en>Fshtea: /* Relationship to other complexity classes */

2013-09-16T13:30:06Z

Relationship to other complexity classes

← Older revision		Revision as of 15:30, 16 September 2013
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	~~When in search~~ of the ~~proper hemorrhoid treatment you will need to consider~~ a ~~limited significant factors such as~~, ~~that you you think you want~~, if there is a ~~terrific amount~~ of ~~recovery required~~, ~~and how lengthy~~ it ~~takes~~ to ~~get results~~. ~~In this short article we will learn~~ the ~~answers~~ to ~~all~~ of ~~these issues~~, ~~offering we~~ the ~~answer we require to find~~ the ~~right hemorrhoid treatment.~~<br><br>~~Now which~~ the ~~hemorrhoids aren~~'~~t getting worse~~, it's time ~~to use [http~~://~~hemorrhoidtreatmentfix~~.~~com/hemorrhoid~~-~~relief hemorrhoid relief~~] ~~that might relieve itching and pain~~. ~~Soaking in~~ a a ~~sitz bathtub for about fifteen minutes at~~ a ~~time~~ is ~~one method~~ to ~~discover relief~~. ~~You~~ are ~~able~~ to ~~do this multiple occasions a day~~. ~~You~~ are ~~able to furthermore heat~~ a ~~wet towel plus apply it~~ to ~~the hemorrhoid region. Ice~~ for ~~ten minutes, followed by~~ a ~~warm compress~~ for an ~~equal~~ or ~~slightly longer length~~ of ~~time~~ is ~~another method to relieve a pain plus itching~~.<br><br>~~There are several treatments that may be selected for hemorrhoid~~. ~~The first and the many popular~~ is ~~the cream plus ointment. These~~ are ~~to be rubbed onto the affected part of the anus~~. ~~It helps to soothe the already inflamed blood vessels plus~~ a ~~momentary relief is achieved. There is~~ a ~~relaxation of the tissues of the rectal column thus far~~ the ~~hemorrhoid~~ is ~~not so much bulged~~. ~~If there~~ is ~~a bulge though~~, ~~the pain relief could not do so much to help~~.<br><br>~~Hemorrhoid Surgery~~ - ~~This is just suggested if your hemorrhoids are thus severe that no over the counter or home treatment is providing we any kind of relief~~.<br><br>~~Your wellness food store has a wealth of natural hemorrhoid treatments~~, ~~all of which contain safe ingredients and are effortless~~ on ~~the purse~~. ~~Barberry~~, ~~Butcher~~'~~s Brew~~, ~~Witch Hazel~~, ~~Horse Chestnut and Slippery Elm are all renowned herbal elements for giving relief whenever piles flare up~~.<br><br>~~Believe me I recognize. I recognize how painful~~, ~~inconvenient~~ and ~~embarrassing hemorrhoids~~ is. ~~For me the big problem was the itching~~. ~~I mean what may you possibly do~~ to ~~relieve~~ the ~~itch whenever you~~'~~re sitting down all day long inside a busy workplace surrounded by colleagues?<br><br>It~~ is ~~pretty challenging now~~ to ~~suffer from hemorrhoid~~. ~~You deal with serious redness, irritation~~, ~~swelling~~, and ~~pain~~. ~~You are typically uncomfortable~~. ~~But have belief which you'll feel better with a hemorrhoid treatment~~. ~~You may receive rid~~ of ~~the hemorrhoid in no time perhaps~~ by ~~using petroleum jelly, using ointment phenylephrine or Preparation H, or using soft cotton underwear~~. ~~We really need to provide a try any~~ of ~~them.~~		{{unsolved\|computer science\|What is the relationship between '''BQP''' and '''NP'''?}}
			[[Image:BQP complexity class diagram.svg\|thumb\|The suspected relationship of '''BQP''' to other problem spaces<ref>Michael Nielsen and Isaac Chuang (2000). ''Quantum Computation and Quantum Information''. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.</ref>]]
			In [[computational complexity theory]], '''BQP''' (bounded error quantum polynomial time) is the class of [[decision problems]] solvable by a [[quantum computer]] in [[polynomial time]], with an error probability of at most 1/3 for all instances. It is the quantum analogue of the complexity class '''[[Bounded-error probabilistic polynomial\|BPP]]'''.

