quadratic Functions ◽  
Matrix Formalism ◽  
Hidden Subgroup Problem ◽  
Linear Diophantine Equation ◽  
Subgroup Problem ◽  
Quantum Fourier Transforms

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 polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. Our result generalizes the Gottesman-Knill theorem: in particular, Clifford circuits for $d$-level qudits arise as normalizer circuits over the group ${\mathbf Z}_d^m$. We also highlight connections between normalizer circuits and Shor's factoring algorithm, and the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.

Quantum algorithms for shifted subset problems

2009 ◽  
Vol 9 (5&6) ◽  
pp. 500-512
Author(s):  
A. Montanaro
Keyword(s):  
Abelian Group ◽  
Time Complexity ◽  
Quantum Algorithms ◽  
Quantum States ◽  
Boolean Cube ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem

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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