Quantum computer 25220 move=:edit= 225922586 2008-07-16T01:03:12Z Alksentrs 5041971 /* Bits vs. Qubits */ formatting maths {{Citation style}} [[Image:Blochsphere.svg|thumb|The [[Bloch sphere]] is a representation of a [[qubit]], the fundamental building block of quantum computers.]] A '''quantum computer''' is a device for [[computation]] that makes direct use of distinctively [[quantum mechanical]] [[phenomena]], such as [[quantum superposition|superposition]] and [[quantum entanglement|entanglement]], to perform operations on data. In a classical (or conventional) computer, information is stored as [[bit]]s; in a quantum computer, it is stored as [[qubit]]s ('''qu'''antum '''bi'''nary digi'''ts'''). The basic principle of quantum computation is that the quantum properties can be used to represent and structure data, and that quantum mechanisms can be devised and built to perform [[Instruction (computer science)|operation]]s with this data.<ref>"[http://www.media.mit.edu/physics/publications/papers/98.06.sciam/0698gershenfeld.html Quantum Computing with Molecules]" article in [[Scientific American]] by [[Neil Gershenfeld]] and [[Isaac L. Chuang]] - a generally accessible overview of quantum computing and so on.</ref> Although quantum computing is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of [[qubit]]s. Research in both theoretical and practical areas continues at a frantic pace, and many national government and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as [[cryptanalysis]].<ref>[http://qist.lanl.gov/qcomp_map.shtml Quantum Information Science and Technology Roadmap] for a sense of where the research is heading.</ref> If large-scale quantum computers can be built, they will be able to solve certain problems much faster than any of our current classical computers (for example [[Shor's algorithm]]). Quantum computers are different from other [[computers]] such as [[DNA computing|DNA computers]] and traditional computers based on [[transistor]]s. Some computing architectures such as [[optical computer]]s <ref>[http://espace.library.uq.edu.au/eserv/UQ:10849/n9905086.pdf One photon [[Grover algorithm]] ]</ref> may use classical superposition of electromagnetic waves. Without some specifically quantum mechanical resources such as [[quantum entanglement|entanglement]], it is conjectured that an exponential advantage over classical computers is not possible.<ref name="BCJLPS99"> {{cite journal | author = [[Lieven M.K. Vandersypen]] et al. | title = Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing | journal = Phys. Rev. Lett | volume = 83 | pages = 1054–1057 | date = 1999 | doi = 10.1103/PhysRevLett.83.1054}} </ref> ==The basis of quantum computing== A classical computer has a memory made up of [[bit]]s, where each bit holds either a one or a zero. A quantum computer maintains a sequence of [[qubit]]s. A single qubit can hold a one, a zero, or, crucially, a [[quantum superposition]] of these; moreover, a pair of qubits can be in a quantum superposition of 4 states, and three qubits in a superposition of 8. In general a quantum computer with n qubits can be in up to <math>2^n</math> different states simultaneously (this compares to a normal computer that can only be in ''one'' of these <math>2^n</math> states at any one time). A quantum computer operates by manipulating those qubits with a fixed sequence of [[quantum gate|quantum logic gate]]s. The sequence of gates to be applied is called a ''quantum algorithm''. An example of an implementation of qubits for a quantum computer could start with the use of particles with two [[spin (physics)|spin]] states: "up" and "down" (typically written <math>|{\uparrow}\rangle</math> and <math>|{\downarrow}\rangle</math>, or <math>|0\rangle</math> and <math>|1\rangle</math>). But in fact any system possessing an [[observable]] quantity ''A'' which is ''conserved'' under time evolution and such that ''A'' has at least two discrete and sufficiently spaced consecutive [[eigenvalue]]s, is a suitable candidate for implementing a qubit. This is true because any such system can be mapped onto an effective [[spin-1/2]] system. ==Bits vs. Qubits == [[Image:Quantum computer.jpg|thumb|200px|Qubits are made up of controlled particles and the means of control (e.g. devices that trap particles and switch them from one state to another).