en>Vlad Patryshev: /* Examples */ Rephrased.

2012-12-25T19:40:36Z

Examples: Rephrased.

← Older revision		Revision as of 21:40, 25 December 2012
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	~~Alyson~~ is the title ~~people use~~ to ~~call me~~ and ~~I think~~ it ~~sounds quite good when you say~~ it. ~~To play lacross~~ is ~~something I really enjoy doing~~. ~~Distributing production~~ is ~~how he tends~~ to ~~make~~ a ~~living~~. ~~For years he~~'s ~~been residing~~ in ~~Mississippi~~ and ~~he doesn't strategy on changing it~~.<br><br>~~Also visit my web site free tarot readings~~ (~~[http~~://~~formalarmour~~.~~com~~/~~index~~.~~php?do~~=~~/profile~~-~~26947/info/ http~~:~~//formalarmour.com~~])		In [[computational complexity theory]] and [[quantum computing]], '''Simon's problem''' is a computational problem in the model of [[decision tree complexity]] or query complexity, conceived by Daniel Simon in 1994.<ref name=Simon1994>{{Citation
			\| title = On the power of quantum computation
			\| url = http://citeseer.ist.psu.edu/viewdoc/download;jsessionid=0D01FC6F66C81F349B409BAE3F515CAC?doi=10.1.1.51.5477&rep=rep1&type=pdf
			\| year = 1994
			\| author = Simon, D.R.
			\| journal = Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on
			\| pages = 116–123
			\| accessdate = 2011-06-06
			}}</ref> Simon exhibited a [[quantum algorithm]], usually called '''Simon's algorithm''' that solves the problem exponentially faster than any (deterministic or [[probabilistic algorithms\|probabilistic]]) classical algorithm.

			Simon's algorithm uses <math>O(n)</math> queries to the black box, whereas the best classical probabilistic algorithm necessarily needs at least <math>\Omega(2^{n/2})</math> queries. It is also known that Simon's algorithm is optimal in the sense that ''any'' quantum algorithm to solve this problem requires <math>\Omega(n)</math> queries.<ref>{{Citation
			\| doi = 10.1016/j.tcs.2007.02.057
			\| title = The quantum query complexity of the abelian hidden subgroup problem
			\| url = http://perso.ens-lyon.fr/pascal.koiran/Publis/lip.05-17.ps
			\| year = 2007
			\| journal = Theoretical Computer Science
			\| pages = 115–126
			\| volume = 380
			\| issue = 1-2
			\| last1 = Koiran \| first1 = P.
			\| last2 = Nesme \| first2 = V.
			\| last3 = Portier \| first3 = N.
			\| accessdate = 2011-06-06 }}</ref><ref>{{Citation
			\| title = A quantum lower bound for the query complexity of Simon's Problem
			\| url = http://perso.ens-lyon.fr/pascal.koiran/Publis/icalp05.ps
			\| arxiv = quant-ph/0501060
			\| year = 2005
			\| journal = Proc. ICALP
			\| pages = 1287–1298
			\| volume = 3580
			\| last1 = Koiran \| first1 = P.
			\| last2 = Nesme \| first2 = V.
			\| last3 = Portier \| first3 = N.
			\| accessdate = 2011-06-06
			}}</ref>
			This problem yields an oracle separation between [[BPP (complexity)\|BPP]] and [[BQP]], unlike the separation provided by the [[Deutsch-Jozsa algorithm]], which separates [[P (complexity)\|P]] and [[EQP (complexity)\|EQP]].

			Although the problem itself is of little practical value it is interesting because it provides an exponential speedup over any classical algorithm{{Citation needed\|date=May 2012}}. Moreover, it was also the inspiration for [[Shor's algorithm]]. Both problems are special cases of the abelian [[hidden subgroup problem]], which is now known to have efficient quantum algorithms.

			==Problem description and algorithm==

			The input to the problem is a function (implemented by a black box) <math>f:\{0,1\}^n\rightarrow \{0,1\}^n</math>, promised to satisfy the property that for some <math>s\in \{0,1\}^n</math> we have for all <math>y,z\in\{0,1\}^n </math>, <math>f(y)=f(z)</math> if and only if <math>y=z</math> or <math>y \oplus z =s</math>. Note that the case of <math>s=0^n</math> is allowed, and corresponds to <math>f</math> being a permutation. The problem then is to find ''s''.

			The set of ''n''-bit strings is a <math>\mathbb{Z}_2</math> vector space under bitwise [[XOR]]. Given the promise, the preimage of ''f'' is either empty, or forms [[coset]]s with ''n''-1 dimensions. Using quantum algorithms, we can, with arbitrarily high probability determine the basis vectors spanning this ''n''-1 subspace since ''s'' is a vector orthogonal to all of the basis vectors.

			[[Image:Simons algorithm.svg\|300px\|thumb\|right\|Quantum subroutine in Simon's algorithm]]
			Consider the [[Hilbert space]] consisting of the tensor product of the Hilbert space of input strings, and output strings. Using [[Hadamard_transform#Hadamard_gate_operations\|Hadamard operation]]s, we can prepare the initial state
			:<math>\sum_x \left\| x \right\rangle \left\| 0 \right\rangle</math>
			and then call the oracle to transform this state to
			:<math>\sum_x \left\| x \right\rangle \left\| f(x) \right\rangle</math>
			Hadamard transforms convert this state to
			:<math>\sum_y \sum_x (-1)^{x.y} \left\| y \right\rangle \left\| f(x) \right\rangle</math>
			We perform a simultaneous measurement of both registers. If <math>s \cdot y = 1</math>, we have [[destructive interference]]. So, only the subspace <math>s \cdot y = 0</math> is picked out. Given enough samples of ''y'', we can figure out the ''n''-1 basis vectors, and compute ''s''.

			== See also ==

			*[[Deutsch-Jozsa algorithm]]

			==References==
			{{reflist}}

			{{Quantum computing}}

			[[Category:Quantum algorithms]]

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2007-10-08T00:59:36Z

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en>Vlad Patryshev: /* Examples */ Rephrased.

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