Quantum computing: Difference between revisions

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{{Short description|Computer hardware technology that uses quantum mechanics}}

[[File:Bloch Sphere.svg|thumb|The [[Bloch sphere]] is a representation of a [[qubit]], the fundamental building block of quantum computers.]]

{{Use American English|date=February 2023}}

{{Refimprove|reason=A lot of arXiv papers used as references|date=June 2015}}

{{Use dmy dates|date=February 2021}}

'''Quantum computing''' studies theoretical [[computation]] systems ('''quantum computers''') that make direct use of [[quantum mechanics|quantum-mechanical]] [[phenomena]], such as [[quantum superposition|superposition]] and [[quantum entanglement|entanglement]], to perform [[Instruction (computer science)|operations]] on [[data]].<ref>{{cite journal |url=http://cba.mit.edu/docs/papers/98.06.sciqc.pdf |title=Quantum Computing with Molecules |journal=[[Scientific American]] |first1=Neil |last1=Gershenfeld |authorlink1=Neil Gershenfeld |first2=Isaac L. |last2=Chuang |authorlink2=Isaac L. Chuang |date=June 1998}}</ref> Quantum computers are different from digital computers based on [[transistor]]s. Whereas digital computers require data to be encoded into binary digits ([[bit]]s), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits ([[qubits]]), which can be in [[quantum superposition|superpositions]] of states. A [[quantum Turing machine]] is a theoretical model of such a computer, and is also known as the universal quantum computer. Quantum computers share theoretical similarities with [[Non-deterministic Turing machine|non-deterministic]] and [[probabilistic automaton|probabilistic computers]]. The field of quantum computing was initiated by the work of [[Yuri Manin]] in 1980,<ref name="manin1980vychislimoe">{{cite book| author=Manin, Yu. I.| title=Vychislimoe i nevychislimoe |trans_title=Computable and Noncomputable | year=1980| publisher=Sov.Radio| url=http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5Bdjv%5D.zip| pages=13–15| language=Russian| accessdate=2013-03-04}}</ref> [[Richard Feynman]] in 1982,<ref name="Feynman82">{{cite journal |last=Feynman |first=R. P. |title=Simulating physics with computers |journal=[[International Journal of Theoretical Physics]] |year=1982 |volume=21 |issue=6 |pages=467–488 |doi=10.1007/BF02650179 |bibcode=1982IJTP...21..467F}}</ref> and [[David Deutsch]] in 1985.<ref>{{cite journal|last1=Deutsch|first1=David|title=Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer|journal=Proceedings of the Royal Society of London A|date=1985|volume=400|pages=97-117|ref=Deutsch85}}</ref> A quantum computer with spins as quantum bits was also formulated for use as a quantum [[space–time]] in 1968.<ref>{{cite book |first=David |last=Finkelstein |chapter=Space-Time Structure in High Energy Interactions |title=Fundamental Interactions at High Energy |editor1-first=T. |editor1-last=Gudehus |editor2-first=G. |editor2-last=Kaiser |___location=New York |publisher=Gordon & Breach |year=1968 }}</ref>

[[File:Bloch sphere.svg|thumb|[[Bloch sphere]] representation of a qubit. The state <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle</math> is a point on the surface of the sphere, partway between the poles, <math>|0\rangle</math> and <math>|1\rangle</math>.]]

A '''quantum computer''' is a (real or theoretical) [[computer]] that uses [[quantum mechanical]] phenomena in an essential way: a quantum computer exploits [[quantum superposition|superposed]] and [[quantum entanglement|entangled]] [[quantum state|states]] and the (non-deterministic) outcomes of [[quantum measurement]]s as features of its computation. Ordinary ("classical") computers operate, by contrast, using deterministic rules. Any classical computer can, in principle, be replicated using a ([[classical physics|classical]]) mechanical device such as a [[Turing machine]], with at most a constant-factor slowdown in time—unlike quantum computers, which are believed to require [[Exponential growth|exponentially]] more resources to simulate classically. It is widely believed that a scalable quantum computer could perform ''some'' calculations exponentially faster than any classical computer. Theoretically, a large-scale quantum computer could [[post-quantum cryptography|break some widely used encryption schemes]] and aid physicists in performing [[quantum simulator|physical simulations]]. However, current hardware implementations of quantum computation are largely experimental and only suitable for specialized tasks.

{{As of|2015}}, 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.<ref>{{cite web|last=Gershon|first=Eric|url=http://phys.org/news/2013-01-qubit-bodes-future-quantum.html|title=New qubit control bodes well for future of quantum computing|date=2013-01-14|publisher=Phys.org|accessdate=2014-10-26}}</ref>{{citation needed|reason=The citation given does not speak to this assertion|date=March 2015}} Both practical and theoretical research continues, and many national governments and military agencies are funding quantum computing research in an effort to develop quantum [[computer]]s for civilian, business, trade, and national security purposes, such as [[cryptanalysis]].<ref>[http://qist.lanl.gov/qcomp_map.shtml Quantum Information Science and Technology Roadmap] for a sense of where the research is heading.</ref>

<!-- Basic principles of quantum computing -->

Large-scale quantum computers will be able to solve certain problems much more quickly than any classical computers that use even the best currently known [[algorithm]]s, like [[integer factorization]] using [[Shor's algorithm]] or the [[Quantum algorithm#Quantum simulation|simulation of quantum many-body systems]]. There exist [[quantum algorithm]]s, such as [[Simon's algorithm]], that run faster than any possible probabilistic classical algorithm.<ref name=Simon1994>{{cite journal

The basic [[unit of information]] in quantum computing, the [[qubit]] (or "quantum bit"), serves the same function as the [[bit]] in ordinary or "classical" computing.{{sfn|Mermin|2007|p=1}} However, unlike a classical bit, which can be in one of two states (a [[Binary number|binary]]), a qubit can exist in a [[quantum superposition|superposition]] of its two "basis" states, a state that is in an abstract sense "between" the two basis states. When [[measurement in quantum mechanics|measuring]] a qubit, the result is a [[Born rule|probabilistic output]] of a classical bit. If a quantum computer manipulates the qubit in a particular way, [[wave interference]] effects can amplify the desired measurement results. The design of [[quantum algorithms]] involves creating procedures that allow a quantum computer to perform calculations efficiently and quickly.

| title = On the power of quantum computation

| year = 1994

| author = Simon, D.R.

| journal = Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on

| pages = 116–123

| doi = 10.1109/SFCS.1994.365701

| isbn = 0-8186-6580-7}}</ref>

Given sufficient computational resources, however, a classical computer could be made to simulate any quantum algorithm, as quantum computation does not violate the [[Church–Turing thesis]].

<ref>{{cite book

|last= Nielsen

|first= Michael A.

|author2=Chuang, Isaac L.

|title= Quantum Computation and Quantum Information

|page=202

}}</ref>

<!-- However, the computational basis of 500 qubits, for example, would already be too large to be represented on a classical computer because it would require 2⁵⁰⁰ complex values (2⁵⁰¹ bits) to be stored.

<ref name="Nielsen">{{cite book

|last= Nielsen

|first= Michael A.

|author2=Chuang, Isaac L.

|title= Quantum Computation and Quantum Information

|page=17

}}</ref>

(For comparison, a terabyte of digital information is only 2⁴³ bits.)-->

<!--Physical implementations-->

== Basis ==

Quantum computers are not yet practical for real-world applications. Physically engineering high-quality qubits has proven to be challenging. If a physical qubit is not sufficiently [[isolated system|isolated]] from its environment, it suffers from [[quantum decoherence]], introducing [[noise (signal processing)|noise]] into calculations. National governments have invested heavily in experimental research aimed at developing scalable qubits with longer coherence times and lower error rates. Example implementations include [[superconducting quantum computing|superconductors]] (which isolate an [[electrical current]] by eliminating [[electrical resistance]]) and [[trapped ion quantum computer|ion traps]] (which confine a single [[atom|atomic particle]] using [[electromagnetic fields]]). Researchers have claimed, and are widely believed to be correct, that certain quantum devices can outperform classical computers on narrowly defined tasks, a milestone referred to as quantum advantage or [[quantum supremacy]]. These tasks are not necessarily useful for real-world applications.

A classical computer has a [[Computer memory|memory]] made up of [[bit]]s, where each bit represents either a one or a zero. A quantum computer maintains a sequence of [[qubit]]s. A single qubit can represent a one, a zero, or any [[quantum superposition]] of those two [[Pure qubit state|qubit states]]; a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8 states. In general, a quantum computer with <math>n</math> qubits can be in an arbitrary superposition of up to <math>2^n</math> different states simultaneously (this compares to a normal computer that can only be in ''one'' of these <math>2^n</math> states at any one time). A quantum computer operates by setting the qubits in a controlled initial state that represents the problem at hand and by manipulating those qubits with a fixed sequence of [[quantum gate|quantum logic gates]]. The sequence of gates to be applied is called a [[quantum algorithm]]. The calculation ends with a measurement, collapsing the system of qubits into one of the <math>2^n</math> pure states, where each qubit is zero or one. The outcome can therefore be at most <math>n</math> classical bits of information. Quantum algorithms are often non-deterministic, in that they provide the correct solution only with a certain known probability.

== History ==

An example of an implementation of qubits for a quantum computer could start with the use of particles with two [[spin (physics)|spin]] states: "down" and "up" (typically written <math>|{\downarrow}\rangle</math> and <math>|{\uparrow}\rangle</math>, or <math>|0{\rangle}</math> and <math>|1{\rangle}</math>). But in fact any system possessing an [[observable]] quantity ''A'', which is ''conserved'' under time evolution such that ''A'' has at least two discrete and sufficiently spaced consecutive [[eigenvalue]]s, is a suitable candidate for implementing a qubit. This is true because any such system can be mapped onto an effective [[spin-1/2]] system.

{{For timeline|Timeline of quantum computing and communication}}

For many years, the fields of [[quantum mechanics]] and [[computer science]] formed distinct academic communities.{{sfn|Aaronson|2013|p=132}} [[Modern quantum theory]] developed in the 1920s to explain perplexing physical phenomena observed at atomic scales,<ref name="Zwiebach2022">{{cite book|first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8 |at=§1 |quote=Quantum physics has replaced classical physics as the correct fundamental description of our physical universe. It is used routinely to describe most phenomena that occur at short distances. [...] The era of quantum physics began in earnest in 1925 with the discoveries of Erwin Schrödinger and Werner Heisenberg. The seeds for these discoveries were planted by Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, and others.}}</ref><ref>{{cite book|first=Steven |last=Weinberg |author-link=Steven Weinberg |title=Lectures on Quantum Mechanics |year=2015 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-11166-0|chapter=Historical Introduction |pages=1–30}}</ref> and [[digital computers]] emerged in the following decades to replace [[human computers]] for tedious calculations.<ref>{{Cite book |last=Ceruzzi |first=Paul E. |title=Computing: A Concise History |date=2012 |isbn=978-0-262-31038-3 |publisher=MIT Press |___location=[[Cambridge, Massachusetts]] |pages=3, 46 |language=en-US |oclc=796812982}}</ref> Both disciplines had practical applications during [[World War II]]; computers played a major role in [[World War II cryptography|wartime cryptography]],<ref>{{Cite book |last=Hodges |first=Andrew |author-link=Andrew Hodges |title=Alan Turing: The Enigma |publisher=[[Princeton University Press]] |year=2014 |isbn=9780691164724 |___location=Princeton, New Jersey |page=xviii |language=en-US}}</ref> and quantum physics was essential for [[nuclear physics]] used in the [[Manhattan Project]].<ref>{{Cite journal |last=Mårtensson-Pendrill |first=Ann-Marie |date=2006-11-01 |title=The Manhattan project—a part of physics history |journal=Physics Education |language=en-US |volume=41 |issue=6 |pages=493–501 |bibcode=2006PhyEd..41..493M |doi=10.1088/0031-9120/41/6/001 |issn=0031-9120 |s2cid=120294023}}</ref>

== Mechanics ==

A quantum computer with a given number of qubits is fundamentally different from a classical computer composed of the same number of classical bits. For example, to represent the state of an ''n''-qubit system on a classical computer would require the storage of 2^''n'' [[Complex number|complex]] coefficients. Although this fact may seem to indicate that qubits can hold exponentially more information than their classical counterparts, care must be taken not to overlook the fact that the qubits are only in a probabilistic superposition of all of their states. This means that when the final state of the qubits is measured, they will only be found in one of the possible configurations they were in before measurement. Moreover, it is incorrect to think of the qubits as only being in one particular state before measurement since the fact that they were in a superposition of states before the measurement was made directly affects the possible outcomes of the computation.

