BQP: Difference between revisions

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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 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^−c on the one hand, or requiring error as small as 2^{−n^c} 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 circuits. A language L is in BQP if and only if there exists a polynomial-time uniform family of quantum circuits $\{Q_{n}:n\in \mathbb {N} \}$ , such that

For all $n\in \mathbb {N}$ , Q_n takes n qubits as input and outputs 1 bit
For all x in L, $\mathrm {Pr} (Q_{|x|}(x)=1)\geq {\tfrac {2}{3}}$
For all x not in L, $\mathrm {Pr} (Q_{|x|}(x)=0)\geq {\tfrac {2}{3}}$

Quantum computation

The number of qubits 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 2n qubits (Shor's algorithm).

Usually, computation on a quantum computer ends with a measurement. This leads to a collapse of quantum state to one of the 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)^[2]
Discrete logarithm^[2]
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 BPP. Just like P and BPP, BQP is low for itself, which means BQP^BQP = 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 and BPP and is contained in AWPP,^[3] PP^[4] and PSPACE.^[5] In fact, BQP is 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.

\mathbf {P} \subseteq \mathbf {BPP} \subseteq \mathbf {BQP} \subseteq \mathbf {AWPP} \subseteq \mathbf {PP} \subseteq \mathbf {PSPACE}

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.^[5] The relation between BQP and NP is not known.

Adding postselection to BQP results in the complexity class PostBQP which is equal to PP.^[6]^[7]

References

↑ Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.
↑ ^2.0 ^2.1 arXiv:quant-ph/9508027v2 Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Peter W. Shor
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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↑ L. Adleman, J. DeMarrais, and M.-D. Huang. Quantum computability. SIAM J. Comput., 26(5):1524–1540, 1997.
↑ ^5.0 ^5.1 Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411-1473, 1997. [1]
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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↑ Template:Cite web

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[1] Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.

[Shor-2] 2.0 ^2.1 arXiv:quant-ph/9508027v2 Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Peter W. Shor

[3] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[4] L. Adleman, J. DeMarrais, and M.-D. Huang. Quantum computability. SIAM J. Comput., 26(5):1524–1540, 1997.

[BernVazi-5] 5.0 ^5.1 Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411-1473, 1997. [1]

[PostBQP=PP-6] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang. Preprint available at [2]

[7] Template:Cite web

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+{{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.,
+(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]]

BQP: Difference between revisions

Latest revision as of 15:30, 16 September 2013

Contents

Definition

Quantum computation

Relationship to other complexity classes

References

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BQP: Difference between revisions

Latest revision as of 15:30, 16 September 2013

Definition

Quantum computation

Relationship to other complexity classes

References

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