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| In [[quantum computing]], a '''quantum algorithm''' is an [[algorithm]] which runs on a realistic model of [[quantum computation]], the most commonly used model being the [[quantum circuit]] model of computation.<ref>
| | It is possible for you to to resume all normal activities and exercise (including flying and scuba-diving. When a woman chooses to have breast augmentation surgery, she will be asked by her surgeon which type of implant she would like. And not many company would sent their Pueraria Mirifica for such stringent test. This is the fertility vitamin, essential to breast enlargement, Vitamin E helps the body to use oxygen, which increases physical endurance and stamina and improves heart response. To avoid bruising and discomfort, skin massage oil lubrication is also important. <br><br>In addition, problems include leak or rupture of the implant requiring additional surgery to change the previous one, pain, seroma (fluid retention), infections and a condition called capsular contracture. If you are interested of all-natural product as a supplement or cream, Breast Actives offers positive results in effectively increasing the size your breasts. You have to sleep with it on, go to work with it on, exercise with it on - everything. It is primarily known for it's uses in treating women's problems including lack of sexual desire, the symptoms of menopause, cramps and PMS. Breast enhancement can be achieved by taking capsules, pills and the use of creams which increase both the size and firmness of the breast. <br><br>The major side effect is defuse the variety of breast enlargement without surgery through any types of surgery technique so that the rounding the implant. For those women lucky enough to try the pills, they were so grateful with the results because not only are they inexpensive but the pill's method of enhancing their breast goes in synch with their body's natural ways. Educating oneself about the ingredients found in the pills is of maximum importance. women out there who have successfully enlarged their breasts anywhere from half to two cup sizes (and even more in some cases), by using a breast enhancement pill. Aside from being risk-free and effective, these enhancement pills are also affordable and pocket friendly. <br><br>Keep a written record in a notebook or on your computer with the date, the number and type of exercises you did. Alternatively, with both hands holding the breast meat should be bent and very gently wrung. Healthcare tourism has become an vital scene in Hungary. The surgery usually takes one to two hours to complete. Using natural creams and supplements or pills is a very safe and natural elective to follow other than going under the knife. <br><br>• All breast implant Beverly Hills Surgeons spend significant time listening to their patients, educating them and reviewing breast implant surgery aftercare. If you gain weight, your breasts will doubtlessly get larger or bigger as well. Don quai, one of the ingredients worn in the pills is a known carcinogen. will require you to have time off work and out from your normal. News that she feels more sexy now as more curves have been added on her. <br><br>It is true that a woman's wedding day is very important to them. In addition, it is an effective hormonal regulator. In essence, natural breast enhancement usually takes the type of all natural herbs that are utilized either in tablet, tablet or capsule type or topical creams and serums that trigger the body's own natural capability to improve the breast tissues. This is because of the viscous nature of the solution that causes it to exit any leak gradually and ever so subtly. They know their outcome the best, and they've formulated it to work a detail way, so any diet supplementation, time of day to take it, or lifestyle modifications they suggest should be adhered to as intently as viable to secure you are liability everything viable to make the outcome work for you.<br><br>If you beloved this report and you would like to acquire much more data relating to how to get bigger boobs ([http://www.lucky-house.info/sitemap/ http://www.lucky-house.info/sitemap/]) kindly check out our website. |
| {{cite book
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| | last=Nielsen |first=M.
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| | last2=Chuang |first2=I.
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| | year=2000
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| | title=Quantum Computation and Quantum Information
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| | publisher=[[Cambridge University Press]]
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| | isbn=0-521-63503-9
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| }}</ref><ref>
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| {{cite arxiv
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| | last = Mosca | first = M.
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| | date = 2008
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| | title = Quantum Algorithms
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| | class = quant-ph
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| | eprint = 0808.0369
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| }}</ref> A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical [[computer]]. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a [[quantum computer]]. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as [[quantum superposition]] or [[quantum entanglement]].
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| All problems which can be solved on a quantum computer can be solved on a classical computer. In particular, problems which are [[Undecidable problem|undecidable]] using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms.
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| The most well known algorithms are [[Shor's algorithm]] for factoring, and [[Grover's algorithm]] for searching an unstructured database or an unordered list. Shor's algorithms runs exponentially faster than the best known classical algorithm for factoring, the [[general number field sieve]]. Grover's algorithm runs quadratically faster than the best possible classical algorithm for the same task.