As [[physicists]] applied quantum mechanical models to computational problems and swapped digital [[bit]]s for [[qubits]], the fields of quantum mechanics and computer science began to converge. In 1980, [[Paul Benioff]] introduced the [[quantum Turing machine]], which uses quantum theory to describe a simplified computer.<ref name="The computer as a physical system">{{cite journal|last1=Benioff|first1=Paul |author-link=Paul Benioff |year=1980|title=The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines|journal=Journal of Statistical Physics|volume=22|issue=5|pages=563–591|bibcode=1980JSP....22..563B|doi=10.1007/bf01011339|s2cid=122949592}}</ref>

[[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).<ref>{{cite book |last = Waldner |first = Jean-Baptiste |title = Nanocomputers and Swarm Intelligence |publisher = [[International Society for Technology in Education|ISTE]] |place = London |year = 2007 |page = 157 |isbn = 2-7462-1516-0}}</ref>

When digital computers became faster, physicists faced an [[exponential time|exponential]] increase in overhead when [[simulating quantum dynamics]],<ref>{{Cite journal |last1=Buluta |first1=Iulia |last2=Nori |first2=Franco |date=2009-10-02 |title=Quantum Simulators |journal=Science |language=en |volume=326 |issue=5949 |pages=108–111 |doi=10.1126/science.1177838 |pmid=19797653 |bibcode=2009Sci...326..108B |s2cid=17187000 |issn=0036-8075}}</ref> prompting [[Yuri Manin]] and [[Richard Feynman]] to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation.<ref name="manin1980vychislimoe">{{cite book |author=Manin |first=Yu. I. |author-link=Yuri Manin |url=http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5bdjv-fax%5d.zip |title=Vychislimoe i nevychislimoe |publisher=Soviet Radio |year=1980 |pages=13–15 |language=ru |trans-title=Computable and Noncomputable |access-date=4 March 2013 |archive-url=https://web.archive.org/web/20130510173823/http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5Bdjv%5D.zip |archive-date=10 May 2013 |url-status=dead}}</ref><ref>{{cite journal |last1=Feynman |first1=Richard |author-link=Richard Feynman |title=Simulating Physics with Computers |journal=International Journal of Theoretical Physics |date=June 1982 |volume=21 |issue=6/7 |pages=467–488 |doi=10.1007/BF02650179 |url=https://people.eecs.berkeley.edu/~christos/classics/Feynman.pdf |access-date=28 February 2019 |bibcode=1982IJTP...21..467F |s2cid=124545445 |archive-url=https://web.archive.org/web/20190108115138/https://people.eecs.berkeley.edu/~christos/classics/Feynman.pdf |archive-date=8 January 2019 |url-status=dead}}</ref>{{sfn|Nielsen|Chuang|2010|p=214}}

]]

In a 1984 paper, [[Charles H. Bennett (physicist)|Charles Bennett]] and [[Gilles Brassard]] applied quantum theory to [[cryptography]] protocols and demonstrated that quantum key distribution could enhance [[information security]].<ref name="bb84">{{cite book|first1=C. H. |last1=Bennett |first2=G. |last2=Brassard |chapter=Quantum cryptography: Public key distribution and coin tossing |title=Proceedings of the International Conference on Computers, Systems & Signal Processing, Bangalore, India |volume=1 |pages=175–179 |publisher=IEEE |year=1984 |___location=New York }} Reprinted as {{cite journal|first1=C. H. |last1=Bennett |first2=G. |last2=Brassard |title=Quantum cryptography: Public key distribution and coin tossing |journal=Theoretical Computer Science |series=Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84 |volume=560 |number=1 |date=4 December 2014 |pages=7–11 |doi=10.1016/j.tcs.2014.05.025 |doi-access=free|arxiv=2003.06557 }}</ref><ref name="personal">{{Cite book |last=Brassard |first=G. |title=IEEE Information Theory Workshop on Theory and Practice in Information-Theoretic Security, 2005 |chapter=Brief history of quantum cryptography: A personal perspective |date=2005 |___location=Awaji Island, Japan |publisher=IEEE |pages=19–23 |doi=10.1109/ITWTPI.2005.1543949 |arxiv=quant-ph/0604072 |isbn=978-0-7803-9491-9|s2cid=16118245 }}</ref>

[[Quantum algorithm]]s then emerged for solving [[oracle machine|oracle problems]], such as [[Deutsch's algorithm]] in 1985,<ref>{{Cite journal |date=1985-07-08 |title=Quantum theory, the Church–Turing principle and the universal quantum computer |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |language=en |volume=400 |issue=1818 |pages=97–117 |doi=10.1098/rspa.1985.0070 |bibcode=1985RSPSA.400...97D |s2cid=1438116 |issn=0080-4630|last1=Deutsch |first1=D. }}</ref> the [[Bernstein{{en dash}}Vazirani algorithm]] in 1993,<ref>{{Cite book |last1=Bernstein |first1=Ethan |title=Proceedings of the twenty-fifth annual ACM symposium on Theory of computing – STOC '93 |last2=Vazirani |first2=Umesh |date=1993 |publisher=ACM Press |isbn=978-0-89791-591-5 |___location=San Diego, California, United States |pages=11–20 |language=en |chapter=Quantum complexity theory |doi=10.1145/167088.167097 |chapter-url=http://portal.acm.org/citation.cfm?doid=167088.167097 |s2cid=676378}}</ref> and [[Simon's algorithm]] in 1994.<ref>{{Cite book |last=Simon |first=D. R. |title=Proceedings 35th Annual Symposium on Foundations of Computer Science |chapter=On the power of quantum computation |date=1994 |___location=Santa Fe, New Mexico, USA |publisher=IEEE Comput. Soc. Press |pages=116–123 |doi=10.1109/SFCS.1994.365701 |isbn=978-0-8186-6580-6 |s2cid=7457814}}</ref>

For example: Consider first a classical computer that operates on a three-bit [[Processor register|register]]. The state of the computer at any time is a probability distribution over the <math>2^3=8</math> different three-bit strings <tt>000, 001, 010, 011, 100, 101, 110, 111</tt>. If it is a deterministic computer, then it is in exactly one of these states with probability 1. However, if it is a [[Probabilistic automaton|probabilistic computer]], then there is a possibility of it being in any ''one'' of a number of different states. We can describe this probabilistic state by eight nonnegative numbers ''A'',''B'',''C'',''D'',''E'',''F'',''G'',''H'' (where ''A'' = is the probability that the computer is in state <tt>000</tt>, ''B'' = is the probability that the computer is in state <tt>001</tt>, etc.). There is a restriction that these probabilities sum to 1.

These algorithms did not solve practical problems, but demonstrated mathematically that one could gain more information by querying a [[black box]] with a quantum state in [[quantum superposition|superposition]], sometimes referred to as ''quantum parallelism''.{{sfn|Nielsen|Chuang|2010|p=30-32}}

[[File:Peter Shor 2017 Dirac Medal Award Ceremony.png|thumb|[[Peter Shor]] (pictured here in 2017) showed in 1994 that a scalable quantum computer would be able to break [[RSA encryption]].|upright=0.9]]

The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector (''a'',''b'',''c'',''d'',''e'',''f'',''g'',''h''), called a [[Bra-ket notation|ket]]. Here, however, the coefficients can have [[complex number|complex values]], and it is the sum of the ''squares'' of the coefficients' [[Absolute value#Complex numbers|magnitudes]], <math>|a|^2+|b|^2+\cdots+|h|^2</math>, that must equal 1. These squared magnitudes represent the probability of each of the given states. However, because a complex number encodes not just a magnitude but also a direction in the [[complex plane]], the phase difference between any two coefficients (states) represents a meaningful parameter. This is a fundamental difference between quantum computing and probabilistic classical computing.<ref name="DiVincenzo 1995">{{cite journal |last=DiVincenzo |first=David P. |title=Quantum Computation |journal=Science |year=1995 |volume=270 |issue=5234 |pages=255–261 |doi= 10.1126/science.270.5234.255 |bibcode = 1995Sci...270..255D }} {{subscription required}}</ref>

[[Peter Shor]] built on these results with [[Shor's algorithm|his 1994 algorithm]] for breaking the widely used [[RSA (cryptosystem)|RSA]] and [[Diffie{{en dash}}Hellman]] encryption protocols,{{sfn|Shor|1994}} which drew significant attention to the field of quantum computing. In 1996, [[Grover's algorithm]] established a quantum speedup for the widely applicable [[unstructured data|unstructured]] search problem.<ref>{{Cite conference |last=Grover |first=Lov K. |date=1996 |title=A fast quantum mechanical algorithm for database search |conference=ACM symposium on Theory of computing |language=en |___location=[[Philadelphia]] |publisher=ACM Press |pages=212–219 |doi=10.1145/237814.237866 |isbn=978-0-89791-785-8 |arxiv=quant-ph/9605043}}</ref>{{sfn|Nielsen|Chuang|2010 |p=7}} The same year, [[Seth Lloyd]] proved that quantum computers could simulate quantum systems without the exponential overhead present in classical simulations,<ref name="273.5278.1073">{{Cite journal |last=Lloyd |first=Seth |date=1996-08-23 |title=Universal Quantum Simulators |journal=Science |volume=273 |issue=5278 |pages=1073–1078 |pmid=8688088 |s2cid=43496899 |bibcode=1996Sci...273.1073L |doi=10.1126/science.273.5278.1073 |issn=0036-8075}}</ref> validating Feynman's 1982 conjecture.<ref>{{Cite journal |last1=Cao |first1=Yudong |last2=Romero |first2=Jonathan |last3=Olson |first3=Jonathan P. |last4=Degroote |first4=Matthias |last5=Johnson |first5=Peter D. |last6=Kieferová |first6=Mária |last7=Kivlichan |first7=Ian D. |last8=Menke |first8=Tim |last9=Peropadre |first9=Borja |last10=Sawaya |first10=Nicolas P. D. |last11=Sim |first11=Sukin |last12=Veis |first12=Libor |last13=Aspuru-Guzik |first13=Alán |date=2019-10-09 |title=Quantum Chemistry in the Age of Quantum Computing |journal=Chemical Reviews |language=en-US |volume=119 |issue=19 |pages=10856–10915 |arxiv=1812.09976 |doi=10.1021/acs.chemrev.8b00803 |issn=0009-2665 |pmid=31469277 |s2cid=119417908 |display-authors=5}}</ref>

Over the years, [[experimental physics|experimentalists]] have constructed small-scale quantum computers using [[trapped ion quantum computer|trapped ions]] and superconductors.{{sfn|Grumbling|Horowitz|2019|pp=164-169}}

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 <tt>000</tt> = <math>|a|^2</math>, the probability of measuring <tt>001</tt> = <math>|b|^2</math>, etc..). Thus, measuring a quantum state described by complex coefficients (''a'',''b'',...,''h'') gives the classical probability distribution <math>(|a|^2, |b|^2, \ldots, |h|^2)</math> and we say that the quantum state "collapses" to a classical state as a result of making the measurement.

In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology,<ref>{{cite journal |last1=Chuang |first1=Isaac L. |last2=Gershenfeld |first2=Neil |last3=Kubinec |first3=Markdoi |date=April 1998 |title=Experimental Implementation of Fast Quantum Searching |journal=Physical Review Letters |publisher=[[American Physical Society]] |volume=80 |issue=15 |pages=3408–3411 |bibcode=1998PhRvL..80.3408C |doi=10.1103/PhysRevLett.80.3408}}</ref><ref>{{cite news |last=Holton |first=William Coffeen |title=quantum computer |url=https://www.britannica.com/technology/quantum-computer |access-date=4 Dec 2021 |newspaper=Encyclopedia Britannica |publisher=[[Encyclopædia Britannica]]}}</ref> and subsequent experiments have increased the number of qubits and reduced error rates.{{sfn|Grumbling|Horowitz|2019 |pp=164-169}}

In 2019, [[Google AI]] and [[NASA]] announced that they had achieved [[quantum supremacy]] with a 54-qubit machine, performing a computation that is impossible for any classical computer.<ref>{{Cite journal |last=Gibney |first=Elizabeth |date=2019-10-23 |title=Hello quantum world! Google publishes landmark quantum supremacy claim |journal=Nature |language=en |volume=574 |issue=7779 |pages=461–462 |doi=10.1038/d41586-019-03213-z |pmid=31645740 |bibcode=2019Natur.574..461G |doi-access=free}}</ref><ref name="1910.11333">Lay summary: {{cite journal |url=https://ai.googleblog.com/2019/10/quantum-supremacy-using-programmable.html |title=Quantum Supremacy Using a Programmable Superconducting Processor |journal=Nature |publisher=[[Google AI]] |first1=John |last1=Martinis |first2=Sergio |last2=Boixo |date=October 23, 2019 |volume=574 |issue=7779 |pages=505–510 |doi=10.1038/s41586-019-1666-5 |pmid=31645734 |arxiv=1910.11333 |bibcode=2019Natur.574..505A |s2cid=204836822 |access-date=2022-04-27}}<br />{{*}}Journal article: {{cite journal |last1=Arute |first1=Frank |last2=Arya |first2=Kunal |last3=Babbush |first3=Ryan |last4=Bacon |first4=Dave |last5=Bardin |first5=Joseph C. |last6=Barends |first6=Rami |last7=Biswas |first7=Rupak |last8=Boixo |first8=Sergio |last9=Brandao |first9=Fernando G. S. L. |last10=Buell |first10=David A. |last11=Burkett |first11=Brian |first15=Roberto |first57=Murphy Yuezhen |last64=Rubin |first63=Pedram |last63=Roushan |first62=Eleanor G. |last62=Rieffel |first61=Chris |last61=Quintana |first60=John C. |last60=Platt |first59=Andre |last59=Petukhov |first58=Eric |last58=Ostby |last57=Niu |last65=Sank |first56=Charles |last56=Neill |first55=Matthew |last55=Neeley |first54=Ofer |last54=Naaman |first53=Josh |last53=Mutus |first52=Masoud |last52=Mohseni |first51=Kristel |last51=Michielsen |first50=Xiao |last50=Mi |first64=Nicholas C. |first65=Daniel |last49=Megrant |last74=Yeh |last12=Chen |first12=Yu |last13=Chen |first13=Zijun |last14=Chiaro |first14=Ben |first77=John M. |last77=Martinis |first76=Hartmut |last76=Neven |first75=Adam |last75=Zalcman |first74=Ping |first73=Z. Jamie |last66=Satzinger |last73=Yao |first72=Theodore |last72=White |first71=Benjamin |last71=Villalonga |first70=Amit |last70=Vainsencher |first69=Matthew D. |last69=Trevithick |first68=Kevin J. |last68=Sung |first67=Vadim |last67=Smelyanskiy |first66=Kevin J. |first49=Anthony |first48=Matthew |last16=Courtney |last24=Guerin |first30=Trent |last30=Huang |first29=Markus |last29=Hoffman |first28=Alan |last28=Ho |first27=Michael J. |last27=Hartmann |first26=Matthew P. |last26=Harrigan |first25=Steve |last25=Habegger |first24=Keith |first23=Rob |first31=Travis S. |last23=Graff |first22=Marissa |last22=Giustina |first21=Craig |last21=Gidney |first20=Austin |last20=Fowler |first19=Brooks|last19=Foxen |first18=Edward |last18=Farhi |first17=Andrew |last17=Dunsworsth |first16=William |last31=Humble |last32=Isakov |last48=McEwen |first40=Alexander |first47=Jarrod R. |last47=McClean |first46=Salvatore |last46=Mandrà |first45=Dmitry |last45=Lyakh |first44=Erik |last44=Lucero |first43=Mike |last43=Lindmark |first42=David |last42=Landhuis |first41=Fedor |last15=Collins |last40=Korotov |first32=Sergei V. |first39=Sergey |last39=Knysh |first38=Paul V. |last38=Klimov |first37=Julian |last37=Kelly |first36=Kostyantyn |last36=Kechedzhi |first35=Dvir |last35=Kafri |first34=Zhang |last34=Jiang |first33=Evan |last33=Jeffery |last41=Kostritsa |display-authors=5 |title=Quantum supremacy using a programmable superconducting processor |journal=Nature |date=October 23, 2019 |volume=574 |issue=7779 |pages=505–510 |doi=10.1038/s41586-019-1666-5 |pmid=31645734 |arxiv=1910.11333 |bibcode=2019Natur.574..505A |s2cid=204836822}}</ref><ref>{{Cite news |last=Aaronson |first=Scott |date=2019-10-30 |title=Opinion {{!}} Why Google's Quantum Supremacy Milestone Matters |url=https://www.nytimes.com/2019/10/30/opinion/google-quantum-computer-sycamore.html |access-date=2021-09-25 |work=The New York Times |issn=0362-4331}}</ref><ref>{{cite arXiv |last1=Pan |first1=Feng |last2=Zhang |first2=Pan |date=2021-03-04 |title=Simulating the Sycamore quantum supremacy circuits |class=quant-ph |eprint=2103.03074}}</ref>