			In other words, there is an [[algorithm]] for a quantum computer (a [[quantum algorithm]]) that solves the decision problem with ''high'' probability and is guaranteed to run in polynomial time. On any given run of the algorithm, it has a probability of at most 1/3 that it will give the wrong answer.

			Similarly to other "bounded error" probabilistic classes the choice of 1/3 in the definition is arbitrary. We can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the [[Chernoff bound]]. Detailed analysis shows that the complexity class is unchanged by allowing error as high as 1/2 − ''n''<sup>−''c''</sup> on the one hand, or requiring error as small as 2<sup>−''n<sup>c</sup>''</sup> on the other hand, where ''c'' is any positive constant, and ''n'' is the length of input.

			==Definition==
			'''BQP''' can also be viewed as a bounded-error uniform family of [[quantum circuit]]s. A language ''L'' is in '''BQP''' if and only if there exists a [[Circuit_complexity#Polynomial-time_uniform\|polynomial-time uniform]] family of quantum circuits <math>\{Q_n:n \in \mathbb{N}\}</math>, such that
			* For all <math>n \in \mathbb{N}</math>, ''Q<sub>n</sub>'' takes ''n'' qubits as input and outputs 1 bit
			* For all ''x'' in ''L'', <math>\mathrm{Pr}(Q_{\|x\|}(x)=1)\geq \tfrac{2}{3}</math>
			* For all ''x'' not in ''L'', <math>\mathrm{Pr}(Q_{\|x\|}(x)=0)\geq \tfrac{2}{3}</math>

			== Quantum computation ==
			The number of [[qubit]]s in the computer is allowed to be a [[polynomial function]] of the instance size. For example, algorithms are known for factoring an ''n''-bit integer using just over 2''n'' qubits ([[Shor's algorithm]]).

			Usually, computation on a quantum computer ends with a [[Measurement in quantum mechanics\|measurement]]. This leads to a [[Wavefunction collapse\|collapse]] of quantum state to one of the [[Quantum state\|basis states]]. It can be said that the quantum state is measured to be in the correct state with high probability.

			Quantum computers have gained widespread interest because some problems of practical interest are known to be in '''BQP''', but suspected to be outside '''P'''. Some prominent examples are:
			*[[Integer factorization]] (see [[Shor's algorithm]])<ref name="Shor">[http://www.arxiv.org/abs/quant-ph/9508027 arXiv:quant-ph/9508027v2 ''Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer'', Peter W. Shor]</ref>
			*[[Discrete logarithm]]<ref name="Shor"/>
			*Simulation of quantum systems (see [[universal quantum simulator]])
			*Computing the [[Jones polynomial]] at certain roots of unity

			== Relationship to other complexity classes ==
			This class is defined for a quantum computer and its natural corresponding class for an ordinary computer (or a [[Turing machine]] plus a source of randomness) is '''[[Bounded-error probabilistic polynomial\|BPP]]'''. Just like '''P''' and '''BPP''', '''BQP''' is [[low (complexity)\|low]] for itself, which means '''BQP'''<sup>'''BQP'''</sup> = '''BQP'''. Informally, this is true because polynomial time algorithms are closed under composition. If a polynomial time algorithm calls as a subroutine polynomially many polynomial time algorithms, the resulting algorithm is still polynomial time.

			'''BQP''' contains '''[[P (complexity)\|P]]''' and '''[[Bounded-error probabilistic polynomial\|BPP]]''' and is contained in '''[[Almost Wide Probabilistic Polynomial-Time\|AWPP]]''',<ref>{{Cite journal \| last1=Fortnow \| first1=Lance \| last2=Rogers \| first2=John \| title=Complexity limitations on Quantum computation \| url=http://people.cs.uchicago.edu/~fortnow/papers/quantum.pdf \| publisher=[[Academic Press]] \| location=Boston, MA \| year=1999 \| journal=J. Comput. Syst. Sci. \| issn=0022-0000 \| volume=59 \| issue=2 \| pages=240–252 \| doi=10.1006/jcss.1999.1651 \| postscript=<!--None-->}}</ref> '''[[PP (complexity)\|PP]]'''<ref>L. Adleman, J. DeMarrais, and M.-D. Huang. Quantum computability. SIAM J. Comput.,
			26(5):1524–1540, 1997.</ref> and '''[[PSPACE]]'''.<ref name=BernVazi>Bernstein and Vazirani, Quantum complexity theory, [[SIAM Journal on Computing]], 26(5):1411-1473, 1997. [http://www.cs.berkeley.edu/~vazirani/bv.ps]</ref>
			In fact, '''BQP''' is [[low (complexity)\|low]] for '''PP''', meaning that a '''PP''' machine achieves no benefit from being able to solve '''BQP''' problems instantly, an indication of the possible difference in power between these similar classes.
			:<math>\mathbf{P} \subseteq \mathbf{BPP} \subseteq \mathbf{BQP}\subseteq \mathbf{AWPP} \subseteq \mathbf{PP} \subseteq \mathbf{PSPACE}</math>