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| ==Overview==
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| Quantum algorithms are usually described, in the commonly used circuit model of quantum computation, by a [[quantum circuit]] which acts on some input [[qubit]]s and terminates with a [[measurement]]. A quantum circuit consists of simple [[quantum gate]]s which act on at most a fixed number of qubits, usually 2 or 3. Quantum algorithms may also be stated in other models of quantum computation, such as the [[Hamiltonian oracle model]].<ref name=Hamiltonian_NAND_Tree>
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| {{cite arxiv
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| | last = Farhi | first = E.
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| | last2 = Goldstone |first2=J.
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| | last3 = Gutmann |first3=S.
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| | date = 2007
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| | title = A Quantum Algorithm for the Hamiltonian NAND Tree
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| | class = quant-ph
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| | eprint = quant-ph/0702144
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| }}</ref>
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| Quantum algorithms can be categorized by the main techniques used by the algorithm. Some commonly used techniques/ideas in quantum algorithms include [[phase kick-back]], [[quantum phase estimation algorithm|phase estimation]], the [[quantum Fourier transform]], [[quantum walk]]s, [[amplitude amplification]] and [[topological quantum field theory]]. Quantum algorithms may also be grouped by the type of problem solved, for instance see the survey on quantum algorithms for algebraic problems.<ref>
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| {{cite journal
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| | last = Childs | first = A. M.
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| | last2= van Dam |first2 = W.
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| | year = 2008
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| | title = Quantum algorithms for algebraic problems
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| | journal = [[Reviews of Modern Physics]]
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| | volume = 82 |issue= | pages = 1–52
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| | arxiv = 0812.0380
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| | bibcode = 2010RvMP...82....1C
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| | doi = 10.1103/RevModPhys.82.1
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| }}</ref>
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| | |
| ==Algorithms based on the quantum Fourier transform==
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| | |
| The [[quantum Fourier transform]] is the quantum analogue of the [[discrete Fourier transform]], and is used in several quantum algorithms. The [[Hadamard transform]] is also an example of a quantum Fourier transform over an n-dimensional vector space over the field '''F'''<sub>2</sub>. The quantum Fourier transform can be efficiently implemented on a quantum computer using only a polynomial number of [[quantum gate]]s.
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| ===Deutsch–Jozsa algorithm===
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| {{main|Deutsch–Jozsa algorithm}}
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| The Deutsch–Jozsa algorithm solves a [[black-box]] problem which probably requires exponentially many queries to the black box for any deterministic classical computer, but can be done with exactly 1 query by a quantum computer. If we allow both bounded-error quantum and classical algorithms, then there is no speedup since a classical probabilistic algorithm can solve the problem with a constant number of queries with small probability of error. The algorithm determines whether a function ''f'' is either constant (0 on all inputs or 1 on all inputs) or balanced (returns 1 for half of the input domain and 0 for the other half).
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| ===Simon's algorithm===
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| {{main|Simon's algorithm}}
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| Simon's algorithm solves a black-box problem exponentially faster than any classical algorithm, including bounded-error probabilistic algorithms. This algorithm, which achieves an exponential speedup over all classical algorithms that we consider efficient, was the motivation for Shor's factoring algorithm.
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| === Quantum phase estimation algorithm ===
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| ===Shor's algorithm===
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| {{main|Shor's Algorithm}}
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| Shor's algorithm solves the [[discrete logarithm]] problem and the [[integer factorization]] problem in polynomial time,<ref>
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| {{cite journal
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| | last = Shor | first = P. W.
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| | year = 1997
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| | title = Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
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| | journal = [[SIAM Journal on Scientific and Statistical Computing]]
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| | volume = 26 | page = 1484
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| | arxiv = quant-ph/9508027
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| | bibcode= 1995quant.ph..8027S
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| | doi=
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| }}</ref> whereas the best known classical algorithms take super-polynomial time. These problems are not known to be in [[P (complexity)|P]] or [[NP-complete]]. It is also one of the few quantum algorithms that solves a non–black-box problem in polynomial time where the best-known classical algorithms run in super-polynomial time.
