SUPPORT THE WORK

# GetWiki

### quantum computing

ARTICLE SUBJECTS
news  →
unix  →
wiki  →
ARTICLE TYPES
feed  →
help  →
wiki  →
ARTICLE ORIGINS
quantum computing
[ temporary import ]
- the content below is remote from Wikipedia
- it has been imported raw for GetWiki

## Basics

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, on the other hand, maintains a sequence of qubits, which can represent a one, a zero, or any quantum superposition of those two qubit states;BOOK, Quantum Computation and Quantum Information, Nielsen, Michael A., Chuang, Isaac L., Cambridge University Press, 2010, 978-1-107-00217-3, 2nd, Cambridge, Michael A. Nielsen, Isaac Chuang, Quantum Computation and Quantum Information (book), {{rp|13â€“16}} a pair of qubits can be in any quantum superposition of 4 states,{{rp|16}} and three qubits in any superposition of 8 states. In general, a quantum computer with n qubits can be in any superposition of up to 2^n different states.{{rp|17}} (This compares to a normal computer that can only be in one of these 2^n 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 2^n eigenstates, where each qubit is zero or one, decomposing into a classical state. The outcome can, therefore, be at most n classical bits of information. If the algorithm did not end with a measurement, the result is an unobserved quantum state. (Such unobserved states may be sent to other computers as part of distributed quantum algorithms.)Quantum algorithms are often probabilistic, in that they provide the correct solution only with a certain known probability.WEB,weblink Lecture Notes for Ph219/CS219: Quantum Information Chapter 5, Preskill, John, 2015, 12, 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 |{downarrow}rangle and |{uparrow}rangle, or |0{rangle} and |1{rangle}). This is true because any such system can be mapped onto an effective spin-1/2 system.

## Principles of operation

{{more footnotes|section|date=February 2018}}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. The qubits are in a superposition of states before any measurement is made, which directly affects the possible outcomes of the computation.File:Quantum computer.svg|thumb|200px|Qubits are made up of controlled particles and the means of control (e.g. devices that trap particles and switch them from one state to another).BOOK, Waldner, Jean-Baptiste, Nanocomputers and Swarm Intelligence, International Society for Technology in Education, ISTEInternational Society for Technology in Education, ISTETo 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 2^3=8 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 (a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7) (or a one-dimensional vector witheach vector node holding the amplitude and the state as the bit string of qubits). Here, however, the coefficients a_i are complex numbers, and it is the sum of the squares of the coefficients' absolute values, sum_i |a_i|^2, that must equal 1. For each i, the absolute value squared left|a_i right|^2 gives the probability of the system being found in the i-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.JOURNAL, DiVincenzo, David P., Quantum Computation, Science, 1995, 270, 5234, 255â€“261, 10.1126/science.270.5234.255, 1995Sci...270..255D, 10.1.1.242.2165, {{subscription required}}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 = |a_0|^2, the probability of measuring 001 = |a_1|^2, etc.). Thus, measuring a quantum state described by complex coefficients (a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7) gives the classical probability distribution (|a_0|^2,|a_1|^2,|a_2|^2,|a_3|^2,|a_4|^2,|a_5|^2,|a_6|^2,|a_7|^2) 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, â€¦, 111) is known as the computational basis. Other possible bases are unit-length, orthogonal vectors and the eigenvectors of the Pauli-x operator. Ket notation is often used to make the choice of basis explicit. For example, the state (a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7) in the computational basis can be written as:
a_0,|000rangle + a_1,|001rangle + a_2,|010rangle + a_3,|011rangle + a_4,|100rangle + a_5,|101rangle + a_6,|110rangle + a_7,|111rangle where, e.g., | 010rangle = left(0,0,1,0,0,0,0,0right)
The computational basis for a single qubit (two dimensions) is |0rangle = left(1,0right) and |1rangle = left(0,1right).Using the eigenvectors of the Pauli-x operator, a single qubit is |+rangle = tfrac{1}{sqrt{2}} left(1,1right) and |-rangle = tfrac{1}{sqrt{2}} left(1,-1right).

