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Quantum computing studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. 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 non-deterministic and probabilistic computers. The field of quantum computing was initiated by the work of Paul Benioff  and Yuri Manin in 1980, Richard Feynman in 1982, and David Deutsch in 1985. A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1968.
As of 2015[update], 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. 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, and national security purposes, such as cryptanalysis.
Large-scale quantum computers will 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 many-body systems. There exist quantum algorithms, such as Simon's algorithm, that run faster than any possible probabilistic classical algorithm. Given sufficient computational resources, however, a classical computer could be made to simulate any quantum algorithm, as quantum computation does not violate the Church–Turing thesis.
A classical computer has a memory made up of bits, where each bit represents 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. The outcome can therefore be at most classical bits of information. Quantum algorithms are often non-deterministic, in that they provide the correct solution only with a certain known probability.
An example of an implementation of qubits for 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 spin-1/2 system.
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 n-qubit system on a classical computer would require the storage of 2n complex 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 three-bit register. The state of the computer at any time is a probability distribution over the different three-bit 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 three-qubit quantum computer is similarly described by an eight-dimensional 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.
If you measure the three qubits, you will observe a three-bit 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 eight-dimensional 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 unit-length, orthogonal vectors and the eigenvectors of the Pauli-x 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 Pauli-x operator, a single qubit is and .
|Open problem in physics:
While a classical three-bit state and a quantum three-qubit state are both eight-dimensional 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 all-zeros 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 three-bit register to obtain one definite three-bit string, say 000. Quantum mechanically, we 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. 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.
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 300-digit primes). 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, Diffie-Hellman, and Elliptic curve Diffie-Hellman 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. Some public-key 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. Lattice-based 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 well-studied open problem. 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, meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force 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, 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. For some problems, quantum computers offer a polynomial speedup. The most well-known 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 two-to-one 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.
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.
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. Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.
There are a number of technical challenges in building a large-scale 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:
- 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.
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 nuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is non-unitary, 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. Currently, some quantum computers require their qubits to be cooled to 20 millikelvin in order to prevent significant decoherence.
These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited 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 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 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 1000-bit number, this implies a need for about 104 qubits without error correction. With error correction, the figure would rise to about 107 qubits. Note that computation time is about L2 or about 107 steps and on 1 MHz, about 10 seconds.
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 braid theory to form stable logic gates.
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 few-qubit quantum gates)
- One-way quantum computer (computation decomposed into sequence of one-qubit 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)
- Topological quantum computer (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):
- Superconductor-based quantum computers (including SQUID-based quantum computers) (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, spin-based (e.g. the Loss-DiVincenzo quantum computer ) (qubit given by the spin states of trapped electrons)
- Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)
- Nuclear magnetic resonance on molecules in solution (liquid-state NMR) (qubit provided by nuclear spins within the dissolved molecule)
- Solid-state NMR Kane quantum computers (qubit realized by the nuclear spin state of phosphorus donors in silicon)
- Electrons-on-helium quantum computers (qubit is the electron spin)
- Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
- Molecular magnet (qubit given by spin states)
- Fullerene-based 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)
- Diamond-based quantum computer (qubit realized by electronic or nuclear spin of nitrogen-vacancy centers in diamond)
- Bose–Einstein condensate-based quantum computer
- Transistor-based quantum computer – string quantum computers with entrainment of positive holes using an electrostatic trap
- Rare-earth-metal-ion-doped inorganic crystal based quantum computers (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.
In 2001, researchers demonstrated Shor's algorithm to factor 15 using a 7-qubit NMR computer.
In 2009, researchers at Yale University created the first solid-state quantum processor. The two-qubit superconducting chip had artificial atom qubits made of a billion aluminum atoms that acted like a single atom that could occupy two states.
A team at the University of Bristol, also created a silicon chip based on quantum optics, able to run Shor's algorithm. Further developments were made in 2010. Springer publishes a journal (Quantum Information Processing) devoted to the subject.
In 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.
