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Encyclopedia > Quantum computer
The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.

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 qubits. 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.[2] (See Timeline of quantum computing for details on current and past progress.) To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Cryptanalysis (from the Greek kryptÃ³s, hidden, and analÃ½ein, to loosen or to untie) is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. ... Timeline of quantum computers // 1970 - Stephen Wiesner invents conjugate coding. ...

If large-scale quantum computers can be built, they will be able to solve certain problems exponentially faster than any of our current classical computers (for example Shor's algorithm). Quantum computers are different from other computers such as DNA computers and traditional computers based on transistors. Some computing architectures such as optical computers may use classical superposition of electromagnetic waves, but without some specifically quantum mechanical resources such as entanglement, they have less potential for computational speed-up than quantum computers. Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ... The tower of a personal computer. ... DNA computing is a form of computing which uses DNA and molecular biology, instead of the traditional silicon-based computer technologies. ... Assorted discrete transistors A transistor is a semiconductor device, commonly used as an amplifier or an electrically controlled switch. ... An optical computer is a computer that uses light instead of electricity (i. ... It has been suggested that Quantum coherence be merged into this article or section. ...

## The basis of quantum computing GA_googleFillSlot("encyclopedia_square");

A classical computer has a memory made up of bits, where each bit holds either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can hold a one, a zero, or, crucially, a quantum superposition of these, allowing for an infinite number of states. A quantum computer operates by manipulating those qubits with (possibly a suite of) quantum logic gates. This article is about the unit of information. ... To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits. ...

An example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: "up" and "down" (typically written $|0rangle$ and $|1rangle$). 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 eigenvalues, is a suitable candidate for implementing a qubit. That's because any such system can be mapped onto an effective spin-1/2. In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In quantum mechanics, spin is an intrinsic property of all elementary particles. ...

## Bits vs. qubits

Consider first a classical computer that operates on a 3-bit register. At any given time, the bits in the register are in a definite state, such as 101. In a quantum computer, however, the qubits can be in a superposition of all the classically allowed states. In fact, the register is described by a wavefunction: In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to frequently used valuesâ€”typically, these values are involved in multiple expression evaluations occurring within a small region on the program. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

$|psi rangle = a,|000rangle + b,|001rangle + c,|010rangle + d,|011rangle + e,|100rangle + f,|101rangle + g,|110rangle + h,|111rangle$
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).[3]

where the coefficients a, b, c,..., h are complex numbers whose amplitudes squared are the probabilities to measure the qubits in each state- for example, | c | 2 is the probability to measure the register in the state 010. It is important that these numbers are complex, because the phases of the numbers can constructively and destructively interfere with one another; this is an important feature for quantum algorithms.[4] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... This article is about a portion of a periodic process. ...

Recording the state of a quantum register requires an exponential number of complex numbers (the 3-qubit register above requires 23 = 8 complex numbers). The number of classical bits required even to estimate the complex numbers of some quantum state grows exponentially with the number of qubits. For a 300-qubit quantum register, somewhere on the order of 1090 classical registers are required, more than there are atoms in the observable universe.[5] A quantum register is the quantum computing analogue of a classical processor register. ... See universe for a general discussion of the universe. ...

## Initialization, execution and termination

In our example, the contents of the qubit registers can be thought of as an 8-dimensional complex vector. An algorithm for a quantum computer must initialize this vector in some specified form (dependent on the design of the quantum computer). In each step of the algorithm, that vector is modified by multiplying it by a unitary matrix. The matrix is determined by the physics of the device. The unitary character of the matrix ensures the matrix is invertible (so each step is reversible). In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... The term reversible computing refers to any computational process that is (at least to some close approximation) reversible, i. ...

Upon termination of the algorithm, the 8-dimensional complex vector stored in the register must be somehow read off from the qubit register by a quantum measurement. However, by the laws of quantum mechanics, that measurement will yield a random 3-bit string (and it will destroy the stored state as well). This random string can be used in computing the value of a function because (by design) the probability distribution of the measured output bitstring is skewed in favor of the correct value of the function. By repeated runs of the quantum computer and measurement of the output, the correct value can be determined, to a high probability, by majority polling of the outputs. In brief, quantum computations are probabilistic; see quantum circuit for a more precise formulation. The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Random redirects here. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In quantum mechanics, a quantum circuit is a specific model for a quantum computational device. ...

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, quantum adiabatic algorithm and quantum error correction. Also refer to the growing field of quantum programming. The universal quantum computer or universal quantum Turing machine (UQTM) is a theoretical machine that combines both Church-Turing and quantum principles. ... Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ... Grovers algorithm is a quantum algorithm for searching an unsorted database with N entries in O(N1/2) time and using O(logN) storage space (see big O notation). ... The Deutsch-Jozsa algorithm is a quantum algorithm, proposed by David Deutsch and Richard Jozsa in 1992. ... The quantum Fourier transform is the discrete Fourier transform with a particular decomposition into a product of simpler unitary matrices. ... A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits. ... Quantum error correction is for use in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. ... It has been suggested that Quantum programming language be merged into this article or section. ...

## The power of quantum computers

A way out of this dilemma would be to use some kind of quantum cryptography. The are also are some digital signature schemes that are believed to be secure against quantum computers. See for instance Lamport signatures. Quantum cryptography is an approach based on quantum physics for secure communications. ... A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a signature in digital, rather than written, form. ... In cryptography, a Lamport signature or Lamport one-time signature scheme is a method for constructing a digital signature. ...

This dramatic advantage of quantum computers has only been discovered for these problems so far: factoring, discrete logarithm. 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. 2007 is a common year starting on Monday of the Gregorian calendar. ... Grovers algorithm is a quantum algorithm for searching an unsorted database with N entries in O(N1/2) time and using O(logN) storage space (see big O notation). ...

Consider a problem that has these four properties:

1. The only way to solve it is to guess answers repeatedly and check them,
2. There are n possible answers to check,
3. Every possible answer takes the same amount of time to check, and
4. 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). Password cracking is the process of recovering secret passwords from data that has been stored in or transmitted by a computer system, typically, by repeatedly verifying guesses for the password. ... Encrypt redirects here. ...

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 + 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 ciphers such as Triple DES and 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. A symmetric-key algorithm is an algorithm for cryptography that uses the same cryptographic key to encrypt and decrypt the message. ... In cryptography, Triple DES (also 3DES) is a block cipher formed from the Data Encryption Standard (DES) cipher. ... In cryptography, the Advanced Encryption Standard (AES), also known as Rijndael, is a block cipher adopted as an encryption standard by the U.S. government. ...

There are currently no other practical problems known where quantum computers give a large speedup over classical computers. Research is continuing, and more problems may yet be found.

## 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:[7]

• 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 problem 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 is the general term for the consequences of irreversible quantum entanglement. ...

### 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 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 T2 (terminology used in NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[8] 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. In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... NMR redirects here. ... The mri are a fictional alien species in the Faded Sun Trilogy of C.J. Cherryh. ... Debabrata Goswami, is an Indian spectroscopist, winner of the Wellcome Trust Senior Research Fellow Award (2004), Swarnajayanti Award (2004), presently Associate Professor of Chemistry at Indian Institute of Technology, Kanpur (IITK). ...

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 10−4. 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 L4 and L6, where L is the number of bits in the number to be factored. For a 1000-bit number, this implies a need for 1012 to 1018.[9]

A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on knot theory to form stable logic gates.[10] A topological quantum computer is a theoretical quantum computer that uses quasiparticles called anyons where their world lines form threads that cross over one another to form braids in a two-dimensional world. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... Trefoil knot, the simplest non-trivial knot. ...

### Candidates

There are a number of quantum computing candidates, among those:

1. Superconductor-based quantum computers (including SQUID-based quantum computers)
2. Trapped ion quantum computer
3. Electrons on helium quantum computers
4. "Nuclear magnetic resonance on molecules in solution"-based
5. "Quantum dot on surface"-based (e.g. the Loss-DiVincenzo quantum computer)
6. "Cavity quantum electrodynamics" (CQED)-based
7. "Molecular magnet"-based
8. Fullerene-based ESR quantum computer
9. Solid state NMR Kane quantum computers
10. Optic-based quantum computers (Quantum optics)
11. Topological quantum computer
12. Spin-based quantum computer
14. Diamond-based quantum computer[12]
15. Bose–Einstein condensate-based quantum computer[13]

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. Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... For other uses, see Squid (disambiguation). ... A Trapped ion quantum computer is a type of quantum computer. ... NMR redirects here. ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... Making a saline water solution by dissolving table salt (NaCl) in water This article is about chemical solutions. ... A quantum dot is a semiconductor nanostructure that confines the motion of conduction band electrons, valence band holes, or excitons (bound pairs of conduction band electrons and valence band holes) in all three spatial directions. ... A double quantum dot. ... Molecular magnets are systems where a permanent magnetization and magnetic hysteresis can be achieved (although usually at extremely low temperatures) not through a three-dimensional magnetic ordering, but as a purely one-molecule phenomenon. ... The Icosahedral Fullerene C540 C60 and C-60 redirect here. ... Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a technique for studying chemical species that have one or more unpaired electrons, such as organic and inorganic free radicals or inorganic complexes possessing a transition metal ion. ... The Kane quantum computer is a proposal for a scalable quantum computer proposed by Bruce Kane in 19981, then at the University of New South Wales. ... Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... A topological quantum computer is a theoretical quantum computer that uses quasiparticles called anyons where their world lines form threads that cross over one another to form braids in a two-dimensional world. ... Unsolved problems in physics: Is it possible to construct a practical electronic device that operates on the spin of the electron, rather than its charge? Spintronics (a neologism for spin-based electronics), also known as magnetoelectronics, is an emergent technology which exploits the quantum spin states of electrons as well... A Boseâ€“Einstein condensate (BEC) is a state of matter formed by a system of bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 kelvin or âˆ’273. ...

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.[14] An improved version was made in 2006. The University of Michigan, Ann Arbor (U of M, UM or simply Michigan) is a coeducational public research university in the state of Michigan, and one of the foremost universities in the United States. ... Integrated circuit showing memory blocks, logic and input/output pads around the periphery A monolithic integrated circuit (also known as IC, microchip, silicon chip, computer chip or chip) is a miniaturized electronic circuit (consisting mainly of semiconductor devices, as well as passive components) which has been manufactured in the surface... An ion trap is a combination of electric or magnetic fields that captures ions in a region of a vacuum system or tube. ... Lithography stone and mirror-image print of a map of Munich. ...

D-Wave Systems Inc. claims to be the world’s first — and only — provider of quantum computing systems designed to run commercial applications. On 13 February, 2007 they ran an initial demonstration of their Orion quantum computing system, which is built around a 16-qubit superconducting adiabatic quantum computer processor.[15] However, since D-Wave Systems has not released the full details of Orion to the scientific community, many experts in the field have expressed skepticism.[16] D-Wave Systems, Inc. ... D-Wave Systems, Inc. ... To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... In quantum mechanics, an adiabatic process is an infinitely slow change in the Hamiltonian of a system. ...

## Quantum computing in computational complexity theory

The suspected relationship of BQP to other problem spaces[17]

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. Image File history File links BQP_complexity_class_diagram. ... Image File history File links BQP_complexity_class_diagram. ... As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a computer. ...

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 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.[18] 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, in computational complexity theory, stands for Bounded error, Quantum, Polynomial time. It denotes the class of problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/4 for all instances. ... A randomized algorithm is an algorithm which is allowed to flip a truly random coin. ... This article is about the complexity class. ...

BQP is suspected to be disjoint from NP-complete 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 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. In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... Prime decomposition redirects here. ... In abstract algebra and its applications, the discrete logarithms are defined in group theory in analogy to ordinary logarithms. ...

An operator for a quantum computer can be thought of as changing a vector by multiplying it with a particular matrix. Multiplication by a matrix is a linear operation. Daniel S. Abrams and Seth Lloyd have shown that if a quantum computer could be designed with nonlinear operators, then it could solve NP-complete problems in polynomial time. It could even do so for #P-complete problems. They do not believe that such a machine is possible. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Seth Lloyd is a Professor of Mechanical Engineering at MIT. His research area is the interplay of information with complex systems, especially quantum systems. ... The title given to this article is incorrect due to technical limitations. ...

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.[19] For the test of artificial intelligence, see Turing test. ... Undecidable has more than one meaning: In mathematical logic: A decision problem is undecidable if there is no known algorithm that decides it. ... In computability theory the halting problem is a decision problem which can be stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ... In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ...

Very recently, many researchers have begun to investigate the possibility of using quantum mechanics for hypercomputation - that is, solving undecidable problems. Such claims have been met with considerable skepticism as to whether it is even theoretically possible; see the hypercomputation article for more details. Hypercomputation refers to various proposed methods for the computation of non-Turing-computable functions. ... Hypercomputation refers to various proposed methods for the computation of non-Turing-computable functions. ...

## Quantum computers in fiction

The TV show Code Lyoko features a quantum supercomputer. Code Lyoko is a French animated television series featuring both conventional animation and CGI animation. ...

Quantum computing technology is used in the alternate reality game for the Nine Inch Nails album Year Zero to transmit data to the present from the year 2022. NIN redirects here. ... Year Zero (also known as Halo 24) is the sixth Nine Inch Nails studio album, released on April 16, 2007 in Europe, April 17 in the United States, and April 25, 2007 in Japan. ...

In the 2007 movie Transformers, it is stated that the Decepticons use quantum computing to break the U.S. Army's data signal, therefore being able to access secure files. It is stated that even with the most powerful supercomputer, it would take 22 years to do. For the 1986 animated film, see The Transformers: The Movie. ... The Decepticons (also known as Destrons in Japan) are the enemies of the Autobots, and the villains in the Transformers toyline and related spin-off comics and cartoons. ...

The Michael Crichton novel Timeline also features use of a quantum computer, especially with regards to deconstructing and reconstructing objects being transported across time (or, more accurately, across the multiverse). Timeline is a science fiction novel by Michael Crichton that was published in November 1999. ...

The Robert J. Sawyer trilogy The Neanderthal Parallax involves a quantum computer which unintentionally opens a gateway between different versions of Earth. Robert J. Sawyer is a Canadian science fiction writer, dubbed the dean of Canadian science fiction by the Ottawa Citizen in 1999. ... The Neanderthal Parallax is a trilogy of novels by Robert J. Sawyer. ...

Quantum computers and mechanics are also used in the Noein anime series extensively. Original run 2005-10-12 â€“ 2006-03-29 No. ...

In the MMORPG EvE Online the Caldari State makes use of Quantum computers

In Dan Brown's novel Digital Fortress, the NSA's cypher breaking computer TRANSLTR uses quantum computing to break encryption keys of encrypted E-mails. This article is about the writer. ... Digital Fortress is a novel by American author Dan Brown and published in 1998 by St. ...

A quantum bus is a device which can be used to store or transfer information between independent qubits in a quantum computer, or combine two qubits into a superposition. ... Timeline of quantum computers // 1970 - Stephen Wiesner invents conjugate coding. ...

## Notes

1. ^ "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.
2. ^ Quantum Information Science and Technology Roadmap for a sense of where the research is heading.
3. ^ Waldner, Jean-Baptiste (2007). Nanocomputers and Swarm Intelligence. ISTE, p157. ISBN 2746215160.
4. ^ DiVincenzo, David (October 1995). "Quantum Computation". Science 270 (5234): 255-261. Retrieved on 2007-04-25.
5. ^ Note that the coefficients are not all independent, since the probabilities must sum to 1.
6. ^ http://modular.fas.harvard.edu/edu/Fall2001/124/misc/arjen_lenstra_factoring.pdf
7. ^ David P. DiVincenzo, IBM (2000-04-13). The Physical Implementation of Quantum Computation. Retrieved on 2006-11-17.
8. ^ David P. DiVincenzo, IBM (1995-10-13). Quantum Computation. Retrieved on 2006-11-17.
9. ^ M. I. Dyakonov, Université Montpellier (2006-10-14). Is Fault-Tolerant Quantum Computation Really Possible?. Retrieved on 2007-02-16.
10. ^ Freedman, Michael; Alexei Kitaev, Michael Larsen, Zhenghan Wang (2002-10-20). "Topological Quantum Computation". Bulletin of the American Mathematical Society 40 (1): 31—38. Retrieved on 2007-05-07.
11. ^ William M Kaminsky, MIT (Date Unknown). Scalable Superconducting Architecture for Adiabatic Quantum Computation. Retrieved on 2007-02-19.
12. ^ Wolfgang Gruener, TG Daily (2007-06-01). Research indicates diamonds could be key to quantum storage. Retrieved on 2007-06-04.
13. ^ Rene Millman, IT PRO (2007-08-03). Trapped atoms could advance quantum computing. Retrieved on 2007-07-26.
14. ^ Ann Arbor (2005-12-12). U-M develops scalable and mass-producible quantum computer chip. Retrieved on 2006-11-17.
15. ^ Comment on D-Wave by David Deutsch
16. ^ Jason Pontin (2007). A Giant Leap Forward in Computing? Maybe Not. The New York Times Company. Retrieved on 2007-04-08.
17. ^ Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.
18. ^ (Nielsen & Chuang 2000)
19. ^ Nielsen, Michael and Isaac Chuang (2000), p. ?

## References

• DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". Experimental Proposals for Quantum Computation. arXiv:quant-ph/0002077.
• DiVincenzo, David P. (1995). "Quantum Computation". Science 270 (5234): 255–261.  Table 1 lists switching and dephasing times for various systems.
• Feynman, Richard (1982). "Simulating physics with computers". International Journal of Theoretical Physics 21: 467.
• Jaeger, Gregg (2006). Quantum Information: An Overview. Berlin: Springer. ISBN 0-387-35725-4. "
• Nielsen, Michael and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.
• Singer, Stephanie Frank (2005). Linearity, Symmetry, and Prediction in the Hydrogen Atom. New York: Springer. ISBN 0-387-24637-1.
• http://jquantum.sourceforge.net/jQuantumApplet.html Java quantum computer simulator.

Results from FactBites:

 Quantum computer - Wikipedia, the free encyclopedia (3339 words) In a classical (or conventional) computer, the amount of data is measured by bits; in a quantum computer, it is measured by qubits. The basic principle of quantum computation is that the quantum properties of particles can be used to represent and structure data, and that quantum mechanisms can be devised and built to perform operations with this data. Quantum computers are different from classical computers such as DNA computers and computers based on transistors, even though these may ultimately use some kind of quantum mechanical effect (for example covalent bonds).
 Reference.com/Encyclopedia/Quantum computer (2819 words) In a classical (or conventional) computer, data are measured by bits; in a quantum computer the data are measured by qubits. In quantum mechanics, the state of a physical system (such as an electron or a photon) is described by an element of a mathematical object called a Hilbert space. Qubits for a quantum computer can be implemented using particles with two spin states: "up" and "down"; 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 eigenvalues, is a suitable candidate for implementing a qubit.
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