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Quantum computing studies theoretical computation systems (quantum computers) that make direct use of quantummechanical phenomena, such as superposition and entanglement, to perform operations on data.[1] Quantum computers are different from digital electronic computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits (qubits), which can be in superpositions of states. A quantum Turing machine is a theoretical model of such a computer, and is also known as the universal quantum computer. Quantum computers share theoretical similarities with nondeterministic and probabilistic computers. The field of quantum computing was initiated by the work of Paul Benioff[2] and Yuri Manin in 1980,[3]Richard Feynman in 1982,[4] and David Deutsch in 1985.[5] A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1968.[6]
As of 2016, the development of actual quantum computers is still in its infancy, but experiments have been carried out in which quantum computational operations were executed on a very small number of quantum bits.[7] Both practical and theoretical research continues, and many national governments and military agencies are funding quantum computing research in an effort to develop quantum computers for civilian, business, trade, environmental and national security purposes, such as cryptanalysis.[8]
Largescale quantum computers would be able to solve certain problems much more quickly than any classical computers that use even the best currently known algorithms, like integer factorization using Shor's algorithm or the simulation of quantum manybody systems. There exist quantum algorithms, such as Simon's algorithm, that run faster than any possible probabilistic classical algorithm.[9] Given sufficient computational resources, a classical computer could in theory be made to simulate any quantum algorithm, as quantum computation does not violate the Church–Turing thesis.[10] On the other hand, quantum computers may be able to efficiently solve problems that no classical computer would be able to solve within a reasonable amount of time.
Contents
 1 Basis
 2 Mechanics
 3 Operation
 4 Potential
 4.1 Quantum decoherence
 5 Developments
 5.1 Timeline
 6 Relation to computational complexity theory
 7 See also
 8 References
 9 Further reading
 10 External links
A classical computer has a memory made up of bits, where each bit is represented by either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can represent a one, a zero, or any quantum superposition of those two qubit states; a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8 states. In general, a quantum computer with qubits can be in an arbitrary superposition of up to different states simultaneously (this compares to a normal computer that can only be in one of these states at any one time). A quantum computer operates by setting the qubits in a controlled initial state that represents the problem at hand and by manipulating those qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. The calculation ends with a measurement, collapsing the system of qubits into one of the pure states, where each qubit is zero or one, decomposing into a classical state. The outcome can therefore be at most classical bits of information. Quantum algorithms are often nondeterministic, in that they provide the correct solution only with a certain known probability.
An example of an implementation of qubits of a quantum computer could start with the use of particles with two spin states: "down" and "up" (typically written and , or and ). But in fact any system possessing an observable quantity A, which is conserved under time evolution such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. This is true because any such system can be mapped onto an effective spin1/2 system.
Mechanics[edit]
A quantum computer with a given number of qubits is fundamentally different from a classical computer composed of the same number of classical bits. For example, to represent the state of an nqubit system on a classical computer would require the storage of 2ncomplex coefficients. Although this fact may seem to indicate that qubits can hold exponentially more information than their classical counterparts, care must be taken not to overlook the fact that the qubits are only in a probabilistic superposition of all of their states. This means that when the final state of the qubits is measured, they will only be found in one of the possible configurations they were in before measurement. Moreover, it is incorrect to think of the qubits as only being in one particular state before measurement since the fact that they were in a superposition of states before the measurement was made directly affects the possible outcomes of the computation.
For example: Consider first a classical computer that operates on a threebit register. The state of the computer at any time is a probability distribution over the different threebit strings 000, 001, 010, 011, 100, 101, 110, 111. If it is a deterministic computer, then it is in exactly one of these states with probability 1. However, if it is a probabilistic computer, then there is a possibility of it being in any one of a number of different states. We can describe this probabilistic state by eight nonnegative numbers A,B,C,D,E,F,G,H (where A = is the probability that the computer is in state 000, B = is the probability that the computer is in state 001, etc.). There is a restriction that these probabilities sum to 1.
The state of a threequbit quantum computer is similarly described by an eightdimensional vector (a,b,c,d,e,f,g,h), called a ket. Here, however, the coefficients can have complex values, and it is the sum of the squares of the coefficients' magnitudes, , that must equal 1. These squared magnitudes represent the probability of each of the given states. However, because a complex number encodes not just a magnitude but also a direction in the complex plane, the phase difference between any two coefficients (states) represents a meaningful parameter. This is a fundamental difference between quantum computing and probabilistic classical computing.[12]
If you measure the three qubits, you will observe a threebit string. The probability of measuring a given string is the squared magnitude of that string's coefficient (i.e., the probability of measuring 000 = , the probability of measuring 001 = , etc..). Thus, measuring a quantum state described by complex coefficients (a,b,...,h) gives the classical probability distribution and we say that the quantum state "collapses" to a classical state as a result of making the measurement.