In 2011, D-Wave Systems announced the first commercial quantum annealer, the D-Wave One, claiming a 128 qubit processor. On May 25, 2011 Lockheed Martin agreed to purchase a D-Wave One system. Lockheed and the University of Southern California (USC) will house the D-Wave One at the newly formed USC Lockheed Martin Quantum Computing Center. D-Wave's engineers designed the chips with an empirical approach, focusing on solving particular problems. Investors liked this more than academics, who said D-Wave had not demonstrated they really had a quantum computer. Criticism softened after a D-Wave paper in Nature, that proved the chips have some quantum properties. Two published papers have suggested that the D-Wave machine's operation can be explained classically, rather than requiring quantum models. Later work showed that classical models are insufficient when all available data is considered. Experts remain divided on the ultimate classification of the D-Wave systems though their quantum behavior was established concretely with a demonstration of entanglement.
In November 2011 researchers factorized 143 using 4 qubits.
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 two-qubit 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.
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 modern-day computers. 
In December 2012, the first dedicated quantum computing software company, 1QBit was founded in Vancouver, BC. 1QBit is the first company to focus exclusively on commercializing software applications for commercially available quantum computers, including the D-Wave Two. 1QBit's research demonstrated the ability of superconducting quantum annealing processors to solve real-world problems.
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 512-qubit D-Wave 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.
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 $79.7 million research program (titled "Penetrating Hard Targets") to develop a quantum computer capable of breaking vulnerable encryption.
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 D-Wave Two device, but did not explicitly rule it out.
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.
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.
Relation to computational 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 polynomial-time algorithm, whose probability of error is bounded away from one half. 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 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.
By the same token, it is often believed that quantum computers are generally superior to classical computers, that is, faster or at least as fast for any computing task. However, there is no proof of that, and it might not be true considering the next three observations:
- A quantum superposition of qubit states is basically a distributed representation of data to be processed. For some computing tasks, it can yield an exponential reduction from O(2N) to O(1) in the amount of work and time needed to complete a computing task for input size N.
- In classical computing, various kinds of distributed representations have been devised to reduce work and time for various computing tasks. This reduction is typically from O(2N) to O(N2), but for at least one computing task, there are distributed representations (called hyperstrings) that enable an exponential reduction from O(2N) to O(1).
- In the end, it always is the computing task at hand that determines if to-be-processed data can be gathered in a certain distributed representation, which, in turn, determines how efficiently the task can be solved.
Hence, the interdependence between computing tasks and distributed representations suggests not only that quantum superpositions may be more efficient than classical distributed representations for some computing tasks, but also that the inverse may hold for other computing tasks.
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. A similar fact takes place for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.
Although quantum computers may be faster than classical computers for some problems, those described above can't solve any problems that classical computers can't solve, given enough time and memory (however, those amounts might be practically infeasible). 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. It has been speculated that theories of quantum gravity, such as M-theory 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.
- Chemical computer
- David Deutsch
- DNA computer
- Electronic quantum holography
- List of emerging technologies
- Richard Feynman
- Kane quantum computer
- Natural computing
- Normal mode
- Photonic computing
- Post-quantum cryptography
- Quantum annealing
- Quantum bus
- Quantum cognition
- Quantum gate
- Quantum threshold theorem
- Timeline of quantum computing
- Topological quantum computer
- D-Wave Two
- Gershenfeld, Neil; Chuang, Isaac L. (June 1998). "Quantum Computing with Molecules" (PDF). Scientific American.
- Benioff, Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of statistical physics 22 (5): 563–591.
- Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Retrieved 2013-03-04.
- Feynman, R. P. (1982). "Simulating physics with computers". International Journal of Theoretical Physics 21 (6): 467–488. Bibcode:1982IJTP...21..467F. doi:10.1007/BF02650179.
- Deutsch, David (1985). "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer". Proceedings of the Royal Society of London A 400: 97–117.
- Finkelstein, David (1968). "Space-Time Structure in High Energy Interactions". In Gudehus, T.; Kaiser, G. Fundamental Interactions at High Energy. New York: Gordon & Breach.