			As the problem of '''P''' ≟ '''PSPACE''' has not yet been solved, the proof of inequality between '''BQP''' and classes mentioned above is supposed to be difficult.<ref name=BernVazi/> The relation between '''BQP''' and '''[[NP (complexity)\|NP]]''' is not known.

			Adding [[postselection]] to '''BQP''' results in the complexity class '''[[PostBQP]]''' which is equal to '''[[PP (complexity)\|PP]]'''.<ref name="PostBQP=PP">{{cite journal\|last=Aaronson\|first=Scott\|year=2005\|title=Quantum computing, postselection, and probabilistic polynomial-time\|journal=Proceedings of the Royal Society A\|volume=461\|issue=2063\|pages=3473–3482\|doi=10.1098/rspa.2005.1546}}. Preprint available at [http://arxiv.org/abs/quant-ph/0412187]</ref><ref>{{cite web\|url=http://weblog.fortnow.com/2004/01/complexity-class-of-week-pp-by-guest.html\|title=Complexity Class of the Week: PP\|last=Aaronson\|first=Scott\|date=2004-01-11\|work=Computational Complexity Weblog\|accessdate=2008-05-02}}</ref>

			==References==
			<references />

			{{quantum_computing}}
			{{ComplexityClasses}}

			{{DEFAULTSORT:Bqp}}
			[[Category:Probabilistic complexity classes]]
			[[Category:Quantum complexity theory]]

en>Serketan at 12:06, 14 May 2012

2012-05-14T12:06:52Z

New page

When in search of the proper hemorrhoid treatment you will need to consider a limited significant factors such as, that you you think you want, if there is a terrific amount of recovery required, and how lengthy it takes to get results. In this short article we will learn the answers to all of these issues, offering we the answer we require to find the right hemorrhoid treatment. Now which the hemorrhoids aren't getting worse, it's time to use [http://hemorrhoidtreatmentfix.com/hemorrhoid-relief hemorrhoid relief] that might relieve itching and pain. Soaking in a a sitz bathtub for about fifteen minutes at a time is one method to discover relief. You are able to do this multiple occasions a day. You are able to furthermore heat a wet towel plus apply it to the hemorrhoid region. Ice for ten minutes, followed by a warm compress for an equal or slightly longer length of time is another method to relieve a pain plus itching. There are several treatments that may be selected for hemorrhoid. The first and the many popular is the cream plus ointment. These are to be rubbed onto the affected part of the anus. It helps to soothe the already inflamed blood vessels plus a momentary relief is achieved. There is a relaxation of the tissues of the rectal column thus far the hemorrhoid is not so much bulged. If there is a bulge though, the pain relief could not do so much to help. Hemorrhoid Surgery - This is just suggested if your hemorrhoids are thus severe that no over the counter or home treatment is providing we any kind of relief. Your wellness food store has a wealth of natural hemorrhoid treatments, all of which contain safe ingredients and are effortless on the purse. Barberry, Butcher's Brew, Witch Hazel, Horse Chestnut and Slippery Elm are all renowned herbal elements for giving relief whenever piles flare up. Believe me I recognize. I recognize how painful, inconvenient and embarrassing hemorrhoids is. For me the big problem was the itching. I mean what may you possibly do to relieve the itch whenever you're sitting down all day long inside a busy workplace surrounded by colleagues? It is pretty challenging now to suffer from hemorrhoid. You deal with serious redness, irritation, swelling, and pain. You are typically uncomfortable. But have belief which you'll feel better with a hemorrhoid treatment. You may receive rid of the hemorrhoid in no time perhaps by using petroleum jelly, using ointment phenylephrine or Preparation H, or using soft cotton underwear. We really need to provide a try any of them.

BQP - Revision history

en>Fshtea: /* Relationship to other complexity classes */

en>Serketan at 12:06, 14 May 2012