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Quantum computing is computing using quantum-mechanical phenomena, such as superposition and entanglement. A quantum computer is a device that performs quantum computing. They are different from binary digital electronic computers based on transistors. Whereas common digital computing requires that the data 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, 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. 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.
As of 2018,[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 additional effort to develop quantum computers for civilian, business, trade, environmental and national security purposes, such as cryptanalysis. A small 20-qubit quantum computer exists and is available for experiments via the IBM quantum experience project. D-Wave Systems has been developing their own version of a quantum computer that uses annealing.
Large-scale quantum computers would theoretically 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 (which is a quantum algorithm) and 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. A classical computer could in principle (with exponential resources) simulate a quantum algorithm, as quantum computation does not violate the Church–Turing thesis.:202 On the other hand, quantum computers may be able to efficiently solve problems which are not practically feasible on classical computers.
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;:13–16 a pair of qubits can be in any quantum superposition of 4 states,:16 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:17 (this compares to a normal computer that can only be in one of these states at any one time). A quantum computer operates on its qubits using quantum gates and measurement (which also alters the observed state). An algorithm is composed of a fixed sequence of quantum logic gates and a problem is encoded by setting the initial values of the qubits, similar to how a classical computer works. The calculation usually ends with a measurement, collapsing the system of qubits into one of the eigenstates, where each qubit is zero or one, decomposing into a classical state. The outcome can therefore be at most classical bits of information (or, if the algorithm did not end with a measurement, the result is an unobserved quantum state). Quantum algorithms are often probabilistic, in that they provide the correct solution only with a certain known probability. Note that the term non-deterministic computing must not be used in that case to mean probabilistic (computing), because the term non-deterministic has a different meaning in computer science.
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 ). This is true because any such system can be mapped onto an effective spin-1/2 system.
Principles of operation
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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, representing the state of an n-qubit system on a classical computer requires the storage of 2n complex coefficients, while to characterize the state of a classical n-bit system it is sufficient to provide the values of the n bits, that is, only n numbers. 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 the measurement. It is generally incorrect to think of a system of qubits as being in one particular state before the 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.
To better understand this point, consider a classical computer that operates on a three-bit register. If the exact state of the register at a given time is not known, it can be described as a probability distribution over the different three-bit strings 000, 001, 010, 011, 100, 101, 110, and 111. If there is no uncertainty over its state, 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.
The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector (or a one dimensional vector with each vector node holding the amplitude and the state as the bit string of qubits). Here, however, the coefficients are complex numbers, and it is the sum of the squares of the coefficients' absolute values, , that must equal 1. For each , the absolute value squared gives the probability of the system being found in the -th state after a measurement. 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 gives the classical probability distribution and we say that the quantum state "collapses" to a classical state as a result of making the measurement.
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 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 .
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|Unsolved problem in physics:
While a classical 3-bit state and a quantum 3-qubit state are each 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, one measures 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. 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 break 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.
John Preskill has introduced the term quantum supremacy to refer to the hypothetical speedup advantage that a quantum computer would have over a classical computer in a certain field. Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year, and IBM says that the best classical computers will be beaten on some task within about five years. Quantum supremacy has not been achieved yet, and skeptics like Gil Kalai doubt that it will ever be. Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994. Paul Davies pointed out that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle. Those such as Roger Schlafly have pointed out that the claimed theoretical benefits of quantum computing go beyond the proven theory of quantum mechanics and imply non-standard interpretations, such as multiple worlds and negative probabilities. Schlafly maintains that the Born rule is just "metaphysical fluff" and that quantum mechanics doesn't rely on probability any more than other branches of science but simply calculates the expected values of observables. He also points out that arguments about Turing complexity cannot be run backwards. Those who prefer Bayesian interpretations of quantum mechanics have questioned the physical nature of the mathematical abstractions employed.
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 thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively 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 millikelvins in order to prevent significant decoherence.
As a result, time consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions.
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.