- Gershon, Eric (2013-01-14). "New qubit control bodes well for future of quantum computing". Phys.org. Retrieved 2014-10-26.
- Quantum Information Science and Technology Roadmap for a sense of where the research is heading.
- Simon, D.R. (1994). "On the power of quantum computation". Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on: 116–123. doi:10.1109/SFCS.1994.365701. ISBN 0-8186-6580-7.
- Nielsen, Michael A.; Chuang, Isaac L. Quantum Computation and Quantum Information. p. 202.
- Waldner, Jean-Baptiste (2007). Nanocomputers and Swarm Intelligence. London: ISTE. p. 157. ISBN 2-7462-1516-0.
- DiVincenzo, David P. (1995). "Quantum Computation". Science 270 (5234): 255–261. Bibcode:1995Sci...270..255D. doi:10.1126/science.270.5234.255. (subscription required)
- Lenstra, Arjen K. (2000). "Integer Factoring" (PDF). Designs, Codes and Cryptography 19 (2/3): 101–128. doi:10.1023/A:1008397921377.
- Daniel J. Bernstein, Introduction to Post-Quantum Cryptography. Introduction to Daniel J. Bernstein, Johannes Buchmann, Erik Dahmen (editors). Post-quantum cryptography. Springer, Berlin, 2009. ISBN 978-3-540-88701-0
- See also pqcrypto.org, a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing.
- Robert J. McEliece. "A public-key cryptosystem based on algebraic coding theory." Jet Propulsion Laboratory DSN Progress Report 42–44, 114–116.
- Kobayashi, H.; Gall, F.L. (2006). "Dihedral Hidden Subgroup Problem: A Survey". Information and Media Technologies 1 (1): 178–185.
- Bennett C.H., Bernstein E., Brassard G., Vazirani U., "The strengths and weaknesses of quantum computation". SIAM Journal on Computing 26(5): 1510–1523 (1997).
- Quantum Algorithm Zoo – Stephen Jordan's Homepage
- Rich, Steven; Gellman, Barton (2014-02-01). "NSA seeks to build quantum computer that could crack most types of encryption". Washington Post.
- Norton, Quinn (2007-02-15). "The Father of Quantum Computing". Wired.com.
- Ambainis, Andris (Spring 2014). "What Can We Do with a Quantum Computer?". Institute for Advanced Study.
- DiVincenzo, David P. (2000-04-13). "The Physical Implementation of Quantum Computation". arXiv:quant-ph/0002077 [quant-ph].
- Jones, Nicola (19 June 2013). "Computing: The quantum company". Nature International Weekly Journal of Science 498 (7454): 286–288. doi:10.1038/498286a.
- Dyakonov, M. I. (2006-10-14). "Is Fault-Tolerant Quantum Computation Really Possible?". In: Future Trends in Microelectronics. Up the Nano Creek, S. Luryi, J. Xu, and A. Zaslavsky (eds), Wiley, pp.: 4–18. arXiv:quant-ph/0610117.
- Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2003). "Topological quantum computation". Bulletin of the American Mathematical Society 40 (1): 31–38. arXiv:quant-ph/0101025. doi:10.1090/S0273-0979-02-00964-3. MR 1943131.
- Monroe, Don (2008-10-01). "Anyons: The breakthrough quantum computing needs?". New Scientist.
- Das, A.; Chakrabarti, B. K. (2008). "Quantum Annealing and Analog Quantum Computation". Rev. Mod. Phys. 80 (3): 1061–1081. Bibcode:2008RvMP...80.1061D. doi:10.1103/RevModPhys.80.1061
- Nayak, Chetan; Simon, Steven; Stern, Ady; Das Sarma, Sankar (2008). "Nonabelian Anyons and Quantum Computation". Rev Mod Phys 80 (3): 1083. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/RevModPhys.80.1083.