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| ===Hidden subgroup problem===
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| The [[Abelian group|abelian]] [[hidden subgroup problem]] is a generalization of many problems that can be solved by a quantum computer, such as Simon's problem, solving [[Pell's equation]], testing the [[principal ideal]] of a [[ring (mathematics)|ring]] R and [[integer factorization|factoring]]. There are efficient quantum algorithms known for the Abelian hidden subgroup problem.<ref>
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| {{cite conference
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| |last=Boneh |first=D.
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| |last2=Lipton |first2=R. J.
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| |year=1995
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| |title=Quantum cryptoanalysis of hidden linear functions
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| |editor-last=Coppersmith |editor-first=D.
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| |booktitle=Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
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| |pages=424–437
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| |publisher=[[Springer-Verlag]]
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| |isbn=3-540-60221-6
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| }}</ref> The more general hidden subgroup problem, where the group isn't necessarily abelian, is a generalization of the previously mentioned problems and [[graph isomorphism]] and certain [[lattice problems]]. Efficient quantum algorithms are known for certain non-abelian groups. However, no efficient algorithms are known for the [[symmetric group]], which would give an efficient algorithm for graph isomorphism<ref>
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| {{cite arxiv
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| |last1=Moore |first1=C.|author1-link=Cris Moore
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| |last2=Russell |first2=A.
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| |last3=Schulman |first3=L. J.
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| |year=2005
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| |title=The Symmetric Group Defies Strong Fourier Sampling: Part I
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| |class=quant-ph
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| |eprint=quant-ph/0501056
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| |author2=Alexander Russell
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| }}</ref> and the [[dihedral group]], which would solve certain lattice problems.<ref>
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| {{cite arxiv
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| | last = Regev | first = O.
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| | date = 2003
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| | title = Quantum Computation and Lattice Problems
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| | class = cs.DS
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| | eprint = cs/0304005
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| }}</ref>
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| ===Estimating Gauss sums===
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| A [[Gauss sum]] is a type of [[exponential sum]]. The best known classical algorithm for estimating these sums takes exponential time. Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely. However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.<ref>
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| {{Cite arxiv
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| | last1=van Dam |first=W.
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| | last2=Seroussi |first2=G.
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| | year =2002
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| | title = Efficient Quantum Algorithms for Estimating Gauss Sums
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| | class = quant-ph
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| | eprint = quant-ph/0207131
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| | author2=Gadiel Seroussi
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| }}</ref>
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| | |
| === Fourier fishing and Fourier checking ===
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| We have an [[Oracle machine|oracle]] consisting of n random Boolean functions mapping n-bit strings to a Boolean value. We are required to find n n-bit strings z<sub>1</sub>,..., z<sub>n</sub> such that for the Hadamard-Fourier transform, at least 3/4 of the strings satisfy
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| :<math>\left| \tilde{f}\left( z_i \right) \right| \geqslant 1</math>
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| and at least 1/4 satisfies
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| :<math>\left| \tilde{f}\left( z_i \right) \right| \geqslant 2</math>.
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| This can be done in [[BQP]].<ref>
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| {{Cite arxiv
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| | last = Aaronson | first = S.
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| | year = 2009
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| | title = BQP and the Polynomial Hierarchy
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| | class = quant-ph
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| | eprint = 0910.4698
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| }}</ref>
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| ==Algorithms based on amplitude amplification==
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| [[Amplitude amplification]] is a technique that allows the amplification of a chosen subspace of a quantum state. Applications of amplitude amplification usually lead to quadratic speedups over the corresponding classical algorithms. It can be considered to be a generalization of Grover's algorithm.
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| ===Grover's algorithm===
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| {{main|Grover's algorithm}}
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| Grover's algorithm searches an unstructured database (or an unordered list) with N entries, for a marked entry, using only <math>O(\sqrt{N})</math> queries instead of the Ω(''N'') queries required classically.<ref>
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| {{Cite arxiv
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| | last = Grover | first = L. K.
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| | date = 1996
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| | title = A fast quantum mechanical algorithm for database search
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| | class = quant-ph
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| | eprint = quant-ph/9605043
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| }}</ref> Classically, Ω(''N'') queries are required, even if we allow bounded-error probabilistic algorithms.
