Quantum Computing

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## Quantum operations

## Potential applications

### Cryptography

### Quantum search

### Quantum simulation

### Quantum annealing and adiabatic optimization

### Solving linear equations

### Quantum supremacy

## Obstacles

### Quantum decoherence

## Developments

### Quantum computing models

### Physical realizations

## Relation to computability and complexity theory

### Computability theory

### Computational complexity theory

## See also

## References

## Further reading

### Textbooks

### Academic papers

## External links

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Quantum Computing

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**Quantum computing** is the use of quantum-mechanical phenomena such as superposition and entanglement to perform computation. A **quantum computer**, implemented theoretically or physically, is used to perform such computation.^{[1]}^{:I-5} There are currently two main approaches to physically implementing a quantum computer: analog and digital. Analog approaches are further divided into quantum simulation, quantum annealing, and adiabatic quantum computation. Digital quantum computers use quantum logic gates to do computation. Both approaches use quantum bits or qubits.^{[1]}^{:2-13}

Quantum computing began in the early 1980s, when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine.^{[2]} Richard Feynman and Yuri Manin later suggested that a quantum computer had the potential to simulate things that a classical computer could not.^{[3]}^{[4]} In 1994, Peter Shor developed a quantum algorithm for factoring integers that had the potential to decrypt RSA-encrypted communications.^{[5]} Despite ongoing experimental progress since the late 1990s, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream".^{[6]} On 23 October 2019, Google AI, in partnership with the U.S. National Aeronautics and Space Administration (NASA), published a paper in which they claimed to have achieved quantum supremacy.^{[7]} While some have disputed this claim, it is still a significant milestone in the history of quantum computing.^{[8]}

Quantum computing is modeled by quantum circuits. Quantum circuits are based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0 states. However, when qubits are measured the result is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that they were in immediately prior to the measurement. Computation is performed by manipulating qubits with quantum logic gates, which are somewhat analogous to classical logic gates.

The field of quantum computing is a subfield of quantum information science, which includes quantum cryptography and quantum communication.

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.^{[9]}

A memory consisting of bits of information has possible states. A vector representing all memory states thus has entries (one for each state). This vector is viewed as a *probability vector* and represents the fact that the memory is to be found in a particular state.

In the classical view, one entry would have a value of 1 (i.e. a 100% probability of being in this state) and all other entries would be zero. In quantum mechanics, probability vectors are generalized to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. This article focuses on the quantum state vector formalism for simplicity.

We begin by considering a simple memory consisting of only one bit. This memory may be found in one of two states: the zero state or the one state. We may represent the state of this memory using Dirac notation so that

A quantum memory may then be found in any quantum superposition of the two classical states and :

In general, the coefficients and are complex numbers. In this scenario, one qubit of information is said to be encoded into the quantum memory. The state is not itself a probability vector but can be connected with a probability vector via a measurement operation. If the quantum memory is measured to determine if the state is or (this is known as a computational basis measurement), the zero state would be observed with probability and the one state with probability . The numbers and are called quantum amplitudes.

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus and .

The mathematics of single qubit gates can be extended to operate on multiqubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit whilst leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are

The CNOT gate can then be represented using the following matrix:

As a mathematical consequence of this definition, , , , and . In other words, the CNOT applies a NOT gate ( from before) to the second qubit if and only if the first qubit is in the state . If the first qubit is , nothing is done to either qubit.

In summary, a quantum computation can be described as a network of quantum logic gates and measurements. Any measurement can be deferred to the end of a quantum computation, though this deferment may come at a computational cost. Because of this possibility of deferring a measurement, most quantum circuits depict a network consisting only of quantum logic gates and no measurements. More information can be found in the following articles: 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.

Any quantum computation can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.

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).^{[10]} 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.^{[11]}^{[12]} 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.^{[11]}^{[13]}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.^{[14]} It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^{n/2} invocations of the underlying cryptographic algorithm, compared with roughly 2^{n} in the classical case,^{[15]} 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. Quantum-based cryptographic systems could, therefore, be more secure than traditional systems against quantum hacking.^{[16]}

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,^{[17]} 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.^{[18]} However, quantum computers offer polynomial speedup for some problems. 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 that are required by classical algorithms. In this case, the advantage is not only provable but also optimal, it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. 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.