## Operation

{{unreferenced section|date=February 2018}}{{Unsolved|physics|Is a universal quantum computer sufficient to efficiently simulate an arbitrary physical system?}}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, |000rangle, 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, Deutschâ€“Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

## Potential

### Cryptography

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).JOURNAL, Lenstra, Arjen K.,weblink Integer Factoring, Designs, Codes and Cryptography, 19, 101â€“128, 2000, 10.1023/A:1008397921377, 2/3, yes,weblink" title="web.archive.org/web/20150410234239weblink">weblink 2015-04-10, 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, Diffieâ€“Hellman, and elliptic curve Diffieâ€“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.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. 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.Robert J. McEliece. "A public-key cryptosystem based on algebraic coding theory." Jet Propulsion Laboratory DSN Progress Report 42â€“44, 114â€“116. 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.JOURNAL, Kobayashi, H., Gall, F.L., Dihedral Hidden Subgroup Problem: A Survey, 2006, Information and Media Technologies, 1, 1, 178â€“185,weblink 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,Bennett C.H., Bernstein E., Brassard G., Vazirani U., "The strengths and weaknesses of quantum computation". SIAM Journal on Computing 26(5): 1510â€“1523 (1997). 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.JOURNAL,weblink What are quantum computers and how do they work? WIRED explains, Wired UK, 2018-02-16,

### Quantum search

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,Quantum Algorithm Zoo {{Webarchive|url=https://web.archive.org/web/20180429014516weblink |date=2018-04-29 }} â€“ Stephen Jordan's Homepage 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.BOOK, Jon Schiller, Phd,weblink
isbn=9781439243497, 2009-06-19, 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:
1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. 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 appliedARXIV, quant-ph/0504012, Ambainis, Andris, Quantum search algorithms, 2005, 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.JOURNAL, 1804.00200, 10.14569/IJACSA.2018.090354, The Impact of Quantum Computing on Present Cryptography, International Journal of Advanced Computer Science and Applications, 9, 3, March 2018, Mavroeidis, Vasileios, Vishi, Kamer, d, Mateusz, JÃ¸sang, Audun, This application of quantum computing is a major interest of government agencies.NEWS,weblink NSA seeks to build quantum computer that could crack most types of encryption, Steven, Rich, Barton, Gellman, 2014-02-01, Washington Post,

### Quantum simulation

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.JOURNAL,weblink The Father of Quantum Computing, Wired, Quinn, Norton, 2007-02-15, Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.WEB,weblink What Can We Do with a Quantum Computer?, Andris, Ambainis, Spring 2014, Institute for Advanced Study,

### Quantum annealing and adiabatic optimization

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.

### Solving linear equations

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.JOURNAL, 0811.3171, Ambainis, Andris, Hassidim, Avinatan, Lloyd, Seth, Quantum algorithm for solving linear systems of equations, Physical Review Letters, 103, 15, 150502, 2008, 10.1103/PhysRevLett.103.150502, 19905613,

### Quantum supremacy

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.JOURNAL, Characterizing Quantum Supremacy in Near-Term Devices, Nature Physics, 14, 6, 595â€“600, Sergio, Boixo, Sergei V., Isakov, Vadim N., Smelyanskiy, Ryan, Babbush, Nan, Ding, Zhang, Jiang, Michael J., Bremner, John M., Martinis, Hartmut, Neven, 2018, 1608.00263, 10.1038/s41567-018-0124-x, 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.WEB,weblink Quantum Computers Compete for "Supremacy", Neil, Savage, Quantum supremacy has not been achieved yet, and skeptics like Gil Kalai doubt that it will ever be.WEB,weblink Quantum Supremacy and Complexity, 23 April 2016, WEB, Kalai, Gil, The Quantum Computer Puzzle,weblink AMS, Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.JOURNAL, Unruh, Bill, Maintaining coherence in Quantum Computers, Physical Review A, 51, 2, 992â€“997, hep-th/9406058, 1995PhRvA..51..992U, 1995, 10.1103/PhysRevA.51.992, Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.WEB, Davies, Paul, The implications of a holographic universe for quantum information science and the nature of physical law,weblink Macquarie University,