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| ===Quantum counting===
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| Quantum counting solves a generalization of the search problem. It solves the problem of counting the number of marked entries in an unordered list, instead of just detecting if one exists. Specifically, it counts the number of marked entries in an <math>N</math>-element list, with error <math>\epsilon</math> making only <math>\Theta\left(\frac{1}{\epsilon} \sqrt{\frac{N}{k}}\right)</math> queries, where <math>k</math> is the number of marked elements in the list.<ref>
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| {{Cite arxiv
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| |last = Brassard |first = G. | |
| |last2=Hoyer |first2=P. | |
| |last3=Tapp |first3=A.
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| |date = 1998
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| |title = Quantum Counting
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| |class = quant-ph
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| |eprint = quant-ph/9805082
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| }}</ref><ref>
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| {{cite arxiv
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| |last1=Brassard |first1=G.
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| |last2=Hoyer |first2=P.
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| |last3=Mosca |first3=M.
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| |last4=Tapp |first4=A.
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| |year=2000
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| |title=Quantum Amplitude Amplification and Estimation
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| |class=quant-ph
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| |eprint=quant-ph/0005055
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| |author2=Peter Hoyer
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| |author3=Michele Mosca
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| |author4=Alain Tapp
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| }}</ref> More precisely, the algorithm outputs an estimate <math>k'</math> for <math>k</math>, the number of marked entries, with the following accuracy: <math>|k-k'| \leq \epsilon k</math>.
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| | |
| ==Algorithms based on quantum walks==
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| {{main|Quantum walk}}
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| A quantum walk is the quantum analogue of a classical [[random walk]], which can be described by a [[probability distribution]] over some states. A quantum walk can be described by a [[quantum superposition]] over states. Quantum walks are known to give exponential speedups for some black-box problems.<ref>
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| {{cite conference
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| |last1=Childs |first1=A. M.
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| |last2=Cleve |first2=R.
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| |last3=Deotto |first3=E.
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| |last4=Farhi |first4=E.
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| |last5=Gutmann |first5=S.
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| |last6=Spielman |first6=D. A.
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| |year=2003
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| |title=Exponential algorithmic speedup by quantum walk
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| |booktitle=Proceedings of the 35th Symposium on Theory of Computing
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| |pages=59–68
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| |publisher=[[Association for Computing Machinery]]
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| |arxiv=quant-ph/0209131
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| |bibcode=
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| |doi=10.1145/780542.780552
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| |isbn=1-58113-674-9
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| }}</ref><ref>
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| {{cite conference
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| |last1=Childs |first1=A. M.
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| |last2=Schulman |first2=L. J.
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| |last3=Vazirani |first3=U. V.
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| |year=2007
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| |title=Quantum Algorithms for Hidden Nonlinear Structures
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| |booktitle=Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
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| |pages=395–404
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| |publisher=[[IEEE]]
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| |arxiv=0705.2784
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| |doi=10.1109/FOCS.2007.18
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| |isbn=0-7695-3010-9
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| }}</ref> They also provide polynomial speedups for many problems. A framework for the creation quantum walk algorithms exists and is quite a versatile tool.<ref name=Search_via_quantum_walk/>
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| ===Element distinctness problem===
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| {{main|Element distinctness problem}}
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| The element distinctness problem is the problem of determining whether all the elements of a list are distinct. Classically, Ω(''N'') queries are required for a list of size ''N'', since this problem is harder than the search problem which requires Ω(''N'') queries. However, it can be solved in <math>\Theta(N^{2/3})</math> queries on a quantum computer. The optimal algorithm is by [[Andris Ambainis]].<ref>
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| {{cite journal
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| |last=Ambainis |first=A.
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| |year=2007
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| |title=Quantum Walk Algorithm for Element Distinctness
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| |journal=[[SIAM Journal on Computing]]
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| |volume=37 |issue=1 |pages=210–239
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| |arxiv=
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| |bibcode=
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| |doi=10.1137/S0097539705447311
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| }}</ref> [[Scott Aaronson]] and Yaoyun Shi first proved a tight lower bound for a large but restricted class of functions.<ref>
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| {{Cite journal
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| | last1=Aaronson | first1=S.
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| | last2=Shi | first2=Y.
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| | year=2004
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| | title=Quantum lower bounds for the collision and the element distinctness problems
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| | journal=[[Journal of the ACM]]
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| | volume=51 | issue=4 | pages=595–605
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| | arxiv=
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| | bibcode=
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| | doi=10.1145/1008731.1008735
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| }}</ref> Ambainis<ref>
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| {{cite journal
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| |last=Ambainis |first=A.
