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| The '''hidden subgroup problem''' ('''HSP''') is a topic of research in [[mathematics]] and [[theoretical computer science]]. The framework captures problems like [[Integer factorization|factoring]], [[Graph isomorphism problem|graph isomorphism]], and the [[shortest vector problem]]. This makes it especially important in the theory of quantum computing because [[Shor's algorithm|Shor's quantum algorithm]] for factoring is essentially equivalent to the hidden subgroup problem for [[Abelian group#Finite abelian groups|finite Abelian groups]], while the other problems correspond to finite groups that are not Abelian. | | The name of the author is Numbers. For a whilst she's been in South Dakota. To collect cash is a thing that I'm totally addicted to. In her professional lifestyle she is a payroll clerk but she's always needed her own company.<br><br>Here is my blog post :: [http://www.Youronlinepublishers.com/authWiki/AudreaocMalmrw std testing at home] |
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| ==Problem statement==
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| Given a [[group (mathematics)|group]] ''G'', a [[subgroup]] ''H'' ≤ ''G'', and a set ''X'', we say a function ''f'' : ''G'' → ''X'' '''hides''' the subgroup ''H'' if for all ''g''<sub>1</sub>, ''g''<sub>2</sub> ∈ ''G'',
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| ''f''(''g''<sub>1</sub>) = ''f''(''g''<sub>2</sub>) if and only if ''g''<sub>1</sub>''H'' = ''g''<sub>2</sub>''H'' for the [[coset]]s of ''H''. Equivalently, the function ''f'' is constant on the cosets of ''H'', while it is different between the different cosets of ''H''.
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| '''Hidden subgroup problem''': Let ''G'' be a group, ''X'' a finite set, and ''f'' : ''G'' → ''X'' a function that hides a subgroup ''H'' ≤ ''G''. The function ''f'' is given via an [[oracle machine|oracle]], which uses ''O''(log |''G''|+log|''X''|) bits. Using information gained from evaluations of ''f'' via its oracle, determine a generating set for ''H''.
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| A special case is when ''X'' is a group and ''f'' is a [[group homomorphism]] in which case ''H'' corresponds to the [[kernel (algebra)|kernel]] of ''f''.
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| ==Motivation==
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| The Hidden Subgroup Problem is especially important in the theory of [[quantum computer|quantum computing]] for the following reasons.
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| * [[Shor's algorithm|Shor's quantum algorithm]] for factoring and [[discrete logarithm]] (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for [[Abelian group#Finite abelian groups|finite Abelian groups]].
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| * The existence of efficient [[quantum algorithms]] for HSPs for certain [[non-abelian group|non-Abelian groups]] would imply efficient quantum algorithms for two major problems: the [[graph isomorphism problem]] and certain [[shortest vector problem]]s (SVPs) in lattices. More precisely, an efficient quantum algorithm for the HSP for the [[symmetric group]] would give a quantum algorithm for the graph isomorphism.<ref>{{cite arxiv|eprint=quant-ph/9901029|author1=Mark Ettinger|author2=Peter Høyer|title=A quantum observable for the graph isomorphism problem}}</ref> An efficient quantum algorithm for the HSP for the [[dihedral group]] would give a quantum algorithm for the poly(''n'') unique SVP.<ref>{{cite arxiv|eprint=cs/0304005|author=Oded Regev|title=Quantum computation and lattice problems}}</ref>
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| == Algorithms ==
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| There is a [[polynomial time]] quantum algorithm for solving HSP over finite [[Abelian group]]s. (In the case of hidden subgroup problem,
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| "a polynomial time algorithm" means an algorithm whose running time is a polynomial of the logarithm of the size of the group.) Shor's algorithm applies a particular case of this quantum algorithm.
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| For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle.<ref>{{cite arxiv|eprint=quant-ph/0401083|author=Mark Ettinger, Peter Hoyer, Emanuel Knill|title=The quantum query complexity of the hidden subgroup problem is polynomial}}</ref> This result, however, allows the quantum algorithm a running time that is exponential in log|''G''|. To design efficient algorithms for the graph isomorphism and SVP, one needs an algorithm for which both the number of oracle evaluations and the running time are polynomial.
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| The existence of such algorithm for arbitrary groups is open. Quantum polynomial time algorithms exist for certain subclasses of groups, such as semi-direct products of some [[Abelian group]]s.
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| The 'standard' approach to this problem involves: the creation of the quantum state <math>\frac{1}{\sqrt{|G|}}\sum_{x\in G}{|x\rangle \otimes |f(x)\rangle}</math>, a subsequent [[Quantum Fourier transform]] to the left register, after which this register gets sampled. This approach has been shown to be insufficient for the hidden subgroup problem for the symmetric group.<ref>{{cite arxiv|eprint=quant-ph/0511148|author=Sean Hallgren, Martin Roetteler, Pranab Sen|title=Limitations of Quantum Coset States for Graph Isomorphism}}</ref><ref>{{cite arxiv|eprint=quant-ph/0501056|author=Cristopher Moore, Alexander Russell, Leonard J. Schulman|title=The Symmetric Group Defies Strong Fourier Sampling: Part I}}</ref>
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| ==References==
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| <references/>
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| ==External links==
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| *[http://arxiv.org/abs/quant-ph/0012084 Richard Jozsa: Quantum factoring, discrete logarithms and the hidden subgroup problem]
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| * [http://arxiv.org/abs/quant-ph/0411037 Chris Lomont: The Hidden Subgroup Problem - Review and Open Problems]
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| * [http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=quant-ph&level=1&index1=14486 Hidden subgroup problem on arxiv.org]
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| [[Category:Group theory]]
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| [[Category:Quantum algorithms]]
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The name of the author is Numbers. For a whilst she's been in South Dakota. To collect cash is a thing that I'm totally addicted to. In her professional lifestyle she is a payroll clerk but she's always needed her own company.
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