Note that an eightdimensional vector can be specified in many different ways depending on what basis is chosen for the space. The basis of bit strings (e.g., 000, 001, …, 111) is known as the computational basis. Other possible bases are unitlength, orthogonal vectors and the eigenvectors of the Paulix operator. Ket notation is often used to make the choice of basis explicit. For example, the state (a,b,c,d,e,f,g,h) in the computational basis can be written as:
where, e.g.,The computational basis for a single qubit (two dimensions) is and .
Using the eigenvectors of the Paulix operator, a single qubit is and .
Operation[edit]
Unsolved problem in physics:
Is a universal quantum computer sufficient to efficiently simulate an arbitrary physical system? 
While a classical threebit state and a quantum threequbit state are both eightdimensional vectors, they are manipulated quite differently for classical or quantum computation. For computing in either case, the system must be initialized, for example into the allzeros string, , corresponding to the vector (1,0,0,0,0,0,0,0). In classical randomized computation, the system evolves according to the application of stochastic matrices, which preserve that the probabilities add up to one (i.e., preserve the L1 norm). In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively rotations (they preserve that the sum of the squares add up to one, the 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. (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 threebit register to obtain one definite threebit string, say 000. Quantum mechanically, we measure the threequbit 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. Many algorithms will only give the correct answer with a certain probability. However, by repeatedly initializing, running and measuring the quantum computer's results, the probability of getting the correct answer can be increased. In contrast, counterfactual quantum computation allows the correct answer to be inferred when the quantum computer is not actually running in a technical sense, though earlier initialization and frequent measurements are part of the counterfactual computation protocol.
For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch–Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.
Potential[edit]
Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300digit primes).[13] 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 decrypt many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular the RSA, DiffieHellman, and Elliptic curve DiffieHellman algorithms could be broken. 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.
However, other cryptographic algorithms do not appear to be broken by those algorithms.[14][15] Some publickey algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory.[14][16]Latticebased cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a wellstudied open problem.[17] It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case,[18] meaning that symmetric key lengths are effectively halved: AES256 would have the same security against an attack using Grover's algorithm that AES128 has against classical bruteforce search (see Key size). Quantum cryptography could potentially fulfill some of the functions of public key cryptography.
Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems,[19] including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely[citation needed]. For some problems, quantum computers offer a polynomial speedup. The most wellknown example of this is quantum database search, which can be solved by Grover's algorithm using quadratically fewer queries to the database than are required by classical algorithms. In this case the advantage is provable. Several other examples of provable quantum speedups for query problems have subsequently been discovered, such as for finding collisions in twotoone functions and evaluating NAND trees.
Consider a problem that has these four properties:
 The only way to solve it is to guess answers repeatedly and check them,
 The number of possible answers to check is the same as the number of inputs,
 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 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 the number of inputs. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret key.[20]
Grover's algorithm can also be used to obtain a quadratic speedup over a bruteforce search for a class of problems known as NPcomplete.
Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.[21] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[22]
There are a number of technical challenges in building a largescale quantum computer, and thus far quantum computers have yet to solve a problem faster than a classical computer. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:[23]
 scalable physically to increase the number of qubits;
 qubits that can be initialized to arbitrary values;
 quantum gates that are faster than decoherence time;
 universal gate set;
 qubits that can be read easily.
Quantum decoherence[edit]
One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is nonunitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems, in particular the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[12] Currently, some quantum computers require their qubits to be cooled to 20 millikelvin in order to prevent significant decoherence.[24]
These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an oftencited approach to overcoming them is optical pulse shaping. 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 figure for required error rate in each gate is 10−4. This implies that each gate must be able to perform its task in one 10,000th of the coherence 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 bits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, 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 1000bit number, this implies a need for about 104 bits without error correction.[25] With error correction, the figure would rise to about 107 bits. Note that computation time is about L2 or about 107 steps and on 1 MHz, about 10 seconds.
A very different approach to the stabilitydecoherence problem is to create a topological quantum computer with anyons, quasiparticles used as threads and relying on braid theory to form stable logic gates.[26][27]
Developments[edit]
There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are:
 Quantum gate array (computation decomposed into sequence of fewqubit quantum gates)
 Oneway quantum computer (computation decomposed into sequence of onequbit measurements applied to a highly entangled initial state or cluster state)
 Adiabatic quantum computer, based on Quantum annealing (computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution)[28]
 Topological quantum computer[29] (computation decomposed into the braiding of anyons in a 2D lattice)
The Quantum Turing machine is theoretically important but direct implementation of this model is not pursued. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead.
For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):
 Superconductorbased quantum computers (including SQUIDbased quantum computers)[30][31] (qubit implemented by the state of small superconducting circuits (Josephson junctions))
 Trapped ion quantum computer (qubit implemented by the internal state of trapped ions)
 Optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)
 Quantum dot computer, spinbased (e.g. the LossDiVincenzo quantum computer [32]) (qubit given by the spin states of trapped electrons)
 Quantum dot computer, spatialbased (qubit given by electron position in double quantum dot)[33]
 Nuclear magnetic resonance on molecules in solution (liquidstate NMR) (qubit provided by nuclear spins within the dissolved molecule)
 Solidstate NMR Kane quantum computers (qubit realized by the nuclear spin state of phosphorus donors in silicon)
 Electronsonhelium quantum computers (qubit is the electron spin)
 Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to highfinesse cavities)
 Molecular magnet[34] (qubit given by spin states)
 Fullerenebased ESR quantum computer (qubit based on the electronic spin of atoms or molecules encased in fullerenes)
 Linear optical quantum computer (qubits realized by processing states of different modes of light through linear elements e.g. mirrors, beam splitters and phase shifters)[35]
 Diamondbased quantum computer[36][37][38] (qubit realized by electronic or nuclear spin of nitrogenvacancy centers in diamond)
 Bose–Einstein condensatebased quantum computer[39]
 Transistorbased quantum computer – string quantum computers with entrainment of positive holes using an electrostatic trap
 Rareearthmetaliondoped inorganic crystal based quantum computers[40][41] (qubit realized by the internal electronic state of dopants in optical fibers)
The large number of candidates demonstrates that the topic, in spite of rapid progress, is still in its infancy. There is also a vast amount of flexibility.
Timeline[edit]
Main article: Timeline of quantum computing
In 2001, researchers demonstrated Shor's algorithm to factor 15 using a 7qubit NMR computer.[42]
In 2005, researchers at the University of Michigan built a semiconductor chip ion trap. Such devices from standard lithography, may point the way to scalable quantum computing.[43]
In 2009, researchers at Yale University created the first solidstate quantum processor. The twoqubit superconducting chip had artificial atom qubits made of a billion aluminum atoms that acted like a single atom that could occupy two states.[44][45]
A team at the University of Bristol, also created a silicon chip based on quantum optics, able to run Shor's algorithm.[46] Further developments were made in 2010.[47] Springer publishes a journal (Quantum Information Processing) devoted to the subject.[48]
In February 2010, Digital Combinational Circuits like adder, subtractor etc. are designed with the help of Symmetric Functions organized from different quantum gates.[49][50]
April 2011, a team of scientists from Australia and Japan made a breakthrough in quantum teleportation. They successfully transferred a complex set of quantum data with full transmission integrity, without affecting the qubits' superpositions.[51][52]
In 2011, DWave Systems announced the first commercial quantum annealer, the DWave One, claiming a 128 qubit processor.[53] On May 25, 2011 Lockheed Martin agreed to purchase a DWave One system.[54] Lockheed and the University of Southern California (USC) will house the DWave One at the newly formed USC Lockheed Martin Quantum Computing Center.[55] DWave's engineers designed the chips with an empirical approach, focusing on solving particular problems. Investors liked this more than academics, who said DWave had not demonstrated they really had a quantum computer. Criticism softened after a DWave paper in Nature, that proved the chips have some quantum properties.[56][57] Two published papers have suggested that the DWave machine's operation can be explained classically, rather than requiring quantum models.[58][59] Later work showed that classical models are insufficient when all available data is considered.[60] Experts remain divided on the ultimate classification of the DWave systems though their quantum behavior was established concretely with a demonstration of entanglement.[61]
During the same year, researchers at the University of Bristol created an allbulk optics system that ran a version of Shor's algorithm to successfully factor 21.[62]
In September 2011 researchers proved quantum computers can be made with a Von Neumann architecture (separation of RAM).[63]
In November 2011 researchers factorized 143 using 4 qubits.[64]
In February 2012 IBM scientists said that they had made several breakthroughs in quantum computing with superconducting integrated circuits.[65]
In April 2012 a multinational team of researchers from the University of Southern California, Delft University of Technology, the Iowa State University of Science and Technology, and the University of California, Santa Barbara, constructed a twoqubit quantum computer on a doped diamond crystal that can easily be scaled up and is functional at room temperature. Two logical qubit directions of electron spin and nitrogen kernels spin were used, with microwave impulses. This computer ran Grover's algorithm generating the right answer from the first try in 95% of cases.[66]
In September 2012, Australian researchers at the University of New South Wales said the world's first quantum computer was just 5 to 10 years away, after announcing a global breakthrough enabling manufacture of its memory building blocks. A research team led by Australian engineers created the first working qubit based on a single atom in silicon, invoking the same technological platform that forms the building blocks of modernday computers.[67][68]
In October 2012, Nobel Prizes were presented to David J. Wineland and Serge Haroche for their basic work on understanding the quantum world, which may help make quantum computing possible.[69][70]
In November 2012, the first quantum teleportation from one macroscopic object to another was reported.[71][72]
In December 2012, the first dedicated quantum computing software company, 1QBit was founded in Vancouver, BC.[73] 1QBit is the first company to focus exclusively on commercializing software applications for commercially available quantum computers, including the DWave Two. 1QBit's research demonstrated the ability of superconducting quantum annealing processors to solve realworld problems.[74]
In February 2013, a new technique, boson sampling, was reported by two groups using photons in an optical lattice that is not a universal quantum computer but may be good enough for practical problems. Science Feb 15, 2013
In May 2013, Google announced that it was launching the Quantum Artificial Intelligence Lab, hosted by NASA's Ames Research Center, with a 512qubit DWave quantum computer. The USRA (Universities Space Research Association) will invite researchers to share time on it with the goal of studying quantum computing for machine learning.[75]
In early 2014 it was reported, based on documents provided by former NSA contractor Edward Snowden, that the U.S. National Security Agency (NSA) is running a .7 million research program (titled "Penetrating Hard Targets") to develop a quantum computer capable of breaking vulnerable encryption.[76]
In 2014, a group of researchers from ETH Zürich, USC, Google and Microsoft reported a definition of quantum speedup, and were not able to measure quantum speedup with the DWave Two device, but did not explicitly rule it out.[77][78]
In 2014, researchers at University of New South Wales used silicon as a protectant shell around qubits, making them more accurate, increasing the length of time they will hold information and possibly made quantum computers easier to build.[79]
In April 2015 IBM scientists claimed two critical advances towards the realization of a practical quantum computer. They claimed the ability to detect and measure both kinds of quantum errors simultaneously, as well as a new, square quantum bit circuit design that could scale to larger dimensions.[80]
In October 2015 researchers at University of New South Wales built a quantum logic gate in silicon for the first time.[81]
In December 2015 NASA publicly displayed the world's first fully operational million quantum computer made by the Canadian company DWave at the Quantum Artificial Intelligence Laboratory at its Ames Research Center in California's Moffett Field. The device was purchased in 2013 via a partnership with Google and Universities Space Research Association. Despite using quantum effects the algorithm run on the quantum computer does not outperform Selby’s algorithm run on a classical computer.[82]
Relation to computational complexity theory[edit]
Main article: Quantum complexity theory
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 algorithms, so BQP on quantum computers is the counterpart of BPP ("bounded error, probabilistic, polynomial time") on classical computers. It is defined as the set of problems solvable with a polynomialtime algorithm, whose probability of error is bounded away from one half.[84] 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 #P (or more precisely in the associated class of decision problems P#P),[85] which is a subclass of PSPACE.
BQP is suspected to be disjoint from NPcomplete and a strict superset of P, but that is not known. Both integer factorization and 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 NPcomplete. There is a common misconception that quantum computers can solve NPcomplete problems in polynomial time. That is not known to be true, and is generally suspected to be false.[85]
The capacity of a quantum computer to accelerate classical algorithms has rigid limits—upper bounds of quantum computation's complexity. The overwhelming part of classical calculations cannot be accelerated on a quantum computer.[86] A similar fact takes place for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.[87]
Although quantum computers may be faster than classical computers for some problem types, those described above can't solve any problem that classical computers can't already solve. 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.[88] It has been speculated that theories of quantum gravity, such as Mtheory or loop quantum gravity, may allow even faster computers to be built. Currently, defining computation in such theories is an open problem due to the problem of time, i.e., there currently exists no obvious way to describe what it means for an observer to submit input to a computer and later receive output.[89]
See also[edit]
 Chemical computer
 David Deutsch
 DNA computer
 Electronic quantum holography
 List of emerging technologies
 Richard Feynman
 IARPA
 Kane quantum computer
 Natural computing
 Normal mode
 Photonic computing
 Postquantum cryptography
 Quantum annealing
 Quantum bus
 Quantum cognition
 Quantum gate
 Quantum threshold theorem
 Soliton
 Theoretical computer science
 Timeline of quantum computing
 Topological quantum computer
 Valleytronics
 DWave Two
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Further reading[edit]
 Nielsen, Michael; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0521635039. OCLC 174527496.
 Abbot, Derek; Doering, Charles R.; Caves, Carlton M.; Lidar, Daniel M.; Brandt, Howard E.; Hamilton, Alexander R.; Ferry, David K.; GeaBanacloche, Julio; Bezrukov, Sergey M.; Kish, Laszlo B. (2003). "Dreams versus Reality: Plenary Debate Session on Quantum Computing". Quantum Information Processing 2 (6): 449–472. arXiv:quantph/0310130. doi:10.1023/B:QINP.0000042203.24782.9a. hdl:2027.42/45526.
 DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". Experimental Proposals for Quantum Computation. arXiv:quantph/0002077
 DiVincenzo, David P. (1995). "Quantum Computation". Science 270 (5234): 255–261. Bibcode:1995Sci...270..255D. doi:10.1126/science.270.5234.255. Table 1 lists switching and dephasing times for various systems.
 Feynman, Richard (1982). "Simulating physics with computers". International Journal of Theoretical Physics 21 (6–7): 467. Bibcode:1982IJTP...21..467F. doi:10.1007/BF02650179.
 Jaeger, Gregg (2006). Quantum Information: An Overview. Berlin: Springer. ISBN 0387357254. OCLC 255569451.
 Singer, Stephanie Frank (2005). Linearity, Symmetry, and Prediction in the Hydrogen Atom. New York: Springer. ISBN 0387246371. OCLC 253709076.
 Benenti, Giuliano (2004). Principles of Quantum Computation and Information Volume 1. New Jersey: World Scientific. ISBN 9812388303. OCLC 179950736.
 Lomonaco, Sam. Four Lectures on Quantum Computing given at Oxford University in July 2006
 C. Adami, N.J. Cerf. (1998). "Quantum computation with linear optics". arXiv:quantph/9806048v1.
 Stolze, Joachim; Suter, Dieter (2004). Quantum Computing. WileyVCH. ISBN 3527404384.
 Mitchell, Ian (1998). "Computing Power into the 21st Century: Moore's Law and Beyond".
 Landauer, Rolf (1961). "Irreversibility and heat generation in the computing process" (PDF).
 Moore, Gordon E. (1965). Cramming more components onto integrated circuits. Electronics Magazine.
 Keyes, R. W. (1988). Miniaturization of electronics and its limits. IBM Journal of Research and Development.
 Nielsen, M. A.; Knill, E.; Laflamme, R. "Complete Quantum Teleportation By Nuclear Magnetic Resonance".
 Vandersypen, Lieven M.K.; Yannoni, Constantino S.; Chuang, Isaac L. (2000). Liquid state NMR Quantum Computing.
 Hiroshi, Imai; Masahito, Hayashi (2006). Quantum Computation and Information. Berlin: Springer. ISBN 3540331328.
 Berthiaume, Andre (1997). "Quantum Computation".
 Simon, Daniel R. (1994). "On the Power of Quantum Computation". Institute of Electrical and Electronic Engineers Computer Society Press.
 "Seminar Post Quantum Cryptology". Chair for communication security at the RuhrUniversity Bochum.
 Sanders, Laura (2009). "First programmable quantum computer created".
 "New trends in quantum computation".
 Wichert, Andreas (2014). Principles of Quantum Artificial Intelligence. World Scientific Publishing Co. ISBN 9789814566742.
 Akama, Seiki (2014). Elements of Quantum Computing: History, Theories and Engineering Applications. Springer International Publishing. ISBN 9783319082844.
External links[edit]
 Stanford Encyclopedia of Philosophy: "Quantum Computing" by Amit Hagar.
 Quantum Annealing and Computation: A Brief Documentary Note, A. Ghosh and S. Mukherjee
 Maryland University Laboratory for Physical Sciences: conducts researches for the quantum computerbased project led by the NSA, named 'Penetrating Hard Target'.
 Visualized history of quantum computing
 Quantum Annealing and Analog Quantum Computation by Arnab Das and BK Chakrabarti
 Quantum computing for the determined – 22 video lectures by Michael Nielsen
 Video Lectures by David Deutsch
 Lectures at the Institut Henri Poincaré (slides and videos)
 Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008)
 Quantum Computing research by Mikko Möttönen at Aalto University (video) on YouTube
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