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| |year=2005
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| |title=Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
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| |journal=[[Theory of Computing]]
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| |volume=1 |issue=1 |pages=37–46
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| |arxiv=
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| |bibcode=
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| |doi=10.4086/toc.2005.v001a003
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| }}</ref> and Kutin<ref>
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| {{cite journal
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| |last1=Kutin |first1=S.
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| |year=2005
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| |title=Quantum Lower Bound for the Collision Problem with Small Range
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| |journal=[[Theory of Computing]]
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| |volume=1 |issue=1 |pages=29–36
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| |arxiv=
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| |bibcode=
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| |doi=10.4086/toc.2005.v001a002
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| }}</ref> independently (and via different proofs) extended their work to obtain the lower bound for all functions.
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| ===Triangle-finding problem===
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| {{main|Triangle finding problem}}
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| The triangle-finding problem is the problem of determining whether a given graph contains a triangle (a [[clique (graph theory)|clique]] of size 3). The best-known lower bound for quantum algorithms is Ω(''N''), but the best algorithm known requires O(''N''<sup>1.297</sup>) queries,<ref>{{cite arxiv| eprint=1105.4024| author1=Aleksandrs Belovs| title=Span Programs for Functions with Constant-Sized 1-certificates| class=quant-ph| year=2011}}</ref> an improvement over the previous best O(''N''<sup>1.3</sup>) queries.<ref name=Search_via_quantum_walk>
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| {{cite conference
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| |last1=Magniez |first1=F.
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| |last2=Nayak |first2=A.
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| |last3=Roland |first3=J.
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| |last4=Santha |first4=M.
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| |year=2007
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| |title=Search via quantum walk
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| |booktitle=Proceedings of the 39th Annual ACM Symposium on Theory of Computing
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| |publisher=[[Association for Computing Machinery]]
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| |pages=575–584
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| |doi=10.1145/1250790.1250874
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| |isbn=978-1-59593-631-8
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| }}</ref><ref>
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| {{cite journal
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| |last1=Magniez |first1=F.
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| |last2=Santha |first2=M.
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| |last3=Szegedy |first3=M.
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| |year=2007
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| |title=Quantum Algorithms for the Triangle Problem
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| |journal=[[SIAM Journal on Computing]]
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| |volume=37 |issue=2 |pages=413–424
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| |arxiv=
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| |bibcode=
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| |doi=10.1137/050643684
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| }}</ref>
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| | |
| ===Formula evaluation===
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| A formula is a tree with a gate at each internal node and an input bit at each leaf node. The problem is to evaluate the formula, which is the output of the root node, given oracle access to the input.
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| | |
| A well studied formula is the balanced binary tree with only NAND gates.<ref>
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| {{cite web
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| |last=Aaronson |first=S.
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| |date=3 February 2007
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| |title=NAND now for something completely different
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| |url=http://scottaaronson.com/blog/?p=207
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| |work=Shtetl-Optimized
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| |accessdate=2009-12-17
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| }}</ref> This type of formula requires Θ(''N''<sup>c</sup>) queries using randomness<ref>
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| {{cite conference
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| |last1=Saks |first1=M.E.
| |
| |last2=Wigderson |first2=A.
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| |year=1986
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| |title=Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees
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| |url=http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/SW86/SW86.pdf
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| |booktitle=Proceedings of the 27th Annual Symposium on Foundations of Computer Science
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| |pages=29–38
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| |publisher=[[IEEE]]
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| |doi=10.1109/SFCS.1986.44
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| |isbn=0-8186-0740-8
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| }}</ref> (where <math>c = \log_2[(1+\sqrt{33})/4] \approx 0.754</math>), but can be solved in Θ(''N''<sup>0.5</sup>) queries by a quantum algorithm. No better quantum algorithm for this case was known until one was found for the unconventional Hamiltonian oracle model.<ref name=Hamiltonian_NAND_Tree/> The same result for the standard setting soon followed.<ref>
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| {{cite arxiv
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| |last=Ambainis |first=A.
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| |year=2007
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| |title=A nearly optimal discrete query quantum algorithm for evaluating NAND formulas
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| |class=quant-ph
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| |eprint=0704.3628
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| }}</ref>
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| | |
| Fast quantum algorithms for more complicated formulas are also known.<ref>
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| {{cite conference
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| |last1=Reichardt |first1=B. W.