As described in the Quantum threshold theorem, If the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often cited figure for required error rate in each gate for fault tolerant computation is 10−3, assuming the noise is depolarizing.
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 qubits 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 bits without error correction. With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 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):
- Superconducting quantum computing (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)
- Metallic-like carbon nanospheres based quantum computers
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 1981, at a conference co-organized by MIT and IBM, physicist Richard Feynman urged the world to build a quantum computer. He said "Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy."
In 1996, The DiVincenzo's criteria are published which is a list of conditions that are necessary for constructing a quantum computer proposed by the theoretical physicist David P. DiVincenzo in his 2000 paper "The Physical Implementation of Quantum Computation".
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. Google added that they had "already developed some quantum machine learning algorithms" and had "learned some useful principles", such as that "best results" come from "mixing quantum and classical computing".
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 making 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.
In December 2015 NASA publicly displayed the world's first fully operational $15-million quantum computer made by the Canadian company D-Wave 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. The presence and use of quantum effects in the D-Wave quantum processing unit is more widely accepted. In some tests it can be shown that the D-Wave quantum annealing processor outperforms Selby’s algorithm. Only 2 of this computer has been made so far.
In May 2016, IBM Research announced that for the first time ever it is making quantum computing available to members of the public via the cloud, who can access and run experiments on IBM’s quantum processor. The service is called the IBM Quantum Experience. The quantum processor is composed of five superconducting qubits and is housed at the IBM T. J. Watson Research Center in New York.
In October 2016 Basel University described a variant of the electron hole based quantum computer, which instead of manipulating electron spins uses electron holes in a semiconductor at low (mK) temperatures which are a lot less vulnerable to decoherence. This has been dubbed the "positronic" quantum computer as the quasi-particle behaves like it has a positive electrical charge.
In March 2017, IBM announced an industry-first initiative to build commercially available universal quantum computing systems called IBM Q. The company also released a new API (Application Program Interface) for the IBM Quantum Experience that enables developers and programmers to begin building interfaces between its existing five quantum bit (qubit) cloud-based quantum computer and classical computers, without needing a deep background in quantum physics.
In May 2017, IBM announced that it has successfully built and tested its most powerful universal quantum computing processors. The first is a 16 qubit processor that will allow for more complex experimentation than the previously available 5 qubit processor. The second is IBM's first prototype commercial processor with 17 qubits and leverages significant materials, device, and architecture improvements to make it the most powerful quantum processor created to date by IBM.
In July 2017, a group of U.S. researchers announced a quantum simulator with 51 qubits. The announcement was made by Mikhail Lukin of Harvard University at the International Conference on Quantum Technologies in Moscow. A quantum simulator differs from a computer. Lukin’s simulator was designed to solve one equation. Solving a different equation would require building a new system. A computer can solve many different equations.
In September 2017, IBM Research scientists use a 7 qubit device to model the largest molecule, Beryllium hydride, ever by a quantum computer. The results were published as the cover story in the peer-reviewed journal Nature.
In October 2017, IBM Research scientists successfully "broke the 49-qubit simulation barrier" and simulated 49- and 56-qubit short-depth circuits, using the Lawrence Livermore National Laboratory's Vulcan supercomputer, and the University of Illinois' Cyclops Tensor Framework (originally developed at the University of California). The results were published in arxiv.
In November 2017, the University of Sydney research team in Australia successfully made a microwave circulator, an important quantum computer part, 1000 times smaller than a conventional circulator by using topological insulators to slow down the speed of light in a material.
In November 2017, IBM announced the availability of its most-powerful 20 qubit commercial processor, and the first prototype 50 qubit processor. The 20 qubit processor has an industry-leading 90 μs coherence time for the systems' operations.
In December 2017, IBM announced its first IBM Q Network clients. The companies, universities, and labs to explore practical quantum applications, using IBM Q 20 qubit commercial systems, for business and science include: JPMorgan Chase, Daimler AG, Samsung, JSR Corporation, Barclays, Hitachi Metals, Honda, Nagase, Keio University, Oak Ridge National Lab, Oxford University and University of Melbourne.