Note that an eight-dimensional vector can be specified in many different ways depending on what [[Basis (linear algebra)|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 vector|unit-length]], [[orthogonal]] vectors and the eigenvectors of the [[Pauli matrices|Pauli-x operator]]. [[Bra-ket notation|Ket notation]] is often used to make the choice of basis explicit. For example, the state (''a'',''b'',''c'',''d'',''e'',''f'',''g'',''h'') in the computational basis can be written as:

:<math>a\,|000\rangle + b\,|001\rangle + c\,|010\rangle + d\,|011\rangle + e\,|100\rangle + f\,|101\rangle + g\,|110\rangle + h\,|111\rangle</math>

:where, e.g., <math>|010\rangle = \left(0,0,1,0,0,0,0,0\right)</math>

This announcement was met with a rebuttal from Google's direct competitor, IBM. IBM contended that the calculation Google claimed would take 10,000 years could be performed in just 2.5 days on its own Summit supercomputer if its architecture were optimized, sparking a debate over the precise threshold for "quantum supremacy".<ref>{{Cite news |last=Sample |first=Ian |last2=editor |first2=Ian Sample Science |date=2019-10-23 |title=Google claims it has achieved 'quantum supremacy' – but IBM disagrees |url=https://www.theguardian.com/technology/2019/oct/23/google-claims-it-has-achieved-quantum-supremacy-but-ibm-disagrees |access-date=2025-08-01 |work=The Guardian |language=en-GB |issn=0261-3077}}</ref>

The computational basis for a single qubit (two dimensions) is <math>|0\rangle = \left(1,0\right)</math> and <math>|1\rangle = \left(0,1\right)</math>.

== Quantum information processing ==

Using the eigenvectors of the Pauli-x operator, a single qubit is <math>|+\rangle = \tfrac{1}{\sqrt{2}} \left(1,1\right)</math> and <math>|-\rangle = \tfrac{1}{\sqrt{2}} \left(1,-1\right)</math>.

{{See also|Introduction to quantum mechanics}}

[[Computer engineer]]s typically describe a [[modern computer]]'s operation in terms of [[classical electrodynamics]].

== Operation ==

Within these "classical" computers, some components (such as [[semiconductors]] and [[random number generators]]) may rely on quantum behavior, but these components are not [[isolated system|isolated]] from their environment, so any [[quantum information]] quickly [[quantum decoherence|decoheres]].

{{Unsolved|physics|Is a [[universal quantum computer]] sufficient to [[Algorithmic efficiency|efficiently]] [[Dynamical simulation|simulate]] an arbitrary physical system?}}

While [[programmers]] may depend on [[probability theory]] when designing a [[randomized algorithm]], quantum mechanical notions like superposition and [[quantum interference|interference]] are largely irrelevant for [[program analysis]].

While a classical three-bit state and a quantum three-qubit state are both eight-dimensional [[coordinate vector|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, <math>|000\rangle</math>, 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 matrix|stochastic matrices]], which preserve that the probabilities add up to one (i.e., preserve the [[Taxicab geometry|L1 norm]]). In quantum computation, on the other hand, allowed operations are [[unitary matrix|unitary matrices]], which are effectively rotations (they preserve that the sum of the squares add up to one, the [[Euclidean metric|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 computing|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.)

[[Quantum program]]s, in contrast, rely on precise control of [[quantum coherence|coherent]] quantum systems. Physicists [[mathematical formulation of quantum mechanics|describe these systems mathematically]] using [[linear algebra]]. [[Complex number]]s model [[probability amplitude]]s, [[vector (mathematics and physics)|vectors]] model [[quantum state]]s, and [[matrix (mathematics)|matrices]] model the operations that can be performed on these states. Programming a quantum computer is then a matter of [[function composition (computer science)|composing]] operations in such a way that the resulting program computes a useful result in theory and is implementable in practice.

Finally, upon termination of the algorithm, the result needs to be read off. In the case of a classical computer, we ''sample'' from the [[probability distribution]] on the three-bit register to obtain one definite three-bit string, say 000. Quantum mechanically, we ''[[quantum measurement|measure]]'' the three-qubit state, which is equivalent to collapsing the quantum state down to a classical distribution (with the coefficients in the classical state being the squared magnitudes of the coefficients for the quantum state, as described above), followed by sampling from that distribution. Note that this destroys the original quantum state. Many algorithms will only give the correct answer with a certain probability. However, by repeatedly initializing, running and measuring the quantum computer's results, the probability of getting the correct answer can be increased.

As physicist [[Charles H. Bennett (physicist)|Charlie Bennett]] describes the relationship between quantum and classical computers,<ref>{{Cite AV media |url=https://www.youtube.com/live/rslt-LwtDK4&t=4102 |title=Information Is Quantum: How Physics Helped Explain the Nature of Information and What Can Be Done With It |date=2020-07-31 |last=Bennett |first=Charlie |type=Videotape |author-link=Charles H. Bennett (physicist) |time=1:08:22 |via=YouTube}}</ref>

For more details on the sequences of operations used for various [[quantum algorithm]]s, see [[universal quantum computer]], [[Shor's algorithm]], [[Grover's algorithm]], [[Deutsch–Jozsa algorithm]], [[amplitude amplification]], [[quantum Fourier transform]], [[quantum gate]], [[Adiabatic quantum computation|quantum adiabatic algorithm]] and [[quantum error correction]].

{{Blockquote|text=A classical computer is a quantum computer&nbsp;... so we shouldn't be asking about "where do quantum speedups come from?" We should say, "well, all computers are quantum.&nbsp;... Where do classical slowdowns come from?"}}

=== PotentialQuantum information ===

Just as the bit is the basic concept of classical information theory, the ''[[qubit]]'' is the fundamental unit of [[quantum information]]. The same term ''qubit'' is used to refer to an abstract mathematical model and to any physical system that is represented by that model. A classical bit, by definition, exists in either of two physical states, which can be denoted 0 and 1. A qubit is also described by a state, and two states often written <math>|0\rangle</math> and <math>|1\rangle</math> serve as the quantum counterparts of the classical states 0 and 1. However, the quantum states <math>|0\rangle</math> and <math>|1\rangle</math> belong to a [[vector space]], meaning that they can be multiplied by constants and added together, and the result is again a valid quantum state. Such a combination is known as a ''superposition'' of <math>|0\rangle</math> and <math>|1\rangle</math>.{{sfn|Nielsen|Chuang|2010|page=13}}{{sfn|Mermin|2007|p=17}}

[[Integer factorization]] is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few [[prime number]]s (e.g., products of two 300-digit primes).<ref>{{cite journal |last=Lenstra |first=Arjen K. |url=http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf |title=Integer Factoring |journal= Designs, Codes and Cryptography |volume= 19 |pages= 101–128 |year=2000 |doi=10.1023/A:1008397921377 |issue=2/3}}</ref> By comparison, a quantum computer could efficiently solve this problem using [[Shor's algorithm]] to find its factors. This ability would allow a quantum computer to decrypt many of the [[cryptography|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 [[Asymmetric Algorithms|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 (algorithm)|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.

A two-dimensional [[vector (mathematics and physics)|vector]] mathematically represents a qubit state. Physicists typically use [[Dirac notation]] for quantum mechanical [[linear algebra]], writing <math>|\psi\rangle</math> {{gloss|ket [[psi (Greek)|psi]]}} for a vector labeled <math>\psi</math> . Because a qubit is a two-state system, any qubit state takes the form <math>\alpha|0\rangle+\beta|1\rangle</math> , where <math>|0\rangle</math> and <math>|1\rangle</math> are the standard ''basis states'',{{efn|The [[standard basis]] is also the ''computational basis''.{{sfn|Mermin|2007|p=18}}}} and <math>\alpha</math> and <math>\beta</math> are the ''[[probability amplitude]]s,'' which are in general [[complex numbers]].{{sfn|Mermin|2007|p=17}} If either <math>\alpha</math> or <math>\beta</math> is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a [[quantum state vector]] acts similarly to a (classical) [[probability vector]], with one key difference: unlike probabilities, probability {{em|amplitudes}} are not necessarily positive numbers.{{sfn|Aaronson|2013|page=110}} Negative amplitudes allow for destructive wave interference.

However, other cryptographic algorithms do not appear to be broken by those algorithms.<ref name="pqcrypto_survey">Daniel J. Bernstein, [http://pqcrypto.org/www.springer.com/cda/content/document/cda_downloaddocument/9783540887010-c1.pdf ''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</ref><ref>See also [http://pqcrypto.org/ pqcrypto.org], a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing.</ref> 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]].<ref name="pqcrypto_survey" /><ref>Robert J. McEliece. "[http://ipnpr.jpl.nasa.gov/progress_report2/42-44/44N.PDF A public-key cryptosystem based on algebraic coding theory]." Jet Propulsion Laboratory DSN Progress Report 42–44, 114–116.</ref> [[Lattice based cryptography|Lattice-based cryptosystems]] are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the [[dihedral group|dihedral]] [[hidden subgroup problem]], which would break many lattice based cryptosystems, is a well-studied open problem.<ref>{{cite journal | last1=Kobayashi |first1=H. |last2=Gall |first2=F.L. |title=Dihedral Hidden Subgroup Problem: A Survey |year=2006 |journal=Information and Media Technologies |volume=1 |issue=1 |pages=178–185 |url=http://www.jstage.jst.go.jp/article/imt/1/1/1_178/_article}}</ref> It has been proven that applying Grover's algorithm to break a [[Symmetric cryptography|symmetric (secret key) algorithm]] by brute force requires time equal to roughly 2^n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2ⁿ in the classical case,<ref name=bennett_1997>Bennett C.H., Bernstein E., Brassard G., Vazirani U., "[http://www.cs.berkeley.edu/~vazirani/pubs/bbbv.ps The strengths and weaknesses of quantum computation]". ''[[SIAM Journal on Computing]]'' 26(5): 1510–1523 (1997).</ref> 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#Effect of quantum computing attacks on key strength|Key size]]). [[Quantum cryptography]] could potentially fulfill some of the functions of public key cryptography.

When a qubit is [[quantum measurement|measured]] in the [[standard basis]], the result is a classical bit. The [[Born rule]] describes the [[norm-squared]] correspondence between amplitudes and probabilities{{mdash}}when measuring a qubit <math>\alpha|0\rangle+\beta|1\rangle</math>, the state [[wave function collapse|collapses]] to <math>|0\rangle</math> with probability <math>|\alpha|^2</math>, or to <math>|1\rangle</math> with probability <math>|\beta|^2</math>.

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,<ref>[http://math.nist.gov/quantum/zoo/ Quantum Algorithm Zoo] – Stephen Jordan's Homepage</ref> including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of [[Jones polynomial]]s, 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.

Any valid qubit state has coefficients <math>\alpha</math> and <math>\beta</math> such that <math>|\alpha|^2+|\beta|^2 = 1</math>.

As an example, measuring the qubit <math>1/\sqrt {2}|0\rangle+1/\sqrt{2}|1\rangle</math> would produce either <math>|0\rangle</math> or <math>|1\rangle</math> with equal probability.