## Obstacles

There are a number of technical challenges in building a large-scale quantum computer,JOURNAL, Dyakonov, Mikhail,weblink The Case Against Quantum Computing, IEEE Spectrum, 2018-11-15, 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:JOURNAL, quant-ph/0002077, The Physical Implementation of Quantum Computation, DiVincenzo, David P., 2000-04-13, 10.1002/1521-3978(200009)48:9/113.0.CO;2-E, 48, 9â€“11, Fortschritte der Physik, 771â€“783, 2000ForPh..48..771D,
• scalable physically to increase the number of qubits;
• qubits that can be initialized to arbitrary values;
• quantum gates that are faster than decoherence time;
• universal gate set;
• qubits that can be read easily.

### Quantum decoherence

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.JOURNAL, Jones, Nicola, Computing: The quantum company, Nature, 19 June 2013, 498, 7454, 286â€“288, 10.1038/498286a, 23783610, 2013Natur.498..286J, 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.ARXIV, Amy, Matthew, Matteo, Olivia, Gheorghiu, Vlad, Mosca, Michele, Parent, Alex, Schanck, John, Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3, November 30, 2016, 1603.09383, quant-ph, 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 10âˆ’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.JOURNAL, Is Fault-Tolerant Quantum Computation Really Possible?, Dyakonov, M. I., 2006-10-14, 4â€“18, Future Trends in Microelectronics. Up the Nano Creek, S. Luryi, J. Xu, A. Zaslavsky, quant-ph/0610117, 2006quant.ph.10117D, 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.JOURNAL
, Freedman, Michael H., Michael Freedman
, Kitaev, Alexei, Alexei Kitaev
, Larsen, Michael J., Michael J. Larsen
, Wang, Zhenghan
, quant-ph/0101025
, 10.1090/S0273-0979-02-00964-3
, 1
, Bulletin of the American Mathematical Society
, 1943131
, 31â€“38
, Topological quantum computation
, 40
, 2003, JOURNAL, Monroe, Don,weblink Anyons: The breakthrough quantum computing needs?, New Scientist, 2008-10-01,

## Developments

### Quantum computing models

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)JOURNAL, A., Das, B. K., Chakrabarti, Quantum Annealing and Analog Quantum Computation, Reviews of Modern Physics, Rev. Mod. Phys., 80, 3, 1061â€“1081, 2008, 10.1103/RevModPhys.80.1061, 2008RvMP...80.1061D, 10.1.1.563.9990, 0801.2193,
• Topological quantum computerJOURNAL, 0707.1889, Rev Mod Phys, 2008, Nonabelian Anyons and Quantum Computation, Chetan, Sankar, Nayak, Das Sarma, Steven, Simon, Ady, Stern, 80, 1083â€“1159, 10.1103/RevModPhys.80.1083, 2008RvMP...80.1083N, 3, (computation decomposed into the braiding of anyons in a 2D lattice)
The quantum Turing machine is theoretically important but the 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.