<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE]] |place = London |date = 2007 | pages = p157 |isbn = 2746215160}}</ref> ]] Consider first a classical computer that operates on a three-bit [[Processor register|register]]. The state of the computer at any time is a probability distribution over the <math>2^3=8</math> different three-bit strings 000, 001, ..., 111—thus it is described by eight nonnegative numbers (''a'',''b'',''c'',''d'',''e'',''f'',''g'',''h'') adding up to one. The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector (''a'',''b'',''c'',''d'',''e'',''f'',''g'',''h''), called a [[wavefunction]]. However, instead of adding to one, the sum of the ''squares'' of the coefficient magnitudes, <math>|a|^2+|b|^2+...+|h|^2</math>, must equal one. Moreover, the coefficients are [[complex number]]s that need not be nonnegative. The fact that the coefficients can be negative as well as positive allows for cancellation, or [[interference]], between different computational paths, and is a key difference between quantum computing and probabilistic classical computing.<ref name="DiVincenzo 1995">DiVincenzo (1995).</ref> If you measure the three qubits, then you will observe a three-bit string. The probability of measuring a string will equal the squared magnitude of that string's coefficients. Thus a measurement of the quantum state with coefficients (''a'',''b'',...,''h'') gives the classical probability distribution <math>(|a|^2, |b|^2, ..., |h|^2)</math>. We say that the quantum state "collapses" to a classical state. Note that an eight-dimensional vector can be specified in many different ways, depending on what [[Basis (linear algebra)|basis]] you choose for the space. The basis of three-bit strings 000, 001, ..., 111 is known as the computational basis, and is often convenient, but other bases of [[unit vector|unit-length]], [[orthogonal]] vectors can also be used. [[Bra-ket notation|Ket notation]] is often used to make explicit the choice of basis. For example, the state (''a'',''b'',''c'',''d'',''e'',''f'',''g'',''h'') in the computational basis can be written as <math>a\,|000\rangle + b\,|001\rangle + c\,|010\rangle + d\,|011\rangle + e\,|100\rangle + f\,|101\rangle + g\,|110\rangle + h\,|111\rangle</math>, where, e.g., <math>|010\rangle</math> = (0,0,1,0,0,0,0,0). The computational basis for a single qubit (two dimensions) is <math>|0\rangle</math> = (1,0), <math>|1\rangle</math> = (0,1), but another common basis is the Hadamard basis of <math>|+\rangle = \left( \tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}} \right)</math> and <math>|-\rangle = \left( \tfrac{1}{\sqrt{2}}, -\tfrac{1}{\sqrt{2}} \right)</math>. Note that although recording a classical state of ''n'' bits, a 2<sup>''n''</sup>-dimensional probability distribution, requires an exponential number of real numbers, practically we can always think of the system as being exactly one of the ''n''-bit strings—we just don't know which one. Quantumly, this is no longer the case, and all 2<sup>''n''</sup> complex coefficients need to be kept track of to see how the quantum system evolves. For example, a 300-qubit quantum computer has a state described by 2<sup>300</sup> (approximately 10<sup>90</sup>) complex numbers, more than the number of atoms in the [[observable universe]]. == Initialization, execution and termination == While a classical three-bit state and a quantum three-qubit state are both eight-dimensional [[coordinate vector|vector]]s, they are manipulated quite differently for classical or quantum computation, respectively. For computing in either case, the system must be initialized, for example into the all-zeros string, i.e., (1,0,0,0,0,0,0,0) or <math>|000\rangle</math>. In classical randomized computation, the system evolves according to the application of [[Stochastic matrix|stochastic matrices]], which preserve that the probabilities add up to one (i.e., preserve the [[Taxicab geometry|L1 norm]]). In quantum computation, on the other hand, allowed operations are [[unitary matrix|unitary matrices]], which are effectively rotations (they preserve that the sum of the squares add up to one, the [[Euclidean metric|Euclidean or L2 norm]]). (Exactly what unitaries can be applied depend on the physics of the quantum device.) Consequently, since rotations can be undone by rotating backward, quantum computations are [[Reversible computing|reversible]]. (Technically, quantum operations can be probabilistic