In December 2017, Microsoft released a preview version of a "Quantum Development Kit". It includes a programming language, Q#, which can be used to write programs that are run on an emulated quantum computer.
In 2017 D-Wave reported to start selling a 2000 qubit quantum computer.
In April 2018, IBM Research announced eight quantum computing startups joined the IBM Q Network, including: Zapata Computing, Strangeworks, QxBranch, Quantum Benchmark, QC Ware, Q-CTRL, Cambridge Quantum Computing, and 1QBit.
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.
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.
Bohmian Mechanics is a non-local hidden variable interpretation of quantum mechanics. It has been shown that a non-local hidden variable quantum computer could implement a search of an N-item database at most in steps. This is slightly faster than the steps taken by Grover's algorithm. Neither search method will allow quantum computers to solve NP-Complete problems in polynomial time.
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. 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
- DNA computing
- Electronic quantum holography
- Intelligence Advanced Research Projects Activity
- Kane quantum computer
- List of emerging technologies
- List of quantum processors
- Natural computing
- Normal mode
- Photonic computing
- Post-quantum cryptography
- Quantum annealing
- Quantum bus
- Quantum cognition
- Quantum gate
- Quantum machine learning
- Quantum threshold theorem
- Theoretical computer science
- Timeline of quantum computing
- Topological quantum computer
- 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. Bibcode:1980JSP....22..563B. doi:10.1007/BF01011339.
- Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 2013-05-10. Retrieved 2013-03-04.
- Feynman, R. P.u (1982). "Simulating physics with computers" (PDF). 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 (1818): 97–117. Bibcode:1985RSPSA.400...97D. CiteSeerX . doi:10.1098/rspa.1985.0070.
- 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.
- "Explaining the upside and downside of D-Wave's new quantum computer".
- Simon, D.R. (1994). "On the power of quantum computation". Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on: 116–123. CiteSeerX . doi:10.1109/SFCS.1994.365701. ISBN 0-8186-6580-7.
- Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3.
- Preskill, John (2015). "Lecture Notes for Ph219/CS219: Quantum Information Chapter 5" (PDF). p. 12.
- 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. CiteSeerX . 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. Archived from the original (PDF) on 2015-04-10.
- 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
- Jon Schiller, Phd. "Quantum Computers".
- 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.
- Boixo, Sergio; Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan; Jiang, Zhang; Bremner, Michael J.; Martinis, John M.; Neven, Hartmut (31 July 2016). "Characterizing Quantum Supremacy in Near-Term Devices". arXiv: [quant-ph].
- Savage, Neil. "Quantum Computers Compete for "Supremacy"".
- "Quantum Supremacy and Complexity". 23 April 2016.
- Kalai, Gil. "The Quantum Computer Puzzle" (PDF). AMS.
- Unruh, Bill (1995). "Maintaining coherence in Quantum Computers". Physical Review A. 51 (2): 992. arXiv: . Bibcode:1995PhRvA..51..992U. doi:10.1103/PhysRevA.51.992.
- Davies, Paul. "The implications of a holographic universe for quantum information science and the nature of physical law" (PDF). Macquarie University.
- Schlafly, Roger. "Concise argument against quantum computing". Dark Buzz.
- Schlafly, Roger. "Impossibility of quantum computers". Dark Buzz.
- Schlafly, Roger. "No quantum probabilities needed". Dark Buzz.
- Hestenes, David. "Hunting for Snarks in Quantum Mechanics" (PDF). Arizona State University.
- DiVincenzo, David P. (2000-04-13). "The Physical Implementation of Quantum Computation". Fortschritte der Physik. 48 (9–11): 771–783. arXiv: [quant-ph]. Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E.
- Jones, Nicola (19 June 2013). "Computing: The quantum company". Nature. 498 (7454): 286–288. Bibcode:2013Natur.498..286J. doi:10.1038/498286a. PMID 23783610.