### Physical realizations

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):
LAST=KAMINSKY YEAR=2004, (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 computerJOURNAL, Atac, ImamoÄŸlu, D. D., Awschalom, Guido, Burkard, D. P., DiVincenzo, D., Loss, M., Sherwin, A., Small, Quantum information processing using quantum dot spins and cavity-QED, Physical Review Letters, 1999, 83, 4204â€“4207, 10.1103/PhysRevLett.83.4204, 1999PhRvL..83.4204I, 20,weblink quant-ph/9904096, ) (qubit given by the spin states of trapped electrons)
• Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)JOURNAL, Leonid, Fedichkin, Maxim, Yanchenko, Kamil, Valiev, Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot, Quantum Computers and Computing, 2000, 1, 58â€“76,weblink quant-ph/0006097, 2000quant.ph..6097F, yes,weblink" title="web.archive.org/web/20110818071243weblink">weblink 2011-08-18,
• Coupled Quantum Wire (qubit implemented by a pair of Quantum Wires coupled by a Quantum Point Contact)JOURNAL, Bertoni, A., Bordone, P., Brunetti, R., Jacoboni, C., Reggiani, S., 2000-06-19, Quantum Logic Gates based on Coherent Electron Transport in Quantum Wires, Physical Review Letters, 84, 25, 5912â€“5915, 10.1103/PhysRevLett.84.5912, 10991086, 2000PhRvL..84.5912B, JOURNAL, Ionicioiu, Radu, Amaratunga, Gehan, Udrea, Florin, 2001-01-20, Quantum Computation with Ballistic Electrons, International Journal of Modern Physics B, en, 15, 2, 125â€“133, 10.1142/s0217979201003521, 0217-9792, quant-ph/0011051, 2001IJMPB..15..125I, JOURNAL, Ramamoorthy, A., Bird, J. P., Reno, J. L., 2007, Using split-gate structures to explore the implementation of a coupled-electron-waveguide qubit scheme,weblink Journal of Physics: Condensed Matter, en, 19, 27, 276205, 10.1088/0953-8984/19/27/276205, 0953-8984, 2007JPCM...19A6205R,
• 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 magnetJOURNAL, Quantum computing in molecular magnets., Apr 12, 2001, Nature, 10.1038/35071024, 11298441, 410, 6830, 789â€“93, Leuenberger, MN, Loss, D, cond-mat/0011415, 2001Natur.410..789L, (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)JOURNAL, Knill, G. J., Laflamme, Milburn, A scheme for efficient quantum computation with linear optics, Nature, 2001, 409, 10.1038/35051009, 2001Natur.409...46K, R., G. J., 6816, 11343107, 46â€“52,
• Diamond-based quantum computerJOURNAL, Optics and Spectroscopy, August 2005, A quantum computer based on NV centers in diamond: Optically detected nutations of single electron and nuclear spins, Nizovtsev, A. P.
issue = 2, 248â€“260, 10.1134/1.2034610ACCESSDATE=2007-06-04 FIRST=WOLFGANG DEADURL=YES ARCHIVEDATE=2007-06-04, JOURNAL, Science, June 6, 2008, Multipartite Entanglement Among Single Spins in Diamond, Neumann, P., 320, 5881, 1326â€“1329, 10.1126/science.1157233, 18535240, 2008Sci...320.1326N, 1, Mizuochi, N., Rempp, F., Hemmer, P., Watanabe, H., Yamasaki, S., Jacques, V., Gaebel, T., Jelezko, F., (qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond)
• Transistor-based quantum computer â€“ string quantum computers with entrainment of positive holes using an electrostatic trap
• Rare-earth-metal-ion-doped inorganic crystal based quantum computersJOURNAL, Opt. Commun., January 1, 2002, Quantum computer hardware based on rare-earth-ion-doped inorganic crystals, N., Ohlsson, R. K., Mohan, S., KrÃ¶ll, 201, 1â€“3, 71â€“77, 10.1016/S0030-4018(01)01666-2, 2002OptCo.201...71O, JOURNAL, Phys. Rev. Lett., September 23, 2004, Demonstration of conditional quantum phase shift between ions in a solid, J. J., Longdell, M. J., Sellars, N. B., Manson, 93, 13, 130503, 10.1103/PhysRevLett.93.130503, 15524694
bibcode = 2004PhRvL..93m0503L, (qubit realized by the internal electronic state of dopants in optical fibers)
• Metallic-like carbon nanospheres based quantum computersJOURNAL, NÃ¡frÃ¡di, BÃ¡lint, Choucair, Mohammad, Dinse, Klaus-Peter, ForrÃ³, LÃ¡szlÃ³, Room Temperature manipulation of long lifetime spins in metallic-like carbon nanospheres, Nature Communications, July 18, 2016, 12232, 10.1038/ncomms12232,weblink 7, 27426851, 4960311, 1611.07690, 2016NatCo...712232N,
A 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.