combinations of unitaries, so quantum computation really does generalize classical computation. See [[quantum circuit]] for a more precise formulation.) Finally, upon termination of the algorithm, the result needs to be read off. In the case of a classical computer, we ''sample'' from the [[probability distribution]] on the three-bit register to obtain one definite three-bit string, say 000. Quantumly, we ''[[quantum measurement|measure]]'' the three-qubit state, which is equivalent to collapsing the quantum state down to a classical distribution (with the coefficients in the classical state being the squared magnitudes of the coefficients for the quantum state, as described above) followed by sampling from that distribution. Note that this destroys the original quantum state. The output is [[random]], and for most algorithms will only give the correct answer with a certain probability. By repeatedly initializing, running and measuring the quantum computer, the probability of getting the correct answer can be increased. For more details on the sequences of operations used for various algorithms, see [[universal quantum computer]], [[Shor's algorithm]], [[Grover's algorithm]], [[Deutsch-Jozsa algorithm]], [[quantum Fourier transform]], [[quantum gate]], [[Adiabatic quantum computation|quantum adiabatic algorithm]] and [[quantum error correction]]. ==The power of quantum computers== [[Integer factorization]] is believed to be computationally infeasible with an ordinary computer for large integers that are the product of only a few [[prime number]]s (e.g., products of two 300-digit primes).<ref>http://modular.fas.harvard.edu/edu/Fall2001/124/misc/arjen_lenstra_factoring.pdf</ref> By comparison, a quantum computer could efficiently solve this problem using [[Shor's algorithm]] to find its factors. This ability would allow a quantum computer to "break" many of the [[cryptography|cryptographic]] systems in use today, in the sense that there would be a [[polynomial time]] (in the number of bits of the integer) algorithm for solving the problem. In particular, most of the popular [[Asymmetric Algorithms|public key ciphers]] are based on the difficulty of factoring integers (or the related [[discrete logarithm]] problem which can also be solved by Shor's algorithm), including forms of [[RSA]]. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security. The only way to increase the security of an algorithm like [[RSA]] would be to increase the key size and hope that an adversary does not have the resources to build and use a powerful enough quantum computer. A way out of this dilemma would be to use some kind of [[quantum cryptography]]. There are also some [[digital signature]] schemes that are believed to be secure against quantum computers. See for instance [[Lamport signature]]s. This dramatic advantage of quantum computers has only been discovered for factorization and discrete logarithms [[as of 2008|so far]]. However, there is no proof that the advantage is real: an equally fast classical algorithm may still be discovered. There is one other problem where quantum computers have a smaller, though significant (quadratic) advantage. It is ''quantum database search'', and can be solved by [[Grover's algorithm]]. In this case the advantage is provable. This establishes beyond doubt that (ideal) quantum computers are superior to classical computers for at least one problem. Consider a problem that has these four properties: # The only way to solve it is to guess answers repeatedly and check them, # There are ''n'' possible answers to check, # Every possible answer takes the same amount of time to check, and # There are no clues about which answers might be better: generating possibilities randomly is just as good as checking them in some special order. An example of this is a [[password cracker]] that attempts to guess the password for an [[encryption|encrypted]] file (assuming that the password has a maximum possible length). For problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of ''n'' (it would take an average of (''n''&nbsp;+&nbsp;1)/2 guesses to find the answer using a classical computer.) That can be a very large speedup, reducing some problems from years to seconds. It can be used to attack [[symmetric cipher]]s such as [[Triple DES]] and [[Advanced Encryption Standard|AES]] by attempting to guess the secret key. Regardless of whether any of these problems can be shown to have an advantage on a quantum computer, they nonetheless will always have the advantage of being an excellent tool for studying quantum mechanical interactions, which of itself is an enormous value to the scientific community. [[Grover's algorithm]] can also be used to obtain a quadratic speed-up [over a brute-force search] for a class of problems known as [[NP-complete]]. ==Problems and practicality issues== There are a number of practical difficulties in building a quantum computer, and thus far quantum computers have only solved trivial problems. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:<ref>{{cite web | url=http://arxiv.org/abs/quant-ph/0002077 | title=The Physical Implementation of Quantum Computation | accessdate=2006-11-17 | author=David P. DiVincenzo, IBM | date=2000-04-13}}</ref><!-- Is a paper on arxiv really the most reliable source we can find? Has it been published yet? --> * scalable physically to increase the number of qubits * qubits can be initialized to arbitrary values * quantum gates faster than [[decoherence]] time * universal gate set * qubits can be read easily To summarize the problems from the perspective of an engineer, one needs to solve the challenge of building a system which is isolated from everything ''except'' the measurement and manipulation mechanism. Furthermore, one needs to be able to turn off the coupling of the qubits to the measurement so as to not decohere the qubits while performing operations on them. ===Quantum decoherence=== One major problem is keeping the components of the computer in a coherent state, as the slightest interaction with the external world would cause the system to [[quantum decoherence|decohere]]. This effect causes the unitary character (and more specifically, the invertibility) of quantum computational steps to be violated. Decoherence times for candidate systems, in particular the transverse relaxation time T<sub>2</sub> (terminology used in [[Nuclear magnetic resonance|NMR]] and [[MRI]] technology, also called the ''dephasing time''), typically range between nanoseconds and seconds at low temperature.<ref name="DiVincenzo 1995" /> The issue for optical approaches are more difficult as these timescales are orders of magnitude lower and an often cited approach to overcome it uses an optical pulse shaping approach. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time. If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time. An often cited (but rather arbitrary) figure for required error rate in each gate is 10<sup>&minus;4</sup>. This implies that each gate must be able to perform its task 10,000 times faster than the decoherence time of the system. Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between ''L'' and ''L''<sup>2</sup>, where ''L'' is the number of bits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of ''L''. For a 1000-bit number, this implies a need about 10<sup>4</sup> qubits without error correction.<ref>{{cite web | url=http://arxiv.org/abs/quant-ph/0610117 | title=Is Fault-Tolerant Quantum Computation Really Possible? | accessdate=2007-02-16 | author=M. I. Dyakonov, Université Montpellier | date=[[2006-10-14]]}}</ref> With error correction, the figure would rise to about 10<sup>7</sup> qubits. Note that computation time is about <math>L^2</math> or about <math>10^7</math> steps and on 1 M[[Hertz|Hz]], about 10 [[second]]s. A very different approach to the stability-decoherence problem is to create a [[topological quantum computer]] with [[anyon]]s, quasi-particles used as threads and relying on [[braid theory]] to form stable logic gates.<ref>{{cite journal | title = Topological Quantum Computation | journal = Bulletin of the American Mathematical Society | volume = 40 | issue = 1 | pages = 31–38 | last = Freedman | first = Michael | coauthors = Alexei Kitaev, Michael Larsen, Zhenghan Wang | date = [[2002-10-20]] | doi = 10.1090/S0273-0979-02-00964-3}}</ref> ===Candidates=== There are a number of quantum computing candidates, among those: #[[Superconductor]]-based quantum computers (including [[SQUID]]-based quantum computers)<ref name ="ClarkeWilhelm2008">{{citation | url = http://www.nature.com/nature/journal/v453/n7198/full/nature07128.html | journal = Nature | year = 2008 | date = [[June 19]] [[2008]] | title = Superconducting quantum bits | first1 = John | last1 = Clarke | first2 = Frank | last2 = Wilhelm | volume = 453 | pages = 1031&ndash;1042 | doi = 10.1038/nature07128 }}</ref> #[[Trapped ion quantum computer]] #[[Optical lattice]]s #[[Topological quantum computer]] #[[Quantum dot]] on surface (e.g. the [[Loss-DiVincenzo quantum computer]]) #[[Nuclear magnetic resonance]] on molecules in solution (liquid NMR) #Solid state NMR [[Kane quantum computer]]s #Electrons on helium quantum computers #Cavity quantum electrodynamics (CQED) #[[Molecular magnet]] #[[Fullerene]]-based [[Electron paramagnetic resonance|ESR]] quantum computer #Optic-based quantum computers ([[Quantum optics]]) #[[Diamond-based quantum computer]]<ref>{{cite web | url=http://www.tgdaily.com/content/view/32306/118/ | title=Research indicates diamonds could be key to quantum storage | accessdate=2007-06-04 | author=Wolfgang Gruener, TG Daily | date=2007-06-01}}</ref><ref name ="Neumannetal2008">{{citation | url = http://www.sciencemag.org/cgi/content/abstract/320/5881/1326 | journal = Science | year = 2008 | date = [[June 6]] [[2008]] | title = Multipartite Entanglement Among Single Spins in Diamond | first1 = P. | last1 = Neumann | first2 = N. | last2 = Mizuochi | first3 = F. | last3 = Rempp | first4 = P. | last4 = Hemmer | first5 = H. | last5 = Watanabe | first6 = S. | last6 = Yamasaki | first7 = V. | last7 = Jacques | first8 = T. | last8 = Gaebel | first9 = F. | last9 = Jelezko | first10 = J. | last10 = Wrachtrup | volume = 320 | issue = 5881 | pages = 1326&ndash;1329 | doi = 10.1126/science.1157233 | pmid = 18535240 }}</ref> #[[Bose–Einstein condensate|Bose–Einstein condensate-based quantum computer]]<ref>{{cite web | url=http://www.itpro.co.uk/news/121086/trapped-atoms-could-advance-quantum-computing.html | title=Trapped atoms could advance quantum computing | accessdate=2007-07-26 | author=Rene Millman, IT PRO | date=2007-08-03}}</ref> #Transistor-based quantum computer - string quantum computers with entrainment of positive holes using a electrostatic trap #[[Spintronics|Spin-based quantum computer]] #[[Adiabatic quantum computation]]<ref>{{cite web | url=http://arxiv.org/pdf/quant-ph/0403090 | title=Scalable Superconducting Architecture for Adiabatic Quantum Computation | accessdate=2007-02-19 | author=William M Kaminsky, MIT | date=Date Unknown}}</ref> The large number of candidates shows explicitly that the topic, in spite of rapid progress, is still in its infancy. But at the same time there is also a vast amount of flexibility. In 2005, researchers at the [[University of Michigan]] built a [[semiconductor chip]] which functioned as an [[ion trap]]. Such devices, produced by standard [[lithography]] techniques, may point the way to scalable quantum computing tools.<ref>{{cite web | url=http://www.umich.edu/news/index.html?Releases/2005/Dec05/r121205b | title= U-M develops scalable and mass-producible quantum computer chip | accessdate=2006-11-17 | author=Ann Arbor | date=2005-12-12}}</ref> An improved version was made in 2006. ==Quantum computing in computational complexity theory== [[Image:BQP complexity class diagram.svg|thumb|The suspected relationship of BQP to other problem spaces.<ref name="Nielsen Chuang 2000">Nielsen and Chuang (2000).</ref>]] This section surveys what is currently known mathematically about the power of quantum computers. It describes the known results from [[computational complexity theory]] and the [[theory of computation]] dealing with quantum computers. The class of problems that can be efficiently solved by quantum computers is called '''[[BQP]]''', for "bounded error, quantum, polynomial time". Quantum computers only run '''[[Probabilistic algorithm|probabilistic]]''' algorithms, so '''BQP''' on quantum computers is the counterpart of '''[[BPP]]''' on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one quarter.<ref name="Nielsen Chuang 2000" /> A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in '''BQP'''. '''BQP''' is contained in the complexity class ''[[Sharp-P|#P]]'' (or more precisely in the associated class of decision problems ''P<sup>#P</sup>'') <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>, which is a subclass of [[PSPACE]]. '''BQP''' is suspected to be disjoint from [[NP-complete]] and a strict superset of '''[[P (complexity)|P]]''', but that is not known. Both [[integer factorization]] and [[discrete logarithm problem|discrete log]] are in '''BQP'''. Both of these problems are '''NP''' problems suspected to be outside '''BPP''', and hence outside '''P'''. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false. <ref name=BernVazi/> Quantum gates may be viewed as [[linear transformation]]s. Daniel S. Abrams and [[Seth Lloyd]] have shown that if ''nonlinear'' transformations are permitted, then NP-complete problems could be solved in polynomial time. It could even do so for [[Sharp-P-complete|#P-complete]] problems. They do not believe that such a machine is possible. Although quantum computers may be faster than classical computers, those described above can't solve any problems that classical computers can't solve, given enough time and memory (albeit possibly an amount that could never practically be brought to bear). A [[Turing machine]] can simulate these quantum computers, so such a quantum computer could never solve an [[undecidable]] problem like the [[halting problem]]. The existence of "standard" quantum computers does not disprove the [[Church-Turing thesis]].<ref name="Nielsen Chuang 2000" /><!-- Page number or range needed on book sources --> ==See also== {{commons|Quantum computer}} * [[List of emerging technologies]] * [[Quantum bus]] * [[Timeline of quantum computing]] * [[Chemical computer]] * [[DNA computer]] * [[Molecular computer]] ==Notes== <!--This article uses the Cite.php citation mechanism. If you would like more information on how to add references to this article, please see http://meta.wikimedia.org/wiki/Cite/Cite.php --> <div class="references-small" style="-moz-column-count:2; column-count:2;"> <references/> </div> == References == {{reflist}} <div class="references-small"> *DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". ''Experimental Proposals for Quantum Computation''. {{Arxiv | archive=quant-ph | id=0002077}}. *{{cite journal | author=DiVincenzo, David P. | title=Quantum Computation | journal=Science | year=1995 | volume=270 | issue=5234 | pages=255&ndash;261 | doi= 10.1126/science.270.5234.255 }} Table 1 lists switching and dephasing times for various systems. *{{cite journal | author=Feynman, Richard | title=Simulating physics with computers | journal=International Journal of Theoretical Physics | volume=21 | pages=467 | year=1982 | doi = 10.1007/BF02650179 <!--Retrieved from CrossRef by DOI bot-->}} *{{cite book | author=Jaeger, Gregg | title=Quantum Information: An Overview | publisher=Springer | location=Berlin | year=2006 | id=ISBN 0-387-35725-4}}[http://www.springer.com/west/home/physics?SGWID=4-10100-22-173664707-detailsPage=ppmmedia|toc"] *{{cite book | author= Nielsen, Michael and Isaac Chuang | title=Quantum Computation and Quantum Information | publisher=Cambridge University Press | location=Cambridge | year=2000 | id=ISBN 0-521-63503-9}} *{{cite book | author= Singer, Stephanie Frank | title=Linearity, Symmetry, and Prediction in the Hydrogen Atom | publisher=Springer | location=New York | year=2005 | id=ISBN 0-387-24637-1}} *{{cite book | author=Benenti, Giuliano | title=Principles of Quantum Computation and Information Volume 1| publisher=World Scientific | location=New Jersey | year=2004 | id=ISBN 9-812-38830-3}} * http://jquantum.sourceforge.net/jQuantumApplet.html Java quantum computer simulator. * http://www.phys.cs.is.nagoya-u.ac.jp/~watanabe/qcad/index.html Quantum computer emulator. * [https://gna.org/projects/quantumlibrary Quantum Library] : C++ Library that simulates the behaviour of qubits thus permitting the conception of quantum algorithms </div> {{Quantum computing}} * [[Category:Quantum information science| ]] [[Category:Computational models]] [[Category:Cryptography]] [[Category:Information theory]] [[Category:computational complexity theory|*]] [[Category:Classes of computers]] [[Category:Theoretical computer science]] [[ar:حاسوب كمومي]] [[da:Kvantecomputer]] [[de:Quantencomputer]] [[el:Κβαντικός υπολογιστής]] [[es:Computación cuántica]] [[fr:Calculateur quantique]] [[ko:양자 컴퓨터]] [[hr:Kvantno računalo]] [[id:Komputer kuantum]] [[it:Computer quantistico]] [[he:מחשב קוונטי]] [[lv:Kvantu dators]] [[lt:kvantinis kompiuteris]] [[nl:Kwantumcomputer]] [[ja:量子コンピュータ]] [[no:Kvantedatamaskin]] [[pl:Komputer kwantowy]] [[pt:Computador quântico]] [[ru:Квантовый компьютер]] [[sq:Kompjuteri kuantik]] [[sk:Kvantový počítač]] [[sl:Kvantni računalnik]] [[fi:Kvanttitietokone]] [[sv:Kvantdator]] [[vi:Máy tính lượng tử]] [[uk:Квантовий комп'ютер]] [[zh:量子计算机]]