Problems that can be addressed with Grover's algorithm have the following properties:

- There is no searchable structure in the collection of possible answers,
- The number of possible answers to check is the same as the number of inputs to the algorithm, and
- There exists a boolean function which evaluates each input and determines whether it is the correct answer

For problems with all these properties, the running time of Grover's algorithm on a quantum computer will scale as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied^{[19]} is Boolean satisfiability problem. In this instance, the *database* through which the algorithm is iterating is that of all possible answers. An example (and possible) application of this is a password cracker that attempts to guess the password or secret key for an encrypted file or system. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack.^{[]} This application of quantum computing is a major interest of government agencies.^{[20]}

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.^{[21]} Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.^{[22]}

Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process.

The Quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is expected to provide speedup over classical counterparts.^{[23]}

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.^{[24]}Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year though that did not happen. IBM said in 2018 that the best classical computers will be beaten on some practical task within about five years and views the quantum supremacy test only as a potential future benchmark.^{[25]} Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved,^{[26]}^{[27]} in October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy,^{[28]} with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer.^{[29]}Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.^{[30]}Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.^{[31]}

There are a number of technical challenges in building a large-scale quantum computer.^{[32]} Physicist David DiVincenzo has listed the following requirements for a practical quantum computer:^{[33]}

- 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

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables that are only made by a single company in Japan.^{[34]}

One of the greatest challenges involved with constructing quantum computers 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 *T*_{2} (for NMR and MRI technology, also called the *dephasing time*), typically range between nanoseconds and seconds at low temperature.^{[35]} Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.^{[36]}

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.^{[37]}

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 the 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 *L*^{2}, 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 10^{4} bits without error correction.^{[38]} With error correction, the figure would rise to about 10^{7} bits. Computation time is about *L*^{2} or about 10^{7} 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.^{[39]}^{[40]}

Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:

- "So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10
^{300}... Could we ever learn to control the more than 10^{300}continuously variable parameters defining the quantum state of such a system? My answer is simple.*No, never.*"^{[41]}

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 a sequence of few-qubit quantum gates)
- One-way quantum computer (computation decomposed into a 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)
^{[42]} - Topological quantum computer
^{[43]}(computation decomposed into the braiding of anyons in a 2D lattice)

The quantum Turing machine is theoretically important but the physical implementation of this model is not feasible. 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
^{[44]}^{[45]}(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
^{[46]}) (qubit given by the spin states of trapped electrons) - Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)
^{[47]} - Coupled Quantum Wire (qubit implemented by a pair of Quantum Wires coupled by a Quantum Point Contact)
^{[48]}^{[49]}^{[50]} - Nuclear magnetic resonance quantum computer (NMRQC) implemented with the nuclear magnetic resonance of molecules in solution, where qubits are provided by nuclear spins within the dissolved molecule and probed with radio waves
- 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
^{[51]}(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)
^{[52]} - Diamond-based quantum computer
^{[53]}^{[54]}^{[55]}(qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond) - Bose-Einstein condensate-based quantum computer
^{[56]} - 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
^{[57]}^{[58]}(qubit realized by the internal electronic state of dopants in optical fibers) - Metallic-like carbon nanospheres based quantum computers
^{[59]}

A large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.

Any computational problem solvable by a classical computer is also solvable by a quantum computer.^{[60]} Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer, or more formally any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem and the existence of quantum computers does not disprove the Church-Turing thesis.^{[61]}

While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve many problems *faster* than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers. However, the capacity of quantum computers to accelerate classical algorithms has rigid upper bounds, and the overwhelming majority of classical calculations cannot be accelerated by the use of quantum computers.^{[63]}

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". Quantum computers only run probabilistic algorithms, so a quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be efficiently solved by probabilistic classical computers with bounded error.^{[64]} It is known that BPPBQP and widely suspected, but not proven, that BQPBPP, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.^{[65]}

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that PBQPPSPACE; that is, the class of problems that can be efficiently solved by quantum computers includes all problems that can be efficiently solved by deterministic classical computers but does not include any problems that cannot be solved by classical computers with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are NP problems in BQP that are suspected to be outside of P. On the relationship of BQP to NP, little is known. However, it is suspected that BQP does not contain NP; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. Consequently, it is also suspected that BQP is disjoint from the class of NP-complete problems.^{[66]}

BQP is contained in the complexity class *#P* (or more precisely in the associated class of decision problems *P ^{#P}*),

It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an *N*-item database in at most steps, a slight speedup over Grover's algorithm, which runs in steps (however, neither search method would allow quantum computers to solve NP-Complete problems in polynomial time).^{[67]} Theories of quantum gravity, such as M-theory and loop quantum gravity, may allow even faster computers to be built. However, defining computation in such theories is an open problem due to the problem of time, i.e., within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.^{[68]}^{[69]}