Each additional qubit doubles the [[dimension (vector space)|dimension]] of the [[state space (physics)|state space]].{{sfn|Mermin|2007|p=18}}

Consider a problem that has these four properties:

As an example, the vector {{nowrap|{{sfrac|1|√2}}{{ket|00}} + {{sfrac|1|√2}}{{ket|01}}}} represents a two-qubit state, a [[tensor product]] of the qubit {{ket|0}} with the qubit {{nowrap|{{sfrac|1|√2}}{{ket|0}} + {{sfrac|1|√2}}{{ket|1}}}}.

#The only way to solve it is to guess answers repeatedly and check them,

This vector inhabits a four-dimensional [[vector space]] spanned by the basis vectors {{ket|00}}, {{ket|01}}, {{ket|10}}, and {{ket|11}}.

#The number of possible answers to check is the same as the number of inputs,

The [[Bell state]] {{nowrap|{{sfrac|1|√2}}{{ket|00}} + {{sfrac|1|√2}}{{ket|11}}}} is impossible to decompose into the tensor product of two individual qubits{{mdash}}the two qubits are ''[[quantum entanglement|entangled]]'' because neither qubit has a state vector of its own.

#Every possible answer takes the same amount of time to check, and

In general, the vector space for an ''n''-qubit system is 2^''n''-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2¹⁰⁰ classical values.

#There are no clues about which answers might be better: generating possibilities randomly is just as good as checking them in some special order.

An example of this is a [[Password cracking|password cracker]] that attempts to guess the password for an [[encryption|encrypted]] file (assuming that the password has a maximum possible length).

=== Unitary operators<span class="anchor" id="gate-application"></span> ===

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 cipher]]s such as [[Triple DES]] and [[Advanced Encryption Standard|AES]] by attempting to guess the secret key.<ref>{{cite news |url=http://www.washingtonpost.com/world/national-security/nsa-seeks-to-build-quantum-computer-that-could-crack-most-types-of-encryption/2014/01/02/8fff297e-7195-11e3-8def-a33011492df2_story.html?hpid=z1 |title=NSA seeks to build quantum computer that could crack most types of encryption |first1=Steven |last1=Rich |first2=Barton |last2=Gellman |date=2014-02-01 |newspaper=Washington Post}}</ref>

{{See also|Unitarity (physics)}}

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

[[Grover's algorithm]] can also be used to obtain a quadratic speed-up over a brute-force search for a class of problems known as [[NP-complete]].

<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

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

: <math>X|0\rangle = |1\rangle</math> and <math>X|1\rangle = |0\rangle</math>.

The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while 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

<math display="block">

|00\rangle := \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix};\quad

|01\rangle := \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix};\quad

|10\rangle := \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix};\quad

|11\rangle := \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.

</math>

The [[Controlled NOT gate|controlled NOT (CNOT)]] gate can then be represented using the following matrix:

<math display="block">

\operatorname{CNOT} :=

\begin{pmatrix}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 0 & 1 \\

0 & 0 & 1 & 0

\end{pmatrix}.

</math>

As a mathematical consequence of this definition, <math display="inline">\operatorname{CNOT}|00\rangle = |00\rangle</math>, <math display="inline">\operatorname{CNOT}|01\rangle = |01\rangle</math>, <math display="inline">\operatorname{CNOT}|10\rangle = |11\rangle</math>, and <math display="inline">\operatorname{CNOT}|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

In summary, quantum computation can be described as a network of quantum logic gates and measurements. However, any [[deferred measurement principle|measurement can be deferred]] to the end of quantum computation, though this deferment may come at a computational cost, so most [[quantum circuit]]s depict a network consisting only of quantum logic gates and no measurements.

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe [[Universal quantum simulator|quantum simulation]] will be one of the most important applications of quantum computing.<ref>{{cite web |url=http://www.wired.com/science/discoveries/news/2007/02/72734 |title=The Father of Quantum Computing |first=Quinn |last=Norton |date=2007-02-15 |publisher=Wired.com}}</ref> Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a [[collider]].<ref>{{cite web |url=http://www.ias.edu/ias-letter/ambainis-quantum-computing |title=What Can We Do with a Quantum Computer? |first=Andris |last=Ambainis |date=Spring 2014 |publisher=Institute for Advanced Study}}</ref>

=== Quantum parallelism ===

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:<ref>{{cite arXiv| eprint=quant-ph/0002077|title=The Physical Implementation of Quantum Computation|last=DiVincenzo |first=David P.|date=2000-04-13| class=quant-ph}}</ref>

*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 parallelism'' is the heuristic that quantum computers can be thought of as evaluating a function for multiple input values simultaneously. This can be achieved by preparing a quantum system in a superposition of input states and applying a unitary transformation that encodes the function to be evaluated. The resulting state encodes the function's output values for all input values in the superposition, allowing for the computation of multiple outputs simultaneously. This property is key to the speedup of many quantum algorithms. However, "parallelism" in this sense is insufficient to speed up a computation, because the measurement at the end of the computation gives only one value. To be useful, a quantum algorithm must also incorporate some other conceptual ingredient.{{sfn|Nielsen|Chuang|2010|p=30–32}}{{sfn|Mermin|2007|pp=38–39}}

=== 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 nuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems, in particular the transverse relaxation time ''T''₂ (for [[Nuclear magnetic resonance|NMR]] and [[MRI]] technology, also called the ''dephasing time''), typically range between nanoseconds and seconds at low temperature.<ref name="DiVincenzo 1995" /> Currently, some quantum computers require their qubits to be cooled to 20 millikelvin in order to prevent significant decoherence.<ref>{{cite journal|last1=Jones|first1=Nicola|title=Computing: The quantum company|journal=Nature International Weekly Journal of Science|date=19 June 2013|volume=498|issue=7454|pages=286-288|doi=10.1038/498286a}}</ref>

=== Quantum programming<span class="anchor" id="Models of computation for quantum computing"></span> ===

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.

{{Further|Quantum programming}}

There are a number of [[model of computation|models of computation]] for quantum computing, distinguished by the basic elements in which the computation is decomposed.

If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time. An often cited figure for required error rate in each gate is 10⁻⁴. This implies that each gate must be able to perform its task in one 10,000th of the decoherence time of the system.

==== Gate array {{anchor|Quantum circuit|Definition}} ====

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 bits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of ''L''. For a 1000-bit number, this implies a need for about 10⁴ qubits without error correction.<ref>{{cite journal |title=Is Fault-Tolerant Quantum Computation Really Possible? |last=Dyakonov |first=M. I. |date=2006-10-14 |pages=4–18 |journal=In: Future Trends in Microelectronics. Up the Nano Creek, S. Luryi, J. Xu, and A. Zaslavsky (eds), Wiley, pp. | arxiv=quant-ph/0610117}}</ref> With error correction, the figure would rise to about 10⁷ qubits. Note that computation time is about ''L''² or about 10⁷ steps and on 1 M[[Hertz|Hz]], about 10 seconds.

[[File:Quantum Toffoli Gate Implementation.svg|thumb|A quantum circuit diagram implementing a [[Toffoli gate]] from [[Quantum logic gate|more primitive gates]]|upright=1.15]]

A [[quantum circuit|quantum gate array]] decomposes computation into a sequence of few-qubit [[quantum gate]]s. A quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.

A very different approach to the stability-decoherence problem is to create a [[topological quantum computer]] with [[anyon]]s, quasi-particles used as threads and relying on [[braid theory]] to form stable logic gates.<ref>{{cite journal

Any quantum computation (which is, in the above formalism, any [[unitary matrix]] of size <math>2^n \times 2^n</math> over <math>n</math> qubits) 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 [[Quantum logic gate#Universal quantum gates|universal gate set]], since a computer that can run such circuits is a [[universal quantum computer]]. 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|Solovay-Kitaev theorem]]. Implementation of Boolean functions using the few-qubit quantum gates is presented here.<ref>{{Cite book |last1=Kurgalin |first1=Sergei |title=Concise guide to quantum computing: algorithms, exercises, and implementations |last2=Borzunov |first2=Sergei |date=2021 |publisher=Springer |isbn=978-3-030-65054-4 |series=Texts in computer science |___location=Cham}}</ref>

==== Measurement-based quantum computing ====

A [[measurement-based quantum computer]] decomposes computation into a sequence of [[Bell state#Bell state measurement|Bell state measurements]] and single-qubit [[quantum gate]]s applied to a highly entangled initial state (a [[cluster state]]), using a technique called [[quantum gate teleportation]].

==== Adiabatic quantum computing ====

An [[Adiabatic quantum computation|adiabatic quantum computer]], based on [[quantum annealing]], decomposes computation into a slow continuous transformation of an initial [[Hamiltonian (quantum mechanics)|Hamiltonian]] into a final Hamiltonian, whose ground states contain the solution.<ref name="Das 2008 1061–1081">{{cite journal|last1=Das|first1=A.|last2=Chakrabarti|first2=B. K.|year=2008|title=Quantum Annealing and Analog Quantum Computation|journal=[[Reviews of Modern Physics|Rev. Mod. Phys.]]|volume=80|issue=3|pages=1061–1081|arxiv=0801.2193|bibcode=2008RvMP...80.1061D|citeseerx=10.1.1.563.9990|doi=10.1103/RevModPhys.80.1061|s2cid=14255125}}</ref>

==== Neuromorphic quantum computing ====

Neuromorphic quantum computing (abbreviated as 'n.quantum computing') is an unconventional type of computing that uses [[Neuromorphic engineering|neuromorphic computing]] to perform quantum operations. It was suggested that quantum algorithms, which are algorithms that run on a realistic model of quantum computation, can be computed equally efficiently with neuromorphic quantum computing. Both traditional quantum computing and neuromorphic quantum computing are physics-based unconventional computing approaches to computations and do not follow the [[von Neumann architecture]]. They both construct a system (a circuit) that represents the physical problem at hand and then leverage their respective physics properties of the system to seek the "minimum". Neuromorphic quantum computing and quantum computing share similar physical properties during computation.

==== Topological quantum computing ====

A [[topological quantum computer]] decomposes computation into the braiding of [[anyon]]s in a 2D lattice.<ref name="Nayaketal2008">{{cite journal |last1=Nayak |first1=Chetan |last2=Simon |first2=Steven |last3=Stern |first3=Ady |last4=Das Sarma |first4=Sankar |year=2008 |title=Nonabelian Anyons and Quantum Computation |journal=Reviews of Modern Physics |volume=80 |issue=3 |pages=1083–1159 |arxiv=0707.1889 |bibcode=2008RvMP...80.1083N |doi=10.1103/RevModPhys.80.1083 |s2cid=119628297}}</ref>

==== Quantum Turing machine ====

A [[quantum Turing machine]] is the quantum analog of a [[Turing machine]].<ref name="The computer as a physical system"/> All of these models of computation—quantum circuits,<ref>{{Cite book|last=Chi-Chih Yao|first=A.|title=Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science |chapter=Quantum circuit complexity |year=1993|pages=352–361|doi=10.1109/SFCS.1993.366852|isbn=0-8186-4370-6|s2cid=195866146}}</ref> [[One-way quantum computer|one-way quantum computation]],<ref>{{Cite journal |last1=Raussendorf |first1=Robert |last2=Browne |first2=Daniel E. |last3=Briegel |first3=Hans J. |date=2003-08-25 |title=Measurement-based quantum computation on cluster states |journal=Physical Review A |volume=68 |issue=2 |pages=022312 |doi=10.1103/PhysRevA.68.022312 |arxiv=quant-ph/0301052 |bibcode=2003PhRvA..68b2312R |s2cid=6197709}}</ref> adiabatic quantum computation,<ref>{{Cite journal |last1=Aharonov |first1=Dorit |last2=van Dam |first2=Wim |last3=Kempe |first3=Julia |last4=Landau |first4=Zeph |last5=Lloyd |first5=Seth |last6=Regev |first6=Oded |date=2008-01-01 |title=Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation |journal=SIAM Review |volume=50 |issue=4 |pages=755–787 |doi=10.1137/080734479 |arxiv=quant-ph/0405098 |bibcode=2008SIAMR..50..755A |s2cid=1503123 |issn=0036-1445}}</ref> and topological quantum computation<ref name="FLW02">{{Cite journal |last1=Freedman |first1=Michael H. |last2=Larsen |first2=Michael |last3=Wang |first3=Zhenghan |date=2002-06-01 |title=A Modular Functor Which is Universal for Quantum Computation |journal=Communications in Mathematical Physics |volume=227 |issue=3 |pages=605–622 |doi=10.1007/s002200200645 |issn=0010-3616 |arxiv=quant-ph/0001108 |bibcode=2002CMaPh.227..605F |s2cid=8990600}}</ref>—have been shown to be equivalent to the quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical.