## Relation to computational complexity theory

(File:BQP complexity class diagram.svg|thumb|The suspected relationship of BQP to other problem spaces.Nielsen, p. 42)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.Nielsen, p. 41 A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP.BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P#P),JOURNAL, Bernstein, Ethan, Vazirani, Umesh, 10.1137/S0097539796300921, Quantum Complexity Theory, 1997, 1411â€“1473, 26, SIAM Journal on Computing,weblink 5, 10.1.1.144.7852, which is a subclass of PSPACE.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 limitsâ€”upper bounds of quantum computation's complexity. The overwhelming part of classical calculations cannot be accelerated on a quantum computer.JOURNAL, Ozhigov, Yuri, Quantum Computers Speed Up Classical with Probability Zero, 1999, 1707â€“1714, 10, Chaos, Solitons & Fractals, quant-ph/9803064, 1998quant.ph..3064O, 10.1016/S0960-0779(98)00226-4, 10, A similar fact prevails for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.JOURNAL, Ozhigov, Yuri, Lower Bounds of Quantum Search for Extreme Point, 1999, 2165â€“2172, A455, Proceedings of the London Royal Society, quant-ph/9806001, 1999RSPSA.455.2165O, 10.1098/rspa.1999.0397, 1986, 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 O(sqrt[3]{N}) steps. This is slightly faster than the O(sqrt{N}) steps taken by Grover's algorithm. Neither search method will allow quantum computers to solve NP-Complete problems in polynomial time.WEB,weblink Quantum Computing and Hidden Variables, Aaronson, Scott, Although quantum computers may be faster than classical computers for some problem types, those described above cannot solve any problem that classical computers cannot 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 Churchâ€“Turing thesis.Nielsen, p. 126 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.JOURNAL, Scott Aaronson, NP-complete Problems and Physical Reality, ACM SIGACT News, 2005, quant-ph/0502072, 2005, 2005quant.ph..2072A, See section 7 "Quantum Gravity": "[â€¦] to anyone who wantsa 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? [â€¦] until we can say what it means for a 'user' to specify an 'input' andâ€˜later' receive an 'output'â€”there is no such thing as computation, not even theoretically." (emphasis in original)