- Amy, Matthew; Matteo, Olivia; Gheorghiu, Vlad; Mosca, Michele; Parent, Alex; Schanck, John (November 30, 2016). "Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3". arXiv: [quant-ph].
- Dyakonov, M. I. (2006-10-14). S. Luryi, J. Xu, and A. Zaslavsky, eds. "Is Fault-Tolerant Quantum Computation Really Possible?". Future Trends in Microelectronics. Up the Nano Creek: 4–18. arXiv: . Bibcode:2006quant.ph.10117D.
- 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: . 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. arXiv: . Bibcode:2008RvMP...80.1061D. CiteSeerX . 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): 1083–1159. arXiv: . 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].
- 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): 4204–4207. arXiv: . 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: . Bibcode:2000quant.ph..6097F. Archived from the original on 2011-08-18.
- Leuenberger, MN; Loss, D (Apr 12, 2001). "Quantum computing in molecular magnets". Nature. 410 (6830): 789–93. arXiv: . Bibcode:2001Natur.410..789L. 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". Archived from the original on 2007-06-04. 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: . Bibcode:2004PhRvL..93m0503L. doi:10.1103/PhysRevLett.93.130503. PMID 15524694.
- Náfrádi, Bálint; Choucair, Mohammad; Dinse, Klaus-Peter; Forró, László (July 18, 2016). "Room Temperature manipulation of long lifetime spins in metallic-like carbon nanospheres". Nature Communications. 7: 12232. arXiv: . Bibcode:2016NatCo...712232N. doi:10.1038/ncomms12232. PMC . PMID 27426851.
- 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. Bibcode:1980JSP....22..563B. doi:10.1007/bf01011339.
- Manin, Yu I (1980). Vychislimoe i nevychislimoe (Computable and Noncomputable) (in Russian). Sov.Radio. pp. 13–15. Archived from the original Archived 2013-05-10 at the Wayback Machine. on May 10, 2013. Retrieved October 20, 2017.
- Gil, Dario (May 4, 2016). "The Dawn of Quantum Computing is Upon Us". Retrieved May 4, 2016.
- Bennett,, C. H. (29 March 1993). "Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels" (PDF). Physical Review Letters. 70 (13): 1895–1899. Bibcode:1993PhRvL..70.1895B. doi:10.1103/PhysRevLett.70.1895. PMID 10053414.
- 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. arXiv: . Bibcode:2001Natur.414..883V. 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. arXiv: . 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. Archived from the original on 2010-06-11. 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.
- Bhattacharjee, Pijush Kanti (2010). "Digital Combinational Circuits Design by QCA Gates" (PDF). International Journal of Computer and Electrical Engineering. 2 (1): 67–72.
- Bhattacharjee, Pijush Kanti (2010). "Digital Combinational Circuits Design with the Help of Symmetric Functions Considering Heat Dissipation by Each QCA Gate" (PDF). International Journal of Computer and Electrical Engineering. 2 (4): 666–672.
- "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.
- "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.
- Johnson, M. W.; Amin, M. H. S.; Gildert, S.; Lanting, T.; Hamze, F.; Dickson, N.; Harris, R.; Berkley, A. J.; Johansson, J.; Bunyk, P.; Chapple, E. M.; Enderud, C.; Hilton, J. P.; Karimi, K.; Ladizinsky, E.; Ladizinsky, N.; Oh, T.; Perminov, I.; Rich, C.; Thom, M. C.; Tolkacheva, E.; Truncik, C. J. S.; Uchaikin, S.; Wang, J.; Wilson, B.; Rose, G. (12 May 2011). "Quantum annealing with manufactured spins". Nature. 473 (7346): 194–198. Bibcode:2011Natur.473..194J. doi:10.1038/nature10012. PMID 21562559.
- Simonite, Tom (October 4, 2012). "The CIA and Jeff Bezos Bet on Quantum Computing". Technology Review.
- Seung Woo Shin; Smith, Graeme; Smolin, John A.; Vazirani, Umesh (2014-05-02). "How "Quantum" is the D-Wave Machine?". arXiv: [quant-ph].