- Supercomputer
- Chemical computer
- D-Wave Systems
- DNA computing
- Electronic quantum holography
- Intelligence Advanced Research Projects Activity
- Kane quantum computer
- List of emerging technologies
- List of quantum processors
- Magic state distillation
- Natural computing
- Normal mode
- Photonic computing
- Post-quantum cryptography
- Quantum annealing
- Quantum bus
- Quantum cognition
- Quantum cryptography
- Quantum logic gate
- Quantum machine learning
- Quantum threshold theorem
- Rigetti Computing
- Soliton
- Theoretical computer science
- Timeline of quantum computing
- Topological quantum computer
- Valleytronics

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^{a}^{b}Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory".*SIAM Journal on Computing*.**26**(5): 1411-1473. CiteSeerX 10.1.1.144.7852. doi:10.1137/S0097539796300921. **^**Aaronson, Scott. "Quantum Computing and Hidden Variables" (PDF).**^**Scott Aaronson (2005). "NP-complete Problems and Physical Reality".*ACM SIGACT News*.**2005**. arXiv:quant-ph/0502072. 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)**^**"D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". D-Wave. 2011-05-25. Retrieved .

This further reading section may contain inappropriate or excessive suggestions that may not follow Wikipedia's guidelines. Please ensure that only a reasonable number of balanced, topical, reliable, and notable further reading suggestions are given; removing less relevant or redundant publications with the same point of view where appropriate. Consider utilising appropriate texts as inline sources or creating a separate bibliography article. (May 2019) (Learn how and when to remove this template message) |

- Nielsen, Michael; Chuang, Isaac (2000).
*Quantum Computation and Quantum Information*. Cambridge: Cambridge University Press. ISBN 978-0-521-63503-5. OCLC 174527496. - Mermin, N. David (2007).
*Quantum Computer Science: An Introduction*. Cambridge University Press. ISBN 978-0-521-87658-2. - Akama, Seiki (2014).
*Elements of Quantum Computing: History, Theories and Engineering Applications*. Springer International Publishing. ISBN 978-3-319-08284-4. - Benenti, Giuliano (2004).
*Principles of Quantum Computation and Information Volume 1*. New Jersey: World Scientific. ISBN 978-981-238-830-8. OCLC 179950736. - Stolze, Joachim; Suter, Dieter (2004).
*Quantum Computing*. Wiley-VCH. ISBN 978-3-527-40438-4. - Wichert, Andreas (2014).
*Principles of Quantum Artificial Intelligence*. World Scientific Publishing Co. ISBN 978-981-4566-74-2. - Hiroshi, Imai; Masahito, Hayashi (2006).
*Quantum Computation and Information*. Berlin: Springer. ISBN 978-3-540-33132-2. - Jaeger, Gregg (2006).
*Quantum Information: An Overview*. Berlin: Springer. ISBN 978-0-387-35725-6. OCLC 255569451.

- 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. - Ambainis, Andris (1998). "Quantum computation with linear optics". arXiv:quant-ph/9806048.
- Ambainis, Andris (2000). "The Physical Implementation of Quantum Computation".
*Fortschritte der Physik*.**48**(9-11): 771-783. arXiv:quant-ph/0002077. Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. - Berthiaume, Andre (1997). "Quantum Computation".
- 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. - DiVincenzo, David P. (1995). "Quantum Computation".
*Science*.**270**(5234): 255-261. Bibcode:1995Sci...270..255D. CiteSeerX 10.1.1.242.2165. 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. CiteSeerX 10.1.1.45.9310. doi:10.1007/BF02650179. - Keyes, R. W. (1988). "Miniaturization of electronics and its limits".
*IBM Journal of Research and Development*.**32**: 84-88. doi:10.1147/rd.321.0024. - Mitchell, Ian (1998). "Computing Power into the 21st Century: Moore's Law and Beyond".
- Nielsen, M. A.; Knill, E.; Laflamme, R. "Complete Quantum Teleportation By Nuclear Magnetic Resonance".
- Simon, Daniel R. (1994). "On the Power of Quantum Computation". Institute of Electrical and Electronic Engineers Computer Society Press.
- Vandersypen, Lieven M.K.; Yannoni, Constantino S.; Chuang, Isaac L. (2000).
*Liquid state NMR Quantum Computing*. arXiv:quant-ph/0012108. Bibcode:2000quant.ph.12108V.

- Stanford Encyclopedia of Philosophy: "Quantum Computing" by Amit Hagar and Michael E. Cuffaro.
- Hazewinkel, Michiel, ed. (2001) [1994], "Quantum computation, theory of",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Quantum computing for the very curious by Andy Matuschak and Michael Nielsen

- Lectures

- 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)
- Lomonaco, Sam. Four Lectures on Quantum Computing given at Oxford University in July 2006

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