{hide}columns-list|colwidth=20em| {edih}

## References

{{Reflist|30em}}

{{Too much further reading|date=May 2019|reason=No indication why any of these are justified here; if used as references, they should be incorporated into the article; if not, each entry should be justified. This is turning into a dead-tree equivalent of a link farm, primarily to promote the text and papers listed.}}
• JOURNAL, Derek Abbott, Abbot, Derek, Charles R. Doering, Doering, Charles R., Carlton M. Caves, Caves, Carlton M., Daniel Lidar, Lidar, Daniel M., Howard Brandt, Brandt, Howard E., Alexander R. Hamilton, Hamilton, Alexander R., David K. Ferry, Ferry, David K., Julio Gea-Banacloche, Gea-Banacloche, Julio, Sergey M. Bezrukov, Bezrukov, Sergey M., Laszlo B. Kish, Laszlo B., Kish, Dreams versus Reality: Plenary Debate Session on Quantum Computing, Quantum Information Processing, 2003, 2, 6, 449â€“472, 10.1023/B:QINP.0000042203.24782.9a, quant-ph/0310130, 2027.42/45526,
• BOOK, Akama, Seiki, Seiki Akama, 2014, Elements of Quantum Computing: History, Theories and Engineering Applications, Springer International Publishing, 978-3-319-08284-4,
• ARXIV, Ambainis, Andris, quant-ph/9806048, Quantum computation with linear optics, 1998,
• JOURNAL, Ambainis, Andris, The Physical Implementation of Quantum Computation, Fortschritte der Physik, 48, 9â€“11, 771â€“783, 2000, 10.1002/1521-3978(200009)48:9/113.0.CO;2-E, quant-ph/0002077,
• WEB, Berthiaume, Andre, 1997, Quantum Computation,weblink
• JOURNAL, Dibyendu Chatterjee, Arijit Roy, A transmon-based quantum half-adder scheme, Progress of Theoretical and Experimental Physics, 2015, 2015, 9, 093A02(16pages), 10.1093/ptep/ptv122, 2015PTEP.2015i3A02C,
• BOOK, Benenti, Giuliano, Principles of Quantum Computation and Information Volume 1, World Scientific, New Jersey, 2004, 978-981-238-830-8, 179950736,
• JOURNAL, DiVincenzo, David P., Quantum Computation, Science, 1995, 270, 5234, 255â€“261, 10.1126/science.270.5234.255, 1995Sci...270..255D, 10.1.1.242.2165, Table 1 lists switching and dephasing times for various systems.
• JOURNAL, Feynman, Richard, Richard Feynman, Simulating physics with computers, International Journal of Theoretical Physics, 21, 467â€“488, 1982, 10.1007/BF02650179, 1982IJTP...21..467F, 6â€“7, 10.1.1.45.9310,
• BOOK, Hiroshi, Imai, Masahito, Hayashi, 2006, Quantum Computation and Information, Springer, 978-3-540-33132-2, Berlin,
• BOOK, Jaeger, Gregg, Quantum Information: An Overview, Springer, Berlin, 2006, 978-0-387-35725-6, 255569451,
• BOOK, Michael Nielsen, Nielsen, Michael, Isaac L. Chuang, Chuang, Isaac, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000, 978-0-521-63503-5, 174527496,weblink
• JOURNAL, Keyes, R. W., 1988, Miniaturization of electronics and its limits, IBM Journal of Research and Development, 32, 84â€“88, 10.1147/rd.321.0024,
• WEB, Landauer, Rolf, Rolf Landauer, 1961, Irreversibility and heat generation in the computing process,weblink
• Lomonaco, Sam. Four Lectures on Quantum Computing given at Oxford University in July 2006
• WEB, Mitchell, Ian, 1998, Computing Power into the 21st Century: Moore's Law and Beyond,weblink
• JOURNAL, Moore, Gordon E., Gordon E. Moore, 1965, Cramming more components onto integrated circuits, Electronics Magazine,
• WEB, Nielsen, M. A., Michael Nielsen, Knill, E., Raymond Laflamme, Laflamme, R., Complete Quantum Teleportation By Nuclear Magnetic Resonance,weblink
• WEB, Sanders, Laura, 2009, First programmable quantum computer created,weblink
• WEB, Simon, Daniel R., 1994, On the Power of Quantum Computation, Institute of Electrical and Electronic Engineers Computer Society Press,weblink
• WEB, Simons Conference on New Trends in Quantum Computation,weblink November 15-19, 2010, Simons Center for Geometry and Physics, and C. N. Yang Institute for Theoretical Physics,
• BOOK, Singer, Stephanie Frank, Linearity, Symmetry, and Prediction in the Hydrogen Atom, Springer, New York, 2005, 978-0-387-24637-6, 253709076,
• BOOK, Stolze, Joachim, Suter, Dieter, 2004, Quantum Computing, Wiley-VCH, 978-3-527-40438-4,
• BOOK, Vandersypen, Lieven M.K., Yannoni, Constantino S., Chuang, Isaac L., 2000, Liquid state NMR Quantum Computing,
• BOOK, Wichert, Andreas, Andreas Wichert, 2014, Principles of Quantum Artificial Intelligence, World Scientific Publishing Co., 978-981-4566-74-2,
• Indian Science News Association, Special Issue of "Science & Culture" on 'A Quantum Jump in Computation', Vol. 85 (5-6), May-June (2019)

{{Commons|Quantum computer}}

Lectures
{{Quantum computing}}{{Emerging technologies}}{{Computer science}}{{Quantum mechanics topics}}{{Authority control}}

- content above as imported from Wikipedia
- "quantum computing" does not exist on GetWiki (yet)
- time: 6:19am EDT - Sat, Aug 24 2019
[ this remote article is provided by Wikipedia ]
LATEST EDITS [ see all ]
GETWIKI 09 JUL 2019
Eastern Philosophy
History of Philosophy
GETWIKI 09 MAY 2016
GETWIKI 18 OCT 2015
M.R.M. Parrott
Biographies
GETWIKI 20 AUG 2014
GETWIKI 19 AUG 2014
CONNECT