Quantum measurements for hidden subgroup problems with optimal sample complexity

2008 ◽  
Vol 8 (3&4) ◽  
pp. 345-358
Author(s):  
M. Hayashi ◽  
A. Kawachi ◽  
H. Kobayashi
Keyword(s):  
Log P ◽  
Sample Complexity ◽  
Arbitrary Group ◽  
Hidden Subgroup Problem ◽  
Optimal Sample ◽  
Subgroup Problem ◽  
Good Measurement ◽  
Hidden Subgroups ◽  
Decision Version ◽  
Candidate Set

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 the same prime order $p$, which implies that the decision version is at least as hard as the identification version in this case. In particular, it does so for the important \REVISE{cases} such as the dihedral and the symmetric hidden subgroup problems. Moreover, the upper bound of the identification is attained \REVISE{by a variant of the pretty good measurement}. \REVISE{This implies that the concept of the pretty good measurement is quite useful for identification of hidden subgroups over an arbitrary group with optimal sample complexity}.

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 ON THE COMPLEXITY OF THE HIDDEN SUBGROUP PROBLEM

2013 ◽  
Vol 24 (08) ◽  
pp. 1221-1234 ◽  
Author(s):  
STEPHEN FENNER ◽  
YONG ZHANG
Keyword(s):  
Quantum Computing ◽  
Computational Complexity ◽  
Polynomial Time ◽  
Permutation Groups ◽  
Dihedral Groups ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
The Relationship ◽  
Decision Version

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 nonadaptive program checkers for HIDDEN SUBGROUP and its decision version.

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 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.

Limitations of single coset states and quantum algorithms for code equivalence

2015 ◽  
Vol 15 (3&4) ◽  
pp. 260-294
Author(s):  
Hang Dinh ◽  
Cristopher Moore ◽  
Alexander Russell
Keyword(s):  
Linear Codes ◽  
Quantum Algorithms ◽  
Graph Isomorphism ◽  
Hidden Subgroup Problem ◽  
Public Key Cryptosystems ◽  
Negative Results ◽  
Code Equivalence ◽  
Fourier Sampling ◽  
Subgroup Problem ◽  
Hidden Subgroups

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 quantum attacks. In this article we extend previous negative results of quantum Fourier sampling for Graph Isomorphism, which corresponds to hidden subgroups of order two (over S_n, to several cases corresponding to larger hidden subgroups. For one case, we strengthen some results of Kempe, Pyber, and Shalev on the Hidden Subgroup Problem over the symmetric group. In another case, we show the failure of quantum Fourier sampling on the Hidden Subgroup Problem over the general linear group GL_2(\FF_q). The most important case corresponds to Code Equivalence, the problem of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. Our results suggest that for many codes of interest---including generalized Reed Solomon codes, alternant codes, and Reed-Muller codes---solving these instances of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms.

Quantum solution to the hidden subgroup problem for Poly-Near-Hamiltonian groups

2004 ◽  
Vol 4 (3) ◽  
pp. 229-235
Author(s):  
D. Gavinsky
Keyword(s):  
Quantum Computing ◽  
Efficient Algorithm ◽  
Abelian Groups ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
Hamiltonian Groups ◽  
Quantum Solution

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{[G:\MG] = 1} for Abelian groups). This measure was considered by Grigni, Schulman, Vazirani and Vazirani, who showed that whenever \f{[G:\MG]\in\exp(O(\log^{1/2}n))} the corresponding HSP can be solved efficiently (under certain assumptions). We show that whenever \f{[G:\MG]\in\poly(n)} the corresponding HSP can be solved efficiently, under the same assumptions (actually, we solve a slightly more general case of the HSP and also show that some assumptions may be relaxed).

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 $A$.

scholarly journals Hamiltonian simulation with optimal sample complexity

2017 ◽  
Vol 3 (1) ◽  
Author(s):  
Shelby Kimmel ◽  
Cedric Yen-Yu Lin ◽  
Guang Hao Low ◽  
Maris Ozols ◽  
Theodore J. Yoder
Keyword(s):  
Sample Complexity ◽  
Optimal Sample ◽  
Hamiltonian Simulation
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