scholarly journals On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems

2007 ◽  
Vol 7 (5&6) ◽  
pp. 504-521
Author(s):  
A.M. Childs ◽  
P. Wocjan
Keyword(s):  
Standard Method ◽  
Quantum Computer ◽  
Graph Isomorphism ◽  
Hidden Subgroup Problem ◽  
Shift Problem ◽  
Subgroup Problem

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 measurements, an exponential number of hidden shift states are required.

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).

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).

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}.

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