====Noisy intermediate-scale quantum computing====

The [[threshold theorem]] shows how increasing the number of qubits can mitigate errors,{{sfn|Nielsen|Chuang|2010 |p=481}} yet fully fault-tolerant quantum computing remains "a rather distant dream".<ref name="preskill2018"/> According to some researchers, ''noisy intermediate-scale quantum'' ([[NISQ]]) machines may have specialized uses in the near future, but [[noise (signal processing)|noise]] in quantum gates limits their reliability.<ref name="preskill2018">{{Cite journal |last=Preskill |first=John |date=6 August 2018 |title=Quantum Computing in the NISQ era and beyond |journal=Quantum |volume=2 |article-number=79 |arxiv=1801.00862 |doi=10.22331/q-2018-08-06-79 |doi-access=free |bibcode=2018Quant...2...79P |s2cid=44098998}}</ref>

Scientists at [[Harvard University|Harvard]] University successfully created "quantum circuits" that correct errors more efficiently than alternative methods, which may potentially remove a major obstacle to practical quantum computers.<ref>{{Cite journal |last1=Bluvstein |first1=Dolev |last2=Evered |first2=Simon J. |last3=Geim |first3=Alexandra A. |last4=Li |first4=Sophie H. |last5=Zhou |first5=Hengyun |last6=Manovitz |first6=Tom |last7=Ebadi |first7=Sepehr |last8=Cain |first8=Madelyn |last9=Kalinowski |first9=Marcin |last10=Hangleiter |first10=Dominik |last11=Ataides |first11=J. Pablo Bonilla |last12=Maskara |first12=Nishad |last13=Cong |first13=Iris |last14=Gao |first14=Xun |last15=Rodriguez |first15=Pedro Sales |date=2023-12-06 |title=Logical quantum processor based on reconfigurable atom arrays |journal=Nature |volume=626 |issue=7997 |language=en |pages=58–65 |doi=10.1038/s41586-023-06927-3 |pmid=38056497 |pmc=10830422 |issn=1476-4687|arxiv=2312.03982 |s2cid=266052773 }}</ref> The Harvard research team was supported by [[Massachusetts Institute of Technology|MIT]], [[QuEra Computing]], [[California Institute of Technology|Caltech]], and [[Princeton University|Princeton]] University and funded by [[DARPA]]'s Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program.<ref>{{Cite web |date=December 6, 2023 |title=DARPA-Funded Research Leads to Quantum Computing Breakthrough |url=https://www.darpa.mil/news-events/2023-12-06 |access-date=January 5, 2024 |website=darpa.mil}}</ref><ref>{{Cite web |last=Choudhury |first=Rizwan |date=2023-12-30 |title=Top 7 innovation stories of 2023 – Interesting Engineering |url=https://interestingengineering.com/lists/top-7-innovation-stories-of-2023-interesting-engineering |access-date=2024-01-06 |website=interestingengineering.com |language=en-US}}</ref>

==== Quantum cryptography and cybersecurity ====

{{main|Quantum cryptography}}

Digital cryptography allows communications without observation by unauthorized parties. Conventional encryption, the obscuring of a message with a key through an algorithm, relies on the algorithm being difficult to reverse. Encryption is also the basis for digital signatures and authentication mechanisms. Quantum computing may be sufficiently more powerful that difficult reversals are feasible, allowing messages relying on conventional encryption to be read.<ref name=Gisin-2002>{{Cite journal |last=Gisin |first=Nicolas |last2=Ribordy |first2=Grégoire |last3=Tittel |first3=Wolfgang |last4=Zbinden |first4=Hugo |date=2002-03-08 |title=Quantum cryptography |url=https://link.aps.org/doi/10.1103/RevModPhys.74.145 |journal=Reviews of Modern Physics |language=en |volume=74 |issue=1 |pages=145–195 |doi=10.1103/RevModPhys.74.145 |issn=0034-6861}}</ref>

Quantum cryptography replaces conventional algorithms with computations based on quantum computing. In principle, quantum encryption would be impossible to decode even with a quantum computer. This advantage comes at a significant cost in terms of elaborate infrastructure as well as preventing legitimate decoding of messages by governmental security officials.<ref name=Gisin-2002/>

Ongoing research in quantum and [[post-quantum cryptography]] has led to new algorithms for [[quantum key distribution]], initial work on quantum [[random number generation]] and to some early technology demonstrations.<ref name=Pirandola>{{cite journal |last1=Pirandola |first1=S. |last2=Andersen |first2=U. L. |last3=Banchi |first3=L. |last4=Berta |first4=M. |last5=Bunandar |first5=D. |last6=Colbeck |first6=R. |last7=Englund |first7=D. |last8=Gehring |first8=T. |last9=Lupo |first9=C. |last10=Ottaviani |first10=C. |last11=Pereira |first11=J. |last12=Razavi |first12=M. |last13=Shamsul Shaari |first13=J. |last14=Tomamichel |first14=M. |last15=Usenko |first15=V. C. |last16=Vallone |first16=G. |last17=Villoresi |first17=P. |last18=Wallden |first18=P. |year=2020 |title=Advances in quantum cryptography |journal=Advances in Optics and Photonics |volume=12 |issue=4 |doi=10.1364/AOP.361502 |arxiv=1906.01645 |bibcode=2020AdOP...12.1012P}}</ref>{{rp|1012–1036}}

== Communication ==

{{Further|Quantum information science}}

[[Quantum cryptography]] enables new ways to transmit data securely; for example, [[quantum key distribution]] uses entangled quantum states to establish secure [[cryptographic keys]].<ref name=Pirandola/>{{rp|1017}} When a sender and receiver exchange quantum states, they can guarantee that an [[adversary (cryptography)|adversary]] does not intercept the message, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change.<ref>{{Cite journal |last1=Xu |first1=Feihu |last2=Ma |first2=Xiongfeng |last3=Zhang |first3=Qiang |last4=Lo |first4=Hoi-Kwong |last5=Pan |first5=Jian-Wei |date=2020-05-26 |title=Secure quantum key distribution with realistic devices |journal=Reviews of Modern Physics |volume=92 |issue=2 |page=025002{{hyphen}}3 |doi=10.1103/RevModPhys.92.025002|arxiv=1903.09051 |bibcode=2020RvMP...92b5002X |s2cid=210942877 }}</ref> With appropriate [[cryptographic protocols]], the sender and receiver can thus establish shared private information resistant to eavesdropping.<ref name="bb84" /><ref>{{Cite conference |last1=Xu |first1=Guobin |last2=Mao |first2=Jianzhou |last3=Sakk |first3=Eric |last4=Wang |first4=Shuangbao Paul |title=2023 57th Annual Conference on Information Sciences and Systems (CISS) |date=2023-03-22 |chapter=An Overview of Quantum-Safe Approaches: Quantum Key Distribution and Post-Quantum Cryptography |publisher=[[IEEE]] |page=3 |doi=10.1109/CISS56502.2023.10089619 |isbn=978-1-6654-5181-9}}</ref>

Modern [[fiber-optic cables]] can transmit quantum information over relatively short distances. Ongoing experimental research aims to develop more reliable hardware (such as quantum repeaters), hoping to scale this technology to long-distance [[quantum networks]] with end-to-end entanglement. Theoretically, this could enable novel technological applications, such as distributed quantum computing and enhanced [[quantum sensing]].<ref>{{Cite conference |last1=Kozlowski |first1=Wojciech |last2=Wehner |first2=Stephanie |title=Proceedings of the Sixth Annual ACM International Conference on Nanoscale Computing and Communication |date=2019-09-25 |chapter=Towards Large-Scale Quantum Networks |pages=1–7 |language=en |publisher=ACM |doi=10.1145/3345312.3345497 |isbn=978-1-4503-6897-1|arxiv=1909.08396 }}</ref><ref>{{Cite journal |last1=Guo |first1=Xueshi |last2=Breum |first2=Casper R. |last3=Borregaard |first3=Johannes |last4=Izumi |first4=Shuro |last5=Larsen |first5=Mikkel V. |last6=Gehring |first6=Tobias |last7=Christandl |first7=Matthias |last8=Neergaard-Nielsen |first8=Jonas S. |last9=Andersen |first9=Ulrik L. |date=23 December 2019 |title=Distributed quantum sensing in a continuous-variable entangled network |journal=Nature Physics |language=en |volume=16 |issue=3 |pages=281–284 |doi=10.1038/s41567-019-0743-x |arxiv=1905.09408 |s2cid=256703226 |issn=1745-2473}}</ref>

== Algorithms ==

<!-- Overview of quantum algorithms, particularly abstract routines with no explicit application -->

Progress in finding [[quantum algorithms]] typically focuses on this quantum circuit model, though exceptions like the [[Adiabatic quantum computation|quantum adiabatic algorithm]] exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.<ref name="zoo">{{cite web |author=Jordan |first=Stephen |date=14 October 2022 |title=Quantum Algorithm Zoo |url=http://math.nist.gov/quantum/zoo/ |url-status=live |archive-url=https://web.archive.org/web/20180429014516/https://math.nist.gov/quantum/zoo/ |archive-date=29 April 2018 |orig-year=22 April 2011}}</ref>

Quantum algorithms that offer more than a polynomial speedup over the best-known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing [[discrete logarithm]]s, solving [[Pell's equation]], and more generally solving the [[hidden subgroup problem]] for [[Abelian group|abelian]] finite groups.<ref name="zoo"/> These algorithms depend on the primitive of the [[quantum Fourier transform]]. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, but evidence suggests that this is unlikely.<ref>{{Cite conference |last1=Aaronson |first1=Scott |last2=Arkhipov |first2=Alex |date=2011-06-06 |title=The computational complexity of linear optics |book-title=Proceedings of the forty-third annual ACM symposium on Theory of computing |language=en |___location=[[San Jose, California]] |publisher=[[Association for Computing Machinery]] |pages=333–342 |doi=10.1145/1993636.1993682 |isbn=978-1-4503-0691-1 |author-link=Scott Aaronson |arxiv=1011.3245}}</ref> Certain oracle problems like [[Simon's problem]] and the [[Bernstein–Vazirani algorithm|Bernstein–Vazirani problem]] do give provable speedups, though this is in the [[quantum complexity theory#Quantum query complexity|quantum query model]], which is a restricted model where lower bounds are much easier to prove and doesn't necessarily translate to speedups for practical problems.

Other problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain [[Jones polynomial]]s, and the [[quantum algorithm for linear systems of equations]], have quantum algorithms appearing to give super-polynomial speedups and are [[BQP]]-complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that ''no quantum algorithm'' gives a super-polynomial speedup, which is believed to be unlikely.{{sfn|Nielsen|Chuang|2010|p=42}}

Some quantum algorithms, like [[Grover's algorithm]] and [[amplitude amplification]], give polynomial speedups over corresponding classical algorithms.<ref name="zoo"/> Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.{{sfn|Nielsen|Chuang|2010|p=7}}

=== Simulation of quantum systems ===

{{Main|Quantum simulation}}

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, [[Quantum simulator|quantum simulation]] may be an important application of quantum computing.<ref>{{Cite magazine |url=http://archive.wired.com/science/discoveries/news/2007/02/72734 |title=The Father of Quantum Computing |magazine=Wired |first=Quinn |last=Norton |date=15 February 2007 }}</ref> Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a [[collider]].<ref>{{cite web |url=http://www.ias.edu/ias-letter/ambainis-quantum-computing |title=What Can We Do with a Quantum Computer? |first=Andris |last=Ambainis |date=Spring 2014 |publisher=Institute for Advanced Study}}</ref> In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.<ref name="NYT-20230614">{{cite news |last=Chang |first=Kenneth |date=14 June 2023 |title=Quantum Computing Advance Begins New Era, IBM Says – A quantum computer came up with better answers to a physics problem than a conventional supercomputer. |work=[[The New York Times]] |url=https://www.nytimes.com/2023/06/14/science/ibm-quantum-computing.html |url-status=live |accessdate=15 June 2023 |archiveurl=https://archive.today/20230614151835/https://www.nytimes.com/2023/06/14/science/ibm-quantum-computing.html |archivedate=14 June 2023}}</ref><ref name="NAT-20230614">{{cite journal |author=Kim, Youngseok |display-authors=et al. |title=Evidence for the utility of quantum computing before fault tolerance |date=14 June 2023 |journal=[[Nature (journal)|Nature]] |volume=618 |issue=7965 |pages=500–505 |doi=10.1038/s41586-023-06096-3 |pmid=37316724 |pmc=10266970 |bibcode=2023Natur.618..500K }}</ref>

About 2% of the annual global energy output is used for [[nitrogen fixation]] to produce [[ammonia]] for the [[Haber process]] in the agricultural fertilizer industry (even though naturally occurring organisms also produce ammonia). Quantum simulations might be used to understand this process and increase the energy efficiency of production.<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=7susESgnDv8 |archive-url=https://web.archive.org/web/20210215140237/https://www.youtube.com/watch?v=7susESgnDv8 |archive-date=15 February 2021 |url-status=bot: unknown |title=Lunch & Learn: Quantum Computing |publisher=[[Sibos (conference)|Sibos TV]] |via=YouTube |date=21 November 2018 |access-date=4 February 2021 |first=Andrea |last=Morello |author-link=Andrea Morello }}</ref> It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process<ref>{{Cite news |last1=Ruane |first1=Jonathan |last2=McAfee |first2=Andrew |last3=Oliver |first3=William D. |date=2022-01-01 |title=Quantum Computing for Business Leaders |work=Harvard Business Review |url=https://hbr.org/2022/01/quantum-computing-for-business-leaders |access-date=2023-04-12 |issn=0017-8012}}</ref> by the mid-2020s<ref>{{Cite web |last1=Budde |first1=Florian |last2=Volz |first2=Daniel |date=July 12, 2019 |title=Quantum computing and the chemical industry {{!}} McKinsey |url=https://www.mckinsey.com/industries/chemicals/our-insights/the-next-big-thing-quantum-computings-potential-impact-on-chemicals |access-date=2023-04-12 |website=www.mckinsey.com |publisher=McKinsey and Company}}</ref> although some have predicted it will take longer.<ref>{{Cite web |last=Bourzac |first=Katherine |date=October 30, 2017 |title=Chemistry is quantum computing's killer app |url=https://cen.acs.org/articles/95/i43/Chemistry-quantum-computings-killer-app.html |access-date=2023-04-12 |website=cen.acs.org |publisher=American Chemical Society}}</ref>

=== Post-quantum cryptography ===

{{Main|Post-quantum cryptography}}

A notable application of quantum computation is for [[cryptanalysis|attacks]] on cryptographic systems that are currently in use. [[Integer factorization]], which underpins the security of [[public key cryptography|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 number]]s (e.g., products of two 300-digit primes).<ref>{{cite journal |last=Lenstra |first=Arjen K. |url=http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf |title=Integer Factoring |journal=Designs, Codes and Cryptography |volume=19 |pages=101–128 |year=2000 |doi=10.1023/A:1008397921377 |issue=2/3 |s2cid=9816153 |url-status=dead |archive-url=https://web.archive.org/web/20150410234239/http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf |archive-date=10 April 2015 }}</ref> By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors.{{sfn|Nielsen|Chuang|2010|p=216}} This ability would allow a quantum computer to break many of the [[cryptography|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 [[Asymmetric Algorithms|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 (algorithm)|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.

Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of ''post-quantum cryptography''.<ref name="pqcrypto_survey">{{cite book |doi=10.1007/978-3-540-88702-7_1 |chapter=Introduction to post-quantum cryptography |title=Post-Quantum Cryptography |year=2009 |last1=Bernstein |first1=Daniel J. |pages=1–14 |isbn=978-3-540-88701-0 |place=Berlin, Heidelberg |publisher=Springer|s2cid=61401925 }}</ref><ref>See also [http://pqcrypto.org/ pqcrypto.org], a bibliography maintained by Daniel J. Bernstein and [[Tanja Lange]] on cryptography not known to be broken by quantum computing.</ref> 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]].<ref name="pqcrypto_survey" /><ref>{{cite journal |last1=McEliece |first1=R. J. |title=A Public-Key Cryptosystem Based On Algebraic Coding Theory |journal=DSNPR |date=January 1978 |volume=44 |pages=114–116 |url=http://ipnpr.jpl.nasa.gov/progress_report2/42-44/44N.PDF |bibcode=1978DSNPR..44..114M}}</ref> [[Lattice-based cryptography|Lattice-based cryptosystems]] are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the [[dihedral group|dihedral]] [[hidden subgroup problem]], which would break many lattice based cryptosystems, is a well-studied open problem.<ref>{{cite journal |last1=Kobayashi |first1=H. |last2=Gall |first2=F. L. |year=2006 |title=Dihedral Hidden Subgroup Problem: A Survey |journal=Information and Media Technologies |volume=1 |issue=1 |pages=178–185 |doi=10.2197/ipsjdc.1.470 |doi-access=free}}</ref> It has been proven that applying Grover's algorithm to break a [[Symmetric-key algorithm|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,<ref name=bennett_1997>{{cite journal |last1=Bennett |first1=Charles H. |last2=Bernstein |first2=Ethan |last3=Brassard |first3=Gilles |last4=Vazirani |first4=Umesh |title=Strengths and Weaknesses of Quantum Computing |journal=SIAM Journal on Computing |date=October 1997 |volume=26 |issue=5 |pages=1510–1523 |doi=10.1137/s0097539796300933 |arxiv=quant-ph/9701001 |bibcode=1997quant.ph..1001B |s2cid=13403194 }}</ref> 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#Effect of quantum computing attacks on key strength|Key size]]'').

=== Search problems<span class="anchor" id="Quantum search"></span> ===

{{Main|Grover's algorithm}}

The most well-known example of a problem that allows for a polynomial quantum speedup is ''unstructured search'', which involves finding a marked item out of a list of <math>n</math> items in a database. This can be solved by Grover's algorithm using <math>O(\sqrt{n})</math> queries to the database, quadratically fewer than the <math>\Omega(n)</math> queries required for 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. Many examples of provable quantum speedups for query problems are based on Grover's algorithm, including [[BHT algorithm|Brassard, Høyer, and Tapp's algorithm]] for finding collisions in two-to-one functions,<ref>{{cite encyclopedia |year=2016 |title=Quantum Algorithm for the Collision Problem |encyclopedia=Encyclopedia of Algorithms |publisher=Springer |place=New York, New York |editor-last=Kao |editor-first=Ming-Yang |pages=1662–1664 |language=en |arxiv=quant-ph/9705002 |doi=10.1007/978-1-4939-2864-4_304 |isbn=978-1-4939-2864-4 |last2=Høyer |first2=Peter |last3=Tapp |first3=Alain |last1=Brassard |first1=Gilles |s2cid=3116149}}</ref> and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.<ref>{{Cite journal |last1=Farhi |first1=Edward |last2=Goldstone |first2=Jeffrey |last3=Gutmann |first3=Sam |date=23 December 2008 |title=A Quantum Algorithm for the Hamiltonian NAND Tree |journal=Theory of Computing |language=EN |volume=4 |issue=1 |pages=169–190 |doi=10.4086/toc.2008.v004a008 |s2cid=8258191 |issn=1557-2862 |doi-access=free}}</ref>

Problems that can be efficiently addressed with Grover's algorithm have the following properties:<ref>{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |year=2011 |isbn=978-1-84628-887-6 |pages=242–244}}</ref><ref>{{cite arXiv |last=Grover| first=Lov| author-link=Lov Grover |title=A fast quantum mechanical algorithm for database search |date=29 May 1996| eprint=quant-ph/9605043}}</ref>

#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 that 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 scales 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<ref>{{cite journal |last1=Ambainis |first1=Ambainis |title=Quantum search algorithms |journal=ACM SIGACT News |date=June 2004 |volume=35 |issue=2 |pages=22–35 |doi=10.1145/992287.992296 |arxiv=quant-ph/0504012 |bibcode=2005quant.ph..4012A |s2cid=11326499 }}</ref> is a [[Boolean satisfiability problem]], where the ''database'' through which the algorithm iterates is that of all possible answers. An example and possible application of this is a [[Password cracking|password cracker]] that attempts to guess a password. Breaking [[Symmetric-key algorithm|symmetric ciphers]] with this algorithm is of interest to government agencies.<ref>{{cite news |url=https://www.washingtonpost.com/world/national-security/nsa-seeks-to-build-quantum-computer-that-could-crack-most-types-of-encryption/2014/01/02/8fff297e-7195-11e3-8def-a33011492df2_story.html |title=NSA seeks to build quantum computer that could crack most types of encryption |first1=Steven |last1=Rich |first2=Barton |last2=Gellman |date=1 February 2014 |newspaper=The Washington Post}}</ref>

=== Quantum annealing<span class="anchor" id="Quantum annealing and adiabatic optimization"></span> ===

[[Quantum annealing]] relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves 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. Quantum annealing can solve [[Ising model]]s and the (computationally equivalent) [[Quadratic unconstrained binary optimization|QUBO]] problem, which in turn can be used to encode a wide range of [[combinatorial optimization]] problems.<ref>{{Cite journal |last1=Lucas |first1=Andrew |date=2014 |title=Ising formulations of many NP problems |journal=Frontiers in Physics |volume=2 |pages=5|doi=10.3389/fphy.2014.00005 |doi-access=free |arxiv=1302.5843 |bibcode=2014FrP.....2....5L }}</ref> {{anchor|Computational biology}}Adiabatic optimization may be helpful for solving [[computational biology]] problems.<ref>{{cite journal |last1=Outeiral |first1=Carlos| last2=Strahm |first2=Martin |last3=Morris |first3=Garrett |last4=Benjamin |first4=Simon |last5=Deane |first5=Charlotte |last6=Shi |first6=Jiye |title=The prospects of quantum computing in computational molecular biology |journal=WIREs Computational Molecular Science |year=2021|volume=11|doi=10.1002/wcms.1481 |arxiv=2005.12792 |s2cid=218889377 |doi-access=free}}</ref>

=== Machine learning ===

{{Main|Quantum machine learning}}

Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up [[machine learning]] tasks.<ref name="preskill2018"/><ref>{{Cite journal |last1=Biamonte |first1=Jacob |last2=Wittek |first2=Peter |last3=Pancotti |first3=Nicola |last4=Rebentrost |first4=Patrick |last5=Wiebe |first5=Nathan |last6=Lloyd |first6=Seth |date=September 2017 |title=Quantum machine learning |journal=Nature |language=en |volume=549 |issue=7671 |pages=195–202 |doi=10.1038/nature23474 |pmid=28905917 |arxiv=1611.09347 |bibcode=2017Natur.549..195B |s2cid=64536201 |issn=0028-0836}}</ref>

For example, the [[HHL Algorithm]], named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.<ref name="preskill2018"/><ref name="Quantum algorithm for solving linear systems of equations by Harrow et al.">{{Cite journal |arxiv=0811.3171 |last1=Harrow |first1=Aram |last2=Hassidim |first2=Avinatan |last3=Lloyd |first3=Seth |title=Quantum algorithm for solving linear systems of equations |journal=Physical Review Letters |volume=103 |issue=15 |page=150502 |year=2009 |doi=10.1103/PhysRevLett.103.150502 |pmid=19905613 |bibcode=2009PhRvL.103o0502H |s2cid=5187993}}</ref> Some research groups have recently explored the use of quantum annealing hardware for training [[Boltzmann machine]]s and [[deep neural networks]].<ref>{{Cite journal |last1=Benedetti |first1=Marcello |last2=Realpe-Gómez |first2=John |last3=Biswas |first3=Rupak |last4=Perdomo-Ortiz |first4=Alejandro |date=9 August 2016 |title=Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning |journal=Physical Review A |volume=94 |issue=2 |page=022308 |doi=10.1103/PhysRevA.94.022308 |arxiv=1510.07611 |bibcode=2016PhRvA..94b2308B |doi-access=free}}</ref><ref>{{Cite journal |last1=Ajagekar |first1=Akshay |last2=You |first2=Fengqi |date=5 December 2020 |title=Quantum computing assisted deep learning for fault detection and diagnosis in industrial process systems |journal=Computers & Chemical Engineering |language=en |volume=143 |article-number=107119 |arxiv=2003.00264 |s2cid=211678230 |doi=10.1016/j.compchemeng.2020.107119 |issn=0098-1354}}</ref><ref>{{Cite journal |last1=Ajagekar |first1=Akshay |last2=You |first2=Fengqi |date=2021-12-01 |title=Quantum computing based hybrid deep learning for fault diagnosis in electrical power systems |journal=Applied Energy |language=en |volume=303 |article-number=117628 |doi=10.1016/j.apenergy.2021.117628 |issn=0306-2619 |doi-access=free|bibcode=2021ApEn..30317628A }}</ref>

{{anchor|Computer-aided drug design and generative chemistry}}

Deep generative chemistry models emerge as powerful tools to expedite [[drug discovery]]. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems<ref name="273.5278.1073"/> and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models<ref>{{cite journal |last1=Gao |first1=Xun |last2=Anschuetz |first2=Eric R. |last3=Wang |first3=Sheng-Tao |last4=Cirac |first4=J. Ignacio |last5=Lukin |first5=Mikhail D. |title=Enhancing Generative Models via Quantum Correlations |journal=Physical Review X |year=2022 |volume=12 |issue=2 |page=021037 |doi=10.1103/PhysRevX.12.021037 |arxiv=2101.08354 |bibcode=2022PhRvX..12b1037G |s2cid=231662294}}</ref> including quantum GANs<ref>{{cite arXiv |last1=Li |first1=Junde |last2=Topaloglu |first2=Rasit |last3=Ghosh |first3=Swaroop |title=Quantum Generative Models for Small Molecule Drug Discovery |date=9 January 2021 |class=cs.ET |eprint=2101.03438}}</ref> may eventually be developed into ultimate generative chemistry algorithms.

== Engineering<span class="anchor" id="Developing physical quantum computers"></span> ==

[[File:A Wafer of the Latest D-Wave Quantum Computers (39188583425).jpg|thumb|A [[wafer (electronics)|wafer]] of [[adiabatic quantum computer]]s]]

{{As of|2023|post=,}} classical computers outperform quantum computers for all real-world applications. While current quantum computers may speed up solutions to particular mathematical problems, they give no computational advantage for practical tasks. Scientists and engineers are exploring multiple technologies for quantum computing hardware and hope to develop scalable quantum architectures, but serious obstacles remain.<ref name="good-for-nothing" /><ref name="CACM" />

=== Challenges ===

There are a number of technical challenges in building a large-scale quantum computer.<ref>{{cite journal |last=Dyakonov |first=Mikhail |url=https://spectrum.ieee.org/the-case-against-quantum-computing |title=The Case Against Quantum Computing |journal=[[IEEE Spectrum]] |date=15 November 2018}}</ref> Physicist [[David P. DiVincenzo|David DiVincenzo]] has listed [[DiVincenzo's criteria|these requirements]] for a practical quantum computer:<ref>{{cite journal| arxiv=quant-ph/0002077 |title=The Physical Implementation of Quantum Computation |last=DiVincenzo |first=David P. |date=13 April 2000 |doi=10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E |volume=48 |issue=9–11 |journal=Fortschritte der Physik |pages=771–783 |bibcode=2000ForPh..48..771D |s2cid=15439711}}</ref>

* Physically scalable 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. [[Superconducting quantum computing|Superconducting quantum computers]], like those constructed by [[Google]] and [[IBM]], need [[helium-3]], a [[Nuclear physics|nuclear]] research byproduct, and special [[superconducting]] cables made only by the Japanese company Coax Co.<ref>{{cite news |last1=Giles |first1=Martin |date=January 17, 2019 |title=We'd have more quantum computers if it weren't so hard to find the damn cables |language=en-US |publisher=MIT Technology Review |url=https://www.technologyreview.com/s/612760/quantum-computers-component-shortage/ |access-date=May 17, 2021}}</ref>

