Efficient quantum algorithm for identifying hidden polynomials

2009 ◽  
Vol 9 (3&4) ◽  
pp. 215-230
Author(s):  
T. Decker ◽  
J. Draisma ◽  
P. Wocjan
Keyword(s):  
Success Probability ◽  
Quantum Algorithm ◽  
Natural Generalization ◽  
Black Box ◽  
Linear Functions ◽  
Total Degree ◽  
Polynomial Functions ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
Classical Computer

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 contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of $d$ with constant probability and that has a running time that is only polylogarithmic in $d$.

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 $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.

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 On solving systems of random linear disequations

2008 ◽  
Vol 8 (6&7) ◽  
pp. 579-594
Author(s):  
G. Ivanyos
Keyword(s):  
Abelian Groups ◽  
Main Idea ◽  
Constant Factor ◽  
Linear Functions ◽  
Important Special Case ◽  
Hidden Subgroup Problem ◽  
Left Hand ◽  
Shift Problem ◽  
Group A ◽  
Subgroup Problem

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 functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.

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.

scholarly journals Finding hidden Borel subgroups of the general linear group

2012 ◽  
Vol 12 (7&8) ◽  
pp. 661-669
Author(s):  
Gabor Ivanyos
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Quantum Algorithm ◽  
General Linear ◽  
Base Field ◽  
Triangular Matrices ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
Lower Triangular ◽  
Lower Triangular Matrices

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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