- Clarke, John; Wilhelm, Frank (June 19, 2008). "Superconducting quantum bits". Nature 453 (7198): 1031–1042. Bibcode:2008Natur.453.1031C. doi:10.1038/nature07128. PMID 18563154.
- Kaminsky, William M (2004). "Scalable Superconducting Architecture for Adiabatic Quantum Computation". arXiv:quant-ph/0403090 [quant-ph].
- Imamoğlu, Atac; Awschalom, D. D.; Burkard, Guido; DiVincenzo, D. P.; Loss, D.; Sherwin, M.; Small, A. (1999). "Quantum information processing using quantum dot spins and cavity-QED". Physical Review Letters 83 (20): 4204. Bibcode:1999PhRvL..83.4204I. doi:10.1103/PhysRevLett.83.4204.
- Fedichkin, Leonid; Yanchenko, Maxim; Valiev, Kamil (2000). "Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot". Quantum Computers and Computing 1: 58–76. arXiv:quant-ph/0006097. Bibcode:2000quant.ph..6097F.
- "Quantum computing in molecular magnets.". Nature 410 (6830): 789–93. Apr 12, 2001. doi:10.1038/35071024. PMID 11298441.
- Knill, G. J.; Laflamme, R.; Milburn, G. J. (2001). "A scheme for efficient quantum computation with linear optics". Nature 409 (6816): 46–52. Bibcode:2001Natur.409...46K. doi:10.1038/35051009. PMID 11343107.
- Nizovtsev, A. P. (August 2005). "A quantum computer based on NV centers in diamond: Optically detected nutations of single electron and nuclear spins". Optics and Spectroscopy 99 (2): 248–260. Bibcode:2005OptSp..99..233N. doi:10.1134/1.2034610.
- Gruener, Wolfgang (2007-06-01). "Research indicates diamonds could be key to quantum storage". Retrieved 2007-06-04.
- Neumann, P.; et al. (June 6, 2008). "Multipartite Entanglement Among Single Spins in Diamond". Science 320 (5881): 1326–1329. Bibcode:2008Sci...320.1326N. doi:10.1126/science.1157233. PMID 18535240.
- Millman, Rene (2007-08-03). "Trapped atoms could advance quantum computing". ITPro. Archived from the original on 2007-09-27. Retrieved 2007-07-26.
- Ohlsson, N.; Mohan, R. K.; Kröll, S. (January 1, 2002). "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals". Opt. Commun. 201 (1–3): 71–77. Bibcode:2002OptCo.201...71O. doi:10.1016/S0030-4018(01)01666-2.
- Longdell, J. J.; Sellars, M. J.; Manson, N. B. (September 23, 2004). "Demonstration of conditional quantum phase shift between ions in a solid". Phys. Rev. Lett. 93 (13): 130503. arXiv:quant-ph/0404083. Bibcode:2004PhRvL..93m0503L. doi:10.1103/PhysRevLett.93.130503. PMID 15524694.
- Vandersypen, Lieven M. K.; Steffen, Matthias; Breyta, Gregory; Yannoni, Costantino S.; Sherwood, Mark H.; Chuang, Isaac L. (2001). "Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance". Nature 414 (6866): 883–7. doi:10.1038/414883a. PMID 11780055.
- "U-M develops scalable and mass-producible quantum computer chip". University of Michigan. 2005-12-12. Retrieved 2006-11-17.
- DiCarlo, L.; Chow, J. M.; Gambetta, J. M.; Bishop, Lev S.; Johnson, B. R.; Schuster, D. I.; Majer, J.; Blais, A.; Frunzio, L.; S. M. Girvin; R. J. Schoelkopf (9 July 2009). "Demonstration of two-qubit algorithms with a superconducting quantum processor" (PDF). Nature 460 (7252): 240–4. Bibcode:2009Natur.460..240D. doi:10.1038/nature08121. PMID 19561592. Retrieved 2009-07-02.
- "Scientists Create First Electronic Quantum Processor". Yale University. 2009-07-02. Retrieved 2009-07-02.