The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of [[quantum controllers]] that enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.<ref>{{cite journal |vauthors=Pauka SJ, Das K, Kalra B, Moini A, Yang Y, Trainer M, Bousquet A, Cantaloube C, Dick N, Gardner GC, Manfra MJ, Reilly DJ|journal=[[Nature Electronics]]|title=A cryogenic CMOS chip for generating control signals for multiple qubits|year=2021|volume=4|issue=4|pages=64–70 |doi=10.1038/s41928-020-00528-y|url=https://www.nature.com/articles/s41928-020-00528-y|arxiv=1912.01299|s2cid=231715555}}</ref>

==== Decoherence<span class="anchor" id="Quantum decoherence"></span> ====

One of the greatest challenges involved in 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 [[Nuclear magnetic resonance|NMR]] and [[MRI]] technology, also called the ''dephasing time''), typically range between nanoseconds and seconds at low temperatures.<ref name="DiVincenzo 1995">{{cite journal |last=DiVincenzo |first=David P. |title=Quantum Computation |journal=Science |year=1995 |volume=270 |issue=5234 |pages=255–261 |doi=10.1126/science.270.5234.255 |bibcode=1995Sci...270..255D |citeseerx=10.1.1.242.2165 |s2cid=220110562}}</ref> Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a [[dilution refrigerator]]<ref>{{Cite journal |doi=10.1016/j.cryogenics.2021.103390| issn=0011-2275 |title=Development of Dilution refrigerators – A review |journal=Cryogenics| volume=121| year=2022| last1=Zu| first1=H.| last2=Dai| first2=W.| last3=de Waele| first3=A.T.A.M.| s2cid=244005391}}</ref>) in order to prevent significant decoherence.<ref>{{cite journal |last1=Jones |first1=Nicola |title=Computing: The quantum company |journal=Nature |date=19 June 2013 |volume=498 |issue=7454 |pages=286–288 |doi=10.1038/498286a|pmid=23783610|bibcode=2013Natur.498..286J|doi-access=free}}</ref> A 2020 study argues that [[ionizing radiation]] such as [[cosmic rays]] can nevertheless cause certain systems to decohere within milliseconds.<ref>{{cite journal |last1=Vepsäläinen |first1=Antti P. |last2=Karamlou |first2=Amir H. |last3=Orrell |first3=John L. |last4=Dogra |first4=Akshunna S. |last5=Loer |first5=Ben |last6=Vasconcelos |first6=Francisca |last7=Kim |first7=David K. |last8=Melville |first8=Alexander J. |last9=Niedzielski |first9=Bethany M. |last10=Yoder |first10=Jonilyn L. |last11=Gustavsson |first11=Simon |last12=Formaggio |first12=Joseph A. |last13=VanDevender |first13=Brent A. |last14=Oliver |first14=William D. |display-authors=5 |title=Impact of ionizing radiation on superconducting qubit coherence |journal=Nature |date=August 2020 |volume=584 |issue=7822 |pages=551–556 |doi=10.1038/s41586-020-2619-8 |pmid=32848227 |url=https://www.nature.com/articles/s41586-020-2619-8 |language=en |issn=1476-4687|arxiv=2001.09190 |bibcode=2020Natur.584..551V |s2cid=210920566 }}</ref>

As a result, time-consuming tasks may render some quantum algorithms inoperable, as attempting to maintain the state of qubits for a long enough duration will eventually corrupt the superpositions.<ref>{{cite arXiv |last1=Amy |first1=Matthew |last2=Matteo |first2=Olivia |last3=Gheorghiu |first3=Vlad |last4=Mosca |first4=Michele |last5=Parent |first5=Alex |last6=Schanck |first6=John |title=Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3 |date=30 November 2016 |eprint=1603.09383 |class=quant-ph}}</ref>

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 by the [[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 binary digits 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.<ref>{{cite journal |last=Dyakonov |first=M. I. |date=14 October 2006 |editor2=Xu |editor2-first=J. |editor3=Zaslavsky |editor3-first=A. |title=Is Fault-Tolerant Quantum Computation Really Possible? |journal=Future Trends in Microelectronics. Up the Nano Creek |pages=4–18 |arxiv=quant-ph/0610117 |bibcode=2006quant.ph.10117D |editor1=S. Luryi}}</ref> With error correction, the figure would rise to about 10⁷ bits. Computation time is about ''L''² or about 10⁷ steps and at 1{{nbsp}}MHz, about 10 seconds. However, the encoding and error-correction overheads increase the size of a real fault-tolerant quantum computer by several orders of magnitude. Careful estimates<ref name=":1">{{Cite book |last=Ahsan |first=Muhammad |title=Architecture Framework for Trapped-ion Quantum Computer based on Performance Simulation Tool |date=2015 |bibcode=2015PhDT........56A |language=en-US |oclc=923881411}}</ref><ref name=":2">{{Cite journal |last1=Ahsan |first1=Muhammad |last2=Meter |first2=Rodney Van |last3=Kim |first3=Jungsang |date=2016-12-28 |title=Designing a Million-Qubit Quantum Computer Using a Resource Performance Simulator |journal=ACM Journal on Emerging Technologies in Computing Systems |volume=12 |issue=4 |pages=39:1–39:25 |doi=10.1145/2830570 |s2cid=1258374 |issn=1550-4832|doi-access=free |arxiv=1512.00796 }}</ref> show that at least 3{{nbsp}}million physical qubits would factor 2,048-bit integer in 5 months on a fully error-corrected trapped-ion quantum computer. In terms of the number of physical qubits, to date, this remains the lowest estimate<ref>{{Cite journal |last1=Gidney |first1=Craig |last2=Ekerå |first2=Martin |date=2021-04-15 |title=How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits |journal=Quantum |volume=5 |article-number=433 |doi=10.22331/q-2021-04-15-433 |arxiv=1905.09749 |bibcode=2021Quant...5..433G |s2cid=162183806 |issn=2521-327X}}</ref> for practically useful integer factorization problem sizing 1,024-bit or larger.

Another approach to the stability-decoherence problem is to create a [[topological quantum computer]] with [[anyon]]s, [[quasi-particle]]s used as threads, and relying on [[braid theory]] to form stable logic gates.<ref>{{cite journal

| last1 = Freedman | first1 = Michael H. | author1-link = Michael Freedman

| last2 = Kitaev | first2 = Alexei | author2-link = Alexei Kitaev

Line 113 ⟶ 236:

| title = Topological quantum computation

| volume = 40

| year = 2003}}</ref><ref>{{cite journal |last=Monroe |first=Don |url=httphttps://www.newscientist.com/channel/fundamentals/mg20026761.700-anyons-the-breakthrough-quantum-computing-needs.html |title=Anyons: The breakthrough quantum computing needs? |journal=[[New Scientist]] |date=1 October 2008}}</ref> Non-10-01Abelian anyons can, in effect, remember how they have been manipulated, making them potentially useful in quantum computing.<ref name=":0">{{Cite journal |last=Cossins |first=Daniel |date=28 June 2025 |title=How to think about...Quasiparticles |journal=[[New Scientist]] |volume=266 |issue=3549 |page=34}}</ref> As of 2025, Microsoft and other organizations are investing in quasi-particle research.<ref name=":0" />

=== Quantum supremacy ===

== Developments ==

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 circuit|Quantum gate array]]'' (computation decomposed into sequence of few-qubit [[quantum gate]]s)

* ''[[One-way quantum computer]]'' (computation decomposed into sequence of one-qubit measurements applied to a highly entangled initial state or [[cluster state]])

* ''[[Adiabatic quantum computation|Adiabatic quantum computer]]'', based on [[Quantum annealing]] (computation decomposed into a slow continuous transformation of an initial [[Hamiltonian (quantum mechanics)|Hamiltonian]] into a final Hamiltonian, whose ground states contains the solution)<ref>{{cite journal |first=A. |last=Das |first2=B. K. |last2=Chakrabarti |title=Quantum Annealing and Analog Quantum Computation | journal=[[Reviews of Modern Physics|Rev. Mod. Phys.]] |volume=80 |issue=3 |pages=1061–1081 |year=2008 |doi=10.1103/RevModPhys.80.1061 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} |bibcode=2008RvMP...80.1061D}}</ref>

* [[Topological quantum computer]]<ref name="Nayaketal2008">{{cite journal

|arxiv = 0707.1889

|journal = Rev Mod Phys

|year = 2008

|title = Nonabelian Anyons and Quantum Computation

|first1 = Chetan

|first4 = Sankar

|last1 = Nayak

|last4 = Das Sarma

|first2 = Steven

|last2 = Simon

|first3 = Ady

|last3 = Stern

|volume = 80

|page = 1083

|doi = 10.1103/RevModPhys.80.1083

|bibcode = 2008RvMP...80.1083N

|issue = 3 }}</ref> (computation decomposed into the braiding of [[anyon]]s 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.

Physicist [[John Preskill]] coined the term ''quantum supremacy'' to describe the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers.<ref>{{cite arXiv |last=Preskill |first=John |date=2012-03-26 |title=Quantum computing and the entanglement frontier |eprint=1203.5813 |class=quant-ph}}</ref><ref>{{Cite journal|last=Preskill |first=John |date=2018-08-06 |title=Quantum Computing in the NISQ era and beyond |journal=Quantum |volume=2 |article-number=79 |doi=10.22331/q-2018-08-06-79 |arxiv=1801.00862 |bibcode=2018Quant...2...79P |doi-access=free}}</ref><ref>{{Cite journal |title=Characterizing Quantum Supremacy in Near-Term Devices|journal=Nature Physics |volume=14 |issue=6 |pages=595–600 |first1=Sergio |last1=Boixo |first2=Sergei V. |last2=Isakov |first3=Vadim N. |last3=Smelyanskiy |first4=Ryan |last4=Babbush |first5=Nan |last5=Ding |first6=Zhang |last6=Jiang |first7=Michael J. |last7=Bremner |first8=John M. |last8=Martinis |first9=Hartmut |last9=Neven |display-authors=5 |year=2018 |arxiv=1608.00263 |doi=10.1038/s41567-018-0124-x |bibcode=2018NatPh..14..595B |s2cid=4167494}}</ref> The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.<ref>{{cite web |first=Neil |last=Savage |date=5 July 2017 |url=https://www.scientificamerican.com/article/quantum-computers-compete-for-supremacy/ |title=Quantum Computers Compete for "Supremacy" |work=Scientific American}}</ref>

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):

In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the [[Sycamore processor|Sycamore quantum computer]] more than 3,000,000 times faster than they could be done on [[Summit (supercomputer)|Summit]], generally considered the world's fastest computer.<ref name="1910.11333"/><ref>{{cite web |last=Giles |first=Martin |date=September 20, 2019 |title=Google researchers have reportedly achieved 'quantum supremacy' |website=MIT Technology Review |language=en |url=https://www.technologyreview.com/f/614416/google-researchers-have-reportedly-achieved-quantum-supremacy/ |access-date=May 15, 2020}}</ref><ref>{{Cite web |last=Tavares |first=Frank |date=2019-10-23 |title=Google and NASA Achieve Quantum Supremacy |url=http://www.nasa.gov/feature/ames/quantum-supremacy |access-date=2021-11-16 |website=NASA |language=en-US}}</ref> This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,<ref>{{cite arXiv |last1=Pednault |first1=Edwin |last2=Gunnels |first2=John A. |last3=Nannicini |first3=Giacomo |last4=Horesh |first4=Lior |last5=Wisnieff |first5=Robert |date=2019-10-22|title=Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits |class=quant-ph |eprint=1910.09534}}</ref><ref>{{Cite journal |last=Cho |first=Adrian |date=2019-10-23 |title=IBM casts doubt on Google's claims of quantum supremacy |url=https://www.science.org/content/article/ibm-casts-doubt-googles-claims-quantum-supremacy |journal=Science |doi=10.1126/science.aaz6080 |s2cid=211982610 |issn=0036-8075}}</ref> and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Sycamore and classical supercomputers<ref>{{Cite book |last1=Liu |first1=Yong (Alexander) |last2=Liu |first2=Xin (Lucy) |last3=Li |first3=Fang (Nancy) |last4=Fu |first4=Haohuan |last5=Yang |first5=Yuling |last6=Song |first6=Jiawei |last7=Zhao |first7=Pengpeng |last8=Wang |first8=Zhen |last9=Peng |first9=Dajia |last10=Chen |first10=Huarong |last11=Guo |first11=Chu |title=Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis |chapter=Closing the "quantum supremacy" gap |display-authors=5 |date=2021-11-14 |series=SC '21 |___location=New York, New York |publisher=Association for Computing Machinery |pages=1–12 |arxiv=2110.14502 |doi=10.1145/3458817.3487399 |isbn=978-1-4503-8442-1 |s2cid=239036985}}</ref><ref>{{Cite journal |last1=Bulmer |first1=Jacob F. F. |last2=Bell |first2=Bryn A. |last3=Chadwick |first3=Rachel S. |last4=Jones |first4=Alex E. |last5=Moise |first5=Diana |last6=Rigazzi |first6=Alessandro |last7=Thorbecke |first7=Jan |last8=Haus |first8=Utz-Uwe |last9=Van Vaerenbergh |first9=Thomas |last10=Patel |first10=Raj B. |last11=Walmsley |first11=Ian A. |display-authors=5 |date=2022-01-28 |title=The boundary for quantum advantage in Gaussian boson sampling |journal=Science Advances |language=en |volume=8 |issue=4 |pages=eabl9236 |doi=10.1126/sciadv.abl9236 |issn=2375-2548 |pmc=8791606 |pmid=35080972 |arxiv=2108.01622 |bibcode=2022SciA....8.9236B}}</ref><ref>{{Cite journal |last=McCormick |first=Katie |date=2022-02-10 |title=Race Not Over Between Classical and Quantum Computers |url=https://physics.aps.org/articles/v15/19 |journal=Physics |language=en |volume=15|article-number=19 |doi=10.1103/Physics.15.19 |bibcode=2022PhyOJ..15...19M |s2cid=246910085 |doi-access=free }}</ref> and even beating it.<ref>{{Cite journal |title=Solving the Sampling Problem of the Sycamore Quantum Circuits |journal=Physical Review Letters |arxiv=2111.03011 |last1=Pan |first1=Feng |last2=Chen |first2=Keyang |last3=Zhang |first3=Pan |year=2022 |volume=129 |issue=9 |page=090502 |doi=10.1103/PhysRevLett.129.090502 |pmid=36083655 |bibcode=2022PhRvL.129i0502P |s2cid=251755796}}</ref><ref>{{Cite journal |author=Cho |first=Adrian |date=2022-08-02 |title=Ordinary computers can beat Google's quantum computer after all |url=https://www.science.org/content/article/ordinary-computers-can-beat-google-s-quantum-computer-after-all |journal=Science |volume=377 |doi=10.1126/science.ade2364|url-access=subscription }}</ref><ref>{{Cite web |title=Google's 'quantum supremacy' usurped by researchers using ordinary supercomputer |url=https://techcrunch.com/2022/08/05/googles-quantum-supremacy-usurped-by-researchers-using-ordinary-supercomputer/ |access-date=2022-08-07 |website=TechCrunch |date=5 August 2022 |language=en-US}}</ref>