- Boixo, Sergio; Rønnow, Troels F.; Isakov, Sergei V.; Wang, Zhihui; Wecker, David; Lidar, Daniel A.; Martinis, John M.; Troyer, Matthias (2013-04-16). "Quantum Annealing With More Than 100 Qbits". Nature Physics. 10 (3): 218–224. arXiv: . Bibcode:2014NatPh..10..218B. doi:10.1038/nphys2900.
- Albash, Tameem; Rønnow, Troels F.; Troyer, Matthias; Lidar, Daniel A. (2014-09-12). "Reexamining classical and quantum models for the D-Wave One processor". The European Physical Journal Special Topics. 224 (111): 111–129. arXiv: . Bibcode:2015EPJST.224..111A. doi:10.1140/epjst/e2015-02346-0.
- Lanting, T.; Przybysz, A. J.; Smirnov, A. Yu.; Spedalieri, F. M.; Amin, M. H.; Berkley, A. J.; Harris, R.; Altomare, F.; Boixo, S.; Bunyk, P.; Dickson, N.; Enderud, C.; Hilton, J. P.; Hoskinson, E.; Johnson, M. W.; Ladizinsky, E.; Ladizinsky, N.; Neufeld, R.; Oh, T.; Perminov, I.; Rich, C.; Thom, M. C.; Tolkacheva, E.; Uchaikin, S.; Wilson, A. B.; Rose, G. (2014-05-29). "Entanglement in a quantum annealing processor". Physical Review X. prx. 4 (2): 021041. arXiv: . Bibcode:2014PhRvX...4b1041L. doi:10.1103/PhysRevX.4.021041.
- 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: . Bibcode:2012NaPho...6..773M. doi:10.1038/nphoton.2012.259.
- Mariantoni, Matteo; Wang, H.; Yamamoto, T.; Neeley, M.; Bialczak, Radoslaw C.; Chen, Y.; Lenander, M.; Lucero, Erik; O'Connell, A. D.; Sank, D.; Weides, M.; Wenner, J.; Yin, Y.; Zhao, J.; Korotkov, A. N.; Cleland, A. N.; Martinis, John M. (2011). "Quantum computer with Von Neumann architecture". Science. 334 (6052): 61–65. arXiv: . Bibcode:2011Sci...334...61M. doi:10.1126/science.1208517. PMID 21885732.
- Xu, Nanyang; Zhu, Jing; Lu, Dawei; Zhou, Xianyi; Peng, Xinhua; Du, Jiangfeng (2011). "Quantum Factorization of 143 on a Dipolar-Coupling NMR system". Physical Review Letters. 109 (26): 269902. arXiv: . Bibcode:2012PhRvL.109z9902X. doi:10.1103/PhysRevLett.109.269902.
- "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". The New York Times. Retrieved 2012-10-14.
- Overbye, Dennis (October 9, 2012). "A Nobel for Teasing Out the Secret Life of Atoms". The 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". Proceedings of the National Academy of Sciences. 109 (50): 20347–20351. arXiv: . Bibcode:2012PNAS..10920347B. doi:10.1073/pnas.1207329109. PMC . PMID 23144222.
- "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.
We’ve already developed some quantum machine learning algorithms. One produces very compact, efficient recognizers -- very useful when you’re short on power, as on a mobile device. Another can handle highly polluted training data, where a high percentage of the examples are mislabeled, as they often are in the real world. And we’ve learned some useful principles: e.g., you get the best results not with pure quantum computing, but by mixing quantum and classical computing.
- "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 achieves critical steps to first quantum computer". www-03.ibm.com. 29 April 2015.
- Condliffe, Jamie. "World's First Silicon Quantum Logic Gate Brings Quantum Computing One Step Closer".
- "3Q: Scott Aaronson on Google's new quantum-computing paper". MIT News. Retrieved 2016-01-05.