- "Code-breaking quantum algorithm runs on a silicon chip". New Scientist. 2009-09-04. Retrieved 2009-10-14.
- "New Trends in Quantum Computation". Simons Conference on New Trends in Quantum Computation 2010: Program. C.N. Yang Institute for Theoretical Physics.
- "Quantum Information Processing". Springer.com. Retrieved on 2011-05-19.
- "Quantum teleporter breakthrough". University of New South Wales. 2011-04-15. Archived from the original on 2011-04-18.
- Lai, Richard (2011-04-18). "First light wave quantum teleportation achieved, opens door to ultra fast data transmission". Engadget.
- "Learning to program the D-Wave One". Hack The Multiverse. D-Wave. Retrieved 2011-05-11.
- "D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". D-Wave. 2011-05-25. Retrieved 2011-05-30.
- "Operational Quantum Computing Center Established at USC". University of Southern California. 2011-10-29. Retrieved 2011-12-06.
- "Quantum annealing with manufactured spins". Nature 473 (7346): 194–198. 12 May 2011. doi:10.1038/nature10012.
- Simonite, Tom (October 4, 2012). "The CIA and Jeff Bezos Bet on Quantum Computing". Technology Review.
- "How "Quantum" is the D-Wave Machine?". Arxiv. 2014-05-02.
- "Quantum Annealing With More Than 100 Qbits". Arxiv. 2013-04-16.
- "Reexamining classical and quantum models for the D-Wave One processor". Arxiv. 2014-09-12.
- "Entanglement in a quantum annealing processor". prx. 2014-05-29.
- Lopez, Enrique Martin; Laing, Anthony; Lawson, Thomas; Alvarez, Roberto; Zhou, Xiao-Qi; O'Brien, Jeremy L. (2011). "Implementation of an iterative quantum order finding algorithm". Nature Photonics 6 (11): 773–776. arXiv:1111.4147. doi:10.1038/nphoton.2012.259.
- "Quantum computer with Von Neumann architecture". Retrieved 2014-10-26.
- "Quantum Factorization of 143 on a Dipolar-Coupling NMR system". Retrieved 2014-10-26.
- "IBM Says It's 'On the Cusp' of Building a Quantum Computer". PCMAG. Retrieved 2014-10-26.
- "Quantum computer built inside diamond". Futurity. Retrieved 2014-10-26.
- "Australian engineers write quantum computer 'qubit' in global breakthrough". The Australian. Retrieved 2012-10-03.
- "Breakthrough in bid to create first quantum computer". University of New South Wales. Retrieved 2012-10-03.
- Frank, Adam (October 14, 2012). "Cracking the Quantum Safe". New York Times. Retrieved 2012-10-14.
- Overbye, Dennis (October 9, 2012). "A Nobel for Teasing Out the Secret Life of Atoms". New York Times. Retrieved 2012-10-14.
- "First Teleportation from One Macroscopic Object to Another: The Physics arXiv Blog". MIT Technology Review. November 15, 2012. Retrieved 2012-11-17.
- Bao, Xiao-Hui; Xu, Xiao-Fan; Li, Che-Ming; Yuan, Zhen-Sheng; Lu, Chao-Yang; Pan, Jian-wei (November 13, 2012). "Quantum teleportation between remote atomic-ensemble quantum memories". arXiv. arXiv:1211.2892.
- "1QBit Founded". 1QBit.com. Retrieved 2014-06-22.
- "1QBit Research". 1QBit.com. Retrieved 2014-06-22.
- "Launching the Quantum Artificial Intelligence Lab". Research@Google Blog. Retrieved 2013-05-16.
- "NSA seeks to build quantum computer that could crack most types of encryption". Washington Post. January 2, 2014.
- Defining and detecting quantum speedup, Troels F. Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V. Isakov, David Wecker, John M. Martinis, Daniel A. Lidar, Matthias Troyer, 2014-01-13.
- "Quantum Chaos: After a Failed Speed Test, the D-Wave Debate Continues". Scientific American. 2014-06-19.