*[[Superconductor]]-based quantum computers (including [[SQUID]]-based quantum computers)<ref name="ClarkeWilhelm2008">{{cite journal

|url = http://www.nature.com/nature/journal/v453/n7198/full/nature07128.html

|journal = Nature

|date = June 19, 2008

|title = Superconducting quantum bits

|first1 = John

|last1 = Clarke

|first2 = Frank

|last2 = Wilhelm

|volume = 453

|pages = 1031–1042

|doi = 10.1038/nature07128

|pmid = 18563154

|issue = 7198

|bibcode = 2008Natur.453.1031C }}</ref><ref>{{cite arXiv|eprint=quant-ph/0403090|title=Scalable Superconducting Architecture for Adiabatic Quantum Computation|last=Kaminsky |first=William M|class=quant-ph|year=2004}}</ref> (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 lattice]]s (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]] <ref>{{cite journal |first1=Atac |last1=Imamoğlu |first2=D. D. |last2=Awschalom |first3=Guido |last3=Burkard |first4=D. P. |last4=DiVincenzo |first5=D. |last5=Loss |first6=M. |last6=Sherwin |first7=A. |last7=Small |title=Quantum information processing using quantum dot spins and cavity-QED |journal=Physical Review Letters |year=1999 |volume=83 |page=4204 |doi=10.1103/PhysRevLett.83.4204 |bibcode = 1999PhRvL..83.4204I |issue=20 }}</ref>) (qubit given by the spin states of trapped electrons)

*[[Quantum dot]] computer, spatial-based (qubit given by electron position in double quantum dot)<ref>{{cite journal |first1=Leonid |last1=Fedichkin |first2=Maxim |last2=Yanchenko |first3=Kamil |last3=Valiev |title=Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot |journal=Quantum Computers and Computing |year=2000 |volume=1 |pages=58–76 |url=http://ics.org.ru/eng?menu=mi_pubs&abstract=249 |arxiv=quant-ph/0006097 |bibcode = 2000quant.ph..6097F }}</ref>

*[[Nuclear magnetic resonance]] on molecules in solution (liquid-state NMR) (qubit provided by [[nuclear spin]]s within the dissolved molecule)

*Solid-state NMR [[Kane quantum computer]]s (qubit realized by the nuclear spin state of [[phosphorus]] [[Electron donor|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]]<ref>{{Cite journal|url = |title = Quantum computing in molecular magnets.|last = |first = |date = Apr 12, 2001|journal = Nature|accessdate = |doi = 10.1038/35071024|pmid = 11298441 |volume=410 |issue=6830 |pages=789–93}}</ref> (qubit given by spin states)

*[[Fullerene]]-based [[Electron paramagnetic resonance|ESR]] quantum computer (qubit based on the electronic spin of [[Endohedral fullerene|atoms or molecules encased in fullerenes]])

*[[Linear optical quantum computing|Linear optical quantum computer]] (qubits realized by processing states of different [[Normal mode|modes]] of light through linear elements e.g. mirrors, [[beam splitter]]s and [[phase shift module|phase shifters]])<ref name="KLM2001">{{cite journal |last1=Knill |first1=G. J. |last2=Laflamme |last3=Milburn |title=A scheme for efficient quantum computation with linear optics |journal=Nature |year=2001 |volume=409 |doi=10.1038/35051009 |bibcode = 2001Natur.409...46K |first2=R. |first3=G. J. |issue=6816 |pmid=11343107 |pages=46–52 }}</ref>

*[[Diamond-based quantum computer]]<ref name="Nizovtsevetal2004">{{cite journal

|journal = Optics and Spectroscopy

|date = August 2005

|title = A quantum computer based on NV centers in diamond: Optically detected nutations of single electron and nuclear spins

|author = Nizovtsev, A. P.

|volume = 99 |issue = 2

|pages = 248–260

|doi = 10.1134/1.2034610

|bibcode = 2005OptSp..99..233N }}</ref><ref>{{cite web | url=http://www.tgdaily.com/content/view/32306/118/ |title=Research indicates diamonds could be key to quantum storage |accessdate=2007-06-04 |last=Gruener |first=Wolfgang|date=2007-06-01}}</ref><ref name="Neumannetal2008">{{cite journal

|journal = Science

|date = June 6, 2008

|title = Multipartite Entanglement Among Single Spins in Diamond

|author = Neumann, P.

|volume = 320

|issue = 5881

|pages = 1326–1329

|doi = 10.1126/science.1157233

|pmid = 18535240

|bibcode = 2008Sci...320.1326N

|display-authors = 1

|last2 = Mizuochi

|first2 = N.

|last3 = Rempp

|first3 = F.

|last4 = Hemmer

|first4 = P.

|last5 = Watanabe

|first5 = H.

|last6 = Yamasaki

|first6 = S.

|last7 = Jacques

|first7 = V.

|last8 = Gaebel

|first8 = T.

|last9 = Jelezko

|first9 = F. }}</ref> (qubit realized by electronic or nuclear spin of [[nitrogen-vacancy center]]s in diamond)

*[[Bose–Einstein condensate]]-based quantum computer<ref>{{cite web | url=http://www.itpro.co.uk/news/121086/trapped-atoms-could-advance-quantum-computing.html |title=Trapped atoms could advance quantum computing |accessdate=2007-07-26 |last=Millman |first=Rene|date=2007-08-03 |publisher=ITPro |archiveurl=http://web.archive.org/web/20070927191354/http://www.itpro.co.uk/news/121086/trapped-atoms-could-advance-quantum-computing.html |archivedate=2007-09-27}}</ref>

*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<ref name="Ohlsson2002">{{cite journal

|journal = Opt. Commun.

|date = January 1, 2002

|title = Quantum computer hardware based on rare-earth-ion-doped inorganic crystals

|first1 = N.

|last1 = Ohlsson

|first2 = R. K.

|last2 = Mohan

|first3 = S.

|last3 = Kröll

|volume = 201

|issue = 1–3

|pages = 71–77

|doi = 10.1016/S0030-4018(01)01666-2

|bibcode = 2002OptCo.201...71O }}</ref><ref name="Longdell2004">{{cite journal

|journal = Phys. Rev. Lett.

|date = September 23, 2004

|title = Demonstration of conditional quantum phase shift between ions in a solid

|first1 = J. J.

|last1 = Longdell

|first2 = M. J.

|last2 = Sellars

|first3 = N. B.

|last3 = Manson

|volume = 93

|issue = 13

|page = 130503

|doi = 10.1103/PhysRevLett.93.130503

|pmid = 15524694

|arxiv = quant-ph/0404083 |bibcode = 2004PhRvL..93m0503L }}</ref> (qubit realized by the internal electronic state of [[dopant]]s in [[optical fiber]]s)

In December 2020, a group at [[University of Science and Technology of China|USTC]] implemented a type of [[Boson sampling]] on 76 photons with a [[Linear optical quantum computing|photonic quantum computer]], [[Jiuzhang (quantum computer)|Jiuzhang]], to demonstrate quantum supremacy.<ref>{{Cite journal |last=Ball |first=Philip |date=2020-12-03 |title=Physicists in China challenge Google's 'quantum advantage' |journal=Nature |volume=588 |issue=7838 |page=380 |language=en |doi=10.1038/d41586-020-03434-7 |pmid=33273711 |bibcode=2020Natur.588..380B |s2cid=227282052 |doi-access=}}</ref><ref>{{Cite web |last=Garisto |first=Daniel |title=Light-based Quantum Computer Exceeds Fastest Classical Supercomputers |url=https://www.scientificamerican.com/article/light-based-quantum-computer-exceeds-fastest-classical-supercomputers/ |access-date=2020-12-07 |website=Scientific American |language=en}}</ref><ref>{{Cite web |last=Conover |first=Emily |date=2020-12-03 |title=The new light-based quantum computer Jiuzhang has achieved quantum supremacy |url=https://www.sciencenews.org/article/new-light-based-quantum-computer-jiuzhang-supremacy |access-date=2020-12-07 |website=Science News |language=en-US}}</ref> The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.<ref name=":6">{{Cite journal |last1=Zhong |first1=Han-Sen |last2=Wang |first2=Hui |last3=Deng |first3=Yu-Hao |last4=Chen |first4=Ming-Cheng |last5=Peng |first5=Li-Chao |last6=Luo |first6=Yi-Han |last7=Qin |first7=Jian |last8=Wu |first8=Dian |last9=Ding |first9=Xing |last10=Hu |first10=Yi |last11=Hu |first11=Peng |display-authors=5 |date=2020-12-03 |title=Quantum computational advantage using photons |journal=Science |volume=370 |issue=6523 |pages=1460–1463 |language=en |doi=10.1126/science.abe8770 |issn=0036-8075 |pmid=33273064 |arxiv=2012.01625 |bibcode=2020Sci...370.1460Z |s2cid=227254333}}</ref>

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.

Claims of quantum supremacy have generated hype around quantum computing,<ref>{{Cite journal |last=Roberson |first=Tara M. |date=2020-05-21 |title=Can Hype Be a Force for Good? |journal=Public Understanding of Science |language=en |volume=29 |issue=5 |pages=544–552 |doi=10.1177/0963662520923109 |pmid=32438851 |s2cid=218831653 |issn=0963-6625|doi-access=free }}</ref> but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.<ref name="good-for-nothing" /><ref>{{Cite journal |last1=Cavaliere |first1=Fabio |last2=Mattsson |first2=John |last3=Smeets |first3=Ben |date=September 2020 |title=The security implications of quantum cryptography and quantum computing |url=http://www.magonlinelibrary.com/doi/10.1016/S1353-4858%2820%2930105-7 |journal=Network Security |language=en |volume=2020 |issue=9 |pages=9–15 |doi=10.1016/S1353-4858(20)30105-7 |s2cid=222349414 |issn=1353-4858|url-access=subscription }}</ref>

=== Timeline ===

{{Main|Timeline of quantum computing}}

In 2001, researchers demonstrated Shor's algorithm to factor 15 using a 7-qubit NMR computer.<ref>{{cite journal|doi=10.1038/414883a|title=Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance|year=2001|last1=Vandersypen|first1=Lieven M. K.|last2=Steffen|first2=Matthias|last3=Breyta|first3=Gregory|last4=Yannoni|first4=Costantino S.|last5=Sherwood|first5=Mark H.|last6=Chuang|first6=Isaac L.|journal=Nature|volume=414|issue=6866|pages=883–7|pmid=11780055}}</ref>

In January 2024, a study published in ''Physical Review Letters'' provided direct verification of quantum supremacy experiments by computing exact amplitudes for experimentally generated bitstrings using a new-generation Sunway supercomputer, demonstrating a significant leap in simulation capability built on a multiple-amplitude tensor network contraction algorithm. This development underscores the evolving landscape of quantum computing, highlighting both the progress and the complexities involved in validating quantum supremacy claims.<ref>{{Cite journal |last1=Liu |first1=Yong |last2=Chen |first2=Yaojian |last3=Guo |first3=Chu |last4=Song |first4=Jiawei |last5=Shi |first5=Xinmin |last6=Gan |first6=Lin |last7=Wu |first7=Wenzhao |last8=Wu |first8=Wei |last9=Fu |first9=Haohuan |last10=Liu |first10=Xin |last11=Chen |first11=Dexun |last12=Zhao |first12=Zhifeng |last13=Yang |first13=Guangwen |last14=Gao |first14=Jiangang |date=2024-01-16 |title=Verifying Quantum Advantage Experiments with Multiple Amplitude Tensor Network Contraction |url=https://link.aps.org/doi/10.1103/PhysRevLett.132.030601 |journal=Physical Review Letters |language=en |volume=132 |issue=3 |page=030601 |doi=10.1103/PhysRevLett.132.030601 |pmid=38307065 |issn=0031-9007|arxiv=2212.04749 |bibcode=2024PhRvL.132c0601L }}</ref>

In 2005, researchers at the [[University of Michigan]] built a [[semiconductor chip]] [[ion trap]]. Such devices from standard [[photolithography|lithography]], may point the way to scalable quantum computing.<ref>{{cite web | url=http://www.umich.edu/news/index.html?Releases/2005/Dec05/r121205b |title= U-M develops scalable and mass-producible quantum computer chip |publisher=University of Michigan |accessdate