- Benchmarking a quantum annealing processor with the time-to-target metric, James King, Sheir Yarkoni, Mayssam M. Nevisi, Jeremy P. Hilton, Catherine C. McGeoch, 2015-08-20.
- "IBM Makes Quantum Computing Available on IBM Cloud to Accelerate Innovation". May 4, 2016. Retrieved May 4, 2016.
- MacDonald, Fiona. "Researchers have built the first reprogrammable quantum computer". ScienceAlert. Retrieved 8 August 2016.
- "A new Type of Quantum Bit". www.unibas.ch.
- "IBM Builds Its Most Powerful Universal Quantum Computing Processors". 17 May 2017. Retrieved 17 May 2017.
- Reynolds, Matt. "Quantum simulator with 51 qubits is largest ever". New Scientist. Retrieved 23 July 2017.
- "IBM Pioneers New Approach to Simulate Chemistry with Quantum Computing". 13 September 2017. Retrieved 13 September 2017.
- Pednault, Edwin; Gunnels, John A; Nannicini, Giacomo; Horesh, Lior; Magerlein, Thomas; Solomonik, Edgar; Wisnieff, Robert (16 October 2017). "Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits". arXiv: [quant-ph].
- "Key component for quantum computing invented: University of Sydney team develop microcircuit based on Nobel Prize research".
- "IBM Announces Advances to IBM Quantum Systems & Ecosystem". 10 November 2017. Retrieved 10 November 2017.
- "IBM Announces Collaboration with Leading Fortune 500 Companies, Academic Institutions and National Research Labs to Accelerate Quantum Computing". 14 November 2017. Retrieved 14 December 2017.
- Microsoft Mechanics (11 December 2017). "Microsoft Quantum Development Kit: Introduction and step-by-step demo" – via YouTube.
- Temperton, James (26 January 2017). "Got a spare $15 million? Why not buy your very own D-Wave quantum computer". Wired. Retrieved 13 May 2018.
- Hignett, Katherine (16 February 2018). "Physics Creates New Form Of Light That Could Drive The Quantum Computing Revolution". Newsweek. Retrieved 17 February 2018.
- Liang, Qi-Yu; et al. (16 February 2018). "Observation of three-photon bound states in a quantum nonlinear medium". Science. 359 (6377): 783–786. arXiv: . Bibcode:2018Sci...359..783L. doi:10.1126/science.aao7293. Retrieved 17 February 2018.
- "A Preview of Bristlecone, Google's New Quantum Processor". Research Blog. Retrieved 2018-03-06.
- "IBM Collaborating With Top Startups to Accelerate Quantum Computing". 5 April 2018. Retrieved 5 April 2018.
- Nielsen, p. 42
- Nielsen, p. 41
- Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing. 26 (5): 1411–1473. CiteSeerX . doi:10.1137/S0097539796300921.
- Ozhigov, Yuri (1999). "Quantum Computers Speed Up Classical with Probability Zero". Chaos, Solitons & Fractals. 10 (10): 1707–1714. arXiv: . 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: . Bibcode:1999RSPSA.455.2165O. doi:10.1098/rspa.1999.0397.
- Aaronson, Scott. "Quantum Computing and Hidden Variables" (PDF).
- Nielsen, p. 126
- Scott Aaronson (2005). "NP-complete Problems and Physical Reality". ACM SIGACT News, March. 2005. arXiv: . Bibcode:2005quant.ph..2072A. See 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.
- Dibyendu Chatterjee; Arijit Roy (2015). "A transmon-based quantum half-adder scheme". Progress of Theoretical and Experimental Physics. 2015 (9): 093A02(16pages). Bibcode:2015PTEP.2015i3A02C. doi:10.1093/ptep/ptv122.
- 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: . 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. CiteSeerX . 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): 467–488. 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.[permanent dead link]
- 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 978-981-4566-74-2.
- Akama, Seiki (2014). Elements of Quantum Computing: History, Theories and Engineering Applications. Springer International Publishing. ISBN 978-3-319-08284-4.
|Wikimedia Commons has media related to Quantum computer.|
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