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| |last2=Spalek |first2=R.
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| |year=2008
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| |title=Span-program-based quantum algorithm for evaluating formulas
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| |booktitle=Proceedings of the 40th Annual ACM symposium on Theory of Computing
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| |publisher=[[Association for Computing Machinery]]
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| |pages=103–112
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| |isbn=978-1-60558-047-0
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| |doi=10.1145/1374376.1374394
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| }}</ref>
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| | |
| ===Group commutativity===
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| The problem is to determine if a black box [[Group (mathematics)|group]], given by ''k'' generators, is [[Commutativity|commutative]]. A black box group is a group with an oracle function, which must be used to perform the group operations (multiplication, inversion, and comparison with identity). We are interested in the query complexity, which is the number of oracle calls needed to solve the problem. The deterministic and randomized query complexities are <math>\Theta(k^2)</math> and <math>\Theta(k)</math> respectively.<ref>
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| {{cite arxiv
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| |last=Pak |first=I.
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| |year=2000
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| |title=Testing commutativity of a group and the power of randomization
| |
| |class=quant-ph
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| |eprint=quant-ph/0506265
| |
| |author2=Ashwin Nayak
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| }}</ref> A quantum algorithm requires <math>\Omega(k^{2/3})</math> queries but the best known algorithm uses <math>O(k^{2/3} \log k)</math> queries.<ref>
| |
| {{cite journal
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| |last1=Magniez |first1=F.
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| |last2=Nayak |first2=A.
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| |year=2007
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| |title=Quantum Complexity of Testing Group Commutativity
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| |journal=[[Algorithmica]]
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| |volume=48 |issue=3 |pages=221–232
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| |doi=10.1007/s00453-007-0057-8
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| }}</ref>
| |
| | |
| ==BQP-complete problems==
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| | |
| ===Computing knot invariants===
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| | |
| Witten had shown that the [[Chern-Simons]] [[topological quantum field theory]] (TQFT) can be solved in terms of [[Jones polynomial]]s. A quantum computer can simulate a TQFT, and thereby approximate the Jones polynomial,<ref>
| |
| {{Cite conference
| |
| | last = Aharonov | first = D.
| |
| | last2 = Jones | first2 = V.
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| | last3 = Landau | first3 = Z.
| |
| | year = 2006
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| | title = A polynomial quantum algorithm for approximating the Jones polynomial
| |
| | booktitle=Proceedings of the 38th Annual ACM symposium on Theory of Computing
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| | pages = 427–436
| |
| | publisher=[[Association for Computing Machinery]]
| |
| | doi = 10.1145/1132516.1132579
| |
| | isbn=
| |
| }}</ref> which as far as we know, is hard to compute classically in the worst case scenario.
| |
| | |
| ===Quantum simulation===
| |
| | |
| The idea that quantum computers might be more powerful than classical computers originated in Richard Feynman's observation that classical computers seem to require exponential time to simulate many-particle quantum systems.<ref>
| |
| {{Cite journal
| |
| | last1=Feynman | first1=R. P.
| |
| | year=1982
| |
| | title=Simulating physics with computers
| |
| | journal=[[International Journal of Theoretical Physics]]
| |
| | volume=21 | issue=6–7 | page=467
| |
| | arxiv=
| |
| | bibcode = 1982IJTP...21..467F
| |
| | doi = 10.1007/BF02650179
| |
| }}</ref> Since then, the idea that quantum computers can simulate quantum physical processes exponentially faster than classical computers has been greatly fleshed out and elaborated. Efficient (that is, polynomial-time) quantum algorithms have been developed for simulating both Bosonic and Fermionic systems<ref>
| |
| {{Cite journal
| |
| | last1=Abrams |first1=D. S.
| |
| | last2=Lloyd | first2=S.
| |
| | year=1997
| |
| | title=Simulation of many-body Fermi systems on a universal quantum computer
| |
| | journal=[[Physical Review Letters]]
| |
| | volume=79 | issue=13 | pages=2586–2589
| |
| | arxiv = quant-ph/9703054
| |
| | bibcode=1997PhRvL..79.2586A
| |
| | doi=10.1103/PhysRevLett.79.2586
| |
| }}</ref> and in particular, the simulation of chemical reactions beyond the capabilities of current classical supercomputers requires only a few hundred qubits.<ref>
| |
| {{Cite journal
| |
| | last1=Kassal | first1=I.
| |
| | last2=Jordan | first2=S. P.
| |
| | last3=Love | first3=P. J.
| |
| | last4=Mohseni | first4=M.
| |
| | last5=Aspuru-Guzik | first5=A.
| |
| | year=2008
| |
| | title=Polynomial-time quantum algorithm for the simulation of chemical dynamics
| |
| | journal=[[Proceedings of the National Academy of Sciences of the United States of America]]
| |
| | volume=105 |issue=48 | pages=18681–86
| |
| | arxiv= 0801.2986
| |
| | bibcode = 2008PNAS..10518681K
| |
| | doi=10.1073/pnas.0808245105
| |
| | pmc=2596249
| |
| | pmid=19033207
| |
| }}</ref> Quantum computers can also efficiently simulate topological quantum field theories.<ref>
| |
| {{Cite journal
| |
| | last1=Freedman | first1=M.
| |
| | last2=Kitaev | first2=A.
| |
| | last3=Wang | first3=Z.
| |
| | year=2002
| |
| | title=Simulation of Topological Field Theories by Quantum Computers
| |
| | journal=[[Communications in Mathematical Physics]]
| |
| | volume=227 | issue=3 | pages=587–603
| |
| | arxiv = quant-ph/0001071
| |
| | bibcode = 2002CMaPh.227..587F
| |
| | doi=10.1007/s002200200635
| |
| }}</ref> In addition to its intrinsic interest, this result has led to efficient quantum algorithms for estimating "quantum"{{Clarify|What makes these "quantum"? Are they used in QM & don't behave QMly?|date=September 2011}} topological invariants such as [[Jones polynomial|Jones]]<ref>
| |
| {{Cite journal
| |
| | last1=Aharonov | first1=D.
| |
| | last2=Jones | first2=V.
| |
| | last3=Landau | first3=Z.
| |
| | year=2009
| |
| | title=A polynomial quantum algorithm for approximating the Jones polynomial
| |
| | journal=[[Algorithmica]]
| |
| | volume=55 | issue=3 | pages=395–421
| |
| | arxiv=quant-ph/0511096
| |
| | bibcode=
| |
| | doi=10.1007/s00453-008-9168-0
| |
| }}</ref> and [[HOMFLY]] <ref>
| |
| {{Cite journal
| |
| | last1=Wocjan |first1=P.
| |
| | last2=Yard | first2=J.
| |
| | year=2008
| |
| | title=The Jones polynomial: quantum algorithms and applications in quantum complexity theory
| |
| | journal=[[Quantum Information and Computation]]
| |
| | volume=8 | issue=1 | pages=147–180
| |
| | arxiv=quant-ph/0603069
| |
| | doi=
| |
| }}</ref> polynomials, and the [[Turaev-Viro invariant]] of three-dimensional manifolds.<ref>
| |
| {{Cite journal
| |
| |last1=Alagic | first1=G.
| |
| |last2=Jordan | first2=S.P.
| |
| |last3=König | first3=R.
| |
| |last4=Reichardt | first4=B. W.
| |
| |year=2010
| |
| |title=Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation
| |
| |journal=[[Physical Review A]]
| |
| |volume=82 |issue=4 |pages=040302
| |
| |arxiv=1003.0923
| |
| |bibcode=2010PhRvA..82d0302A
| |
| |doi=10.1103/PhysRevA.82.040302
| |
| }}</ref>
| |
| | |
| ==See also==
| |
| * [[Quantum sort]]
| |
| * [[Primality test]]
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| ==References==
| |
| | |
| {{reflist|2}}
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| | |
| ==External links==
| |
| *The [http://math.nist.gov/quantum/zoo/ Quantum Algorithm Zoo]: A comprehensive list of quantum algorithms that provide a speedup over the fastest known classical algorithms.
| |
| | |
| ===Surveys===
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| *{{cite doi | 10.1007/978-3-540-92910-9_43}}
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| *{{cite doi | 10.1103/RevModPhys.82.1}}
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| {{quantum computing}}
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| {{Use dmy dates|date=September 2011}}
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| {{DEFAULTSORT:Quantum Algorithm}}
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| [[Category:Quantum information science]]
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| [[Category:Theoretical computer science]]
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| [[Category:Quantum algorithms| ]]
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