- Gaudin, Sharon (23 October 2014). "Researchers use silicon to push quantum computing toward reality". Computer World.
- IBM Scientists Achieve Critical Steps to Building First Practical Quantum Computer
- World's First Silicon Quantum Logic Gate Brings Quantum Computing One Step Closer
- Nielsen, p. 42
- Nielsen, p. 41
- Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing 26 (5): 1411. doi:10.1137/S0097539796300921.
- van der Helm, P. A. (2004). Transparallel processing by hyperstrings. Proceedings of the National Academy of Sciences USA, 101, 10862—10867. doi:10.1073/pnas.0403402101.
- van der Helm, P. A. (2015). Transparallel mind: Classical computing with quantum power. Artificial Intelligence Review, 44, 341—361. doi:10.1007/s10462-015-9429-7.
- Ozhigov, Yuri (1999). "Quantum Computers Speed Up Classical with Probability Zero". Chaos Solitons Fractals 10 (10): 1707–1714. arXiv:quant-ph/9803064. Bibcode:1998quant.ph..3064O. doi:10.1016/S0960-0779(98)00226-4.
- Ozhigov, Yuri (1999). "Lower Bounds of Quantum Search for Extreme Point". Proceedings of the London Royal Society A455 (1986): 2165–2172. arXiv:quant-ph/9806001. Bibcode:1999RSPSA.455.2165O. doi:10.1098/rspa.1999.0397.
- Nielsen, p. 126
- Scott Aaronson, NP-complete Problems and Physical Reality, ACM SIGACT News, Vol. 36, No. 1. (March 2005), pp. 30–52, section 7 "Quantum Gravity": "[…] to anyone who wants a test or benchmark for a favorite quantum gravity theory,[author's footnote: That is, one without all the bother of making numerical predictions and comparing them to observation] let me humbly propose the following: can you define Quantum Gravity Polynomial-Time? […] until we can say what it means for a 'user' to specify an 'input' and ‘later' receive an 'output'—there is no such thing as computation, not even theoretically." (emphasis in original)
- Nielsen, Michael; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9. OCLC 174527496.
- Abbot, Derek; Doering, Charles R.; Caves, Carlton M.; Lidar, Daniel M.; Brandt, Howard E.; Hamilton, Alexander R.; Ferry, David K.; Gea-Banacloche, 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:quant-ph/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:quant-ph/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 0-387-35725-4. OCLC 255569451.
- Singer, Stephanie Frank (2005). Linearity, Symmetry, and Prediction in the Hydrogen Atom. New York: Springer. ISBN 0-387-24637-1. OCLC 253709076.
- Benenti, Giuliano (2004). Principles of Quantum Computation and Information Volume 1. New Jersey: World Scientific. ISBN 981-238-830-3. 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:quant-ph/9806048v1.
- Stolze, Joachim; Suter, Dieter (2004). Quantum Computing. Wiley-VCH. ISBN 3-527-40438-4.
- 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 3-540-33132-8.
- 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 Ruhr-University Bochum.
- Sanders, Laura (2009). "First programmable quantum computer created".
- "New trends in quantum computation".
|Wikimedia Commons has media related to Quantum computer.|
- Stanford Encyclopedia of Philosophy: "Quantum Computing" by Amit Hagar.
- Quantiki – Wiki and portal with free-content related to quantum information science.
- Scott Aaronson's blog, which features informative and critical commentary on developments in the field
- D-Wave thinks it has built the world's first commercial quantum computer. Mother Nature has other ideas, in the January 2014 issue of Inc. magazine
- Quantum Annealing and Computation: A Brief Documentary Note, A. Ghosh and S. Mukherjee
- Maryland University Laboratory for Physical Sciences: conducts researches for the quantum computer-based 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
- Joseph B. Altepeter (2010). "A tale of two qubits: how quantum computers work". Ars Technica.
- Quantum computers in Next Big Future.
- 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)
- on YouTube
- A discussion of the history of quantum computing research at Bell Labs and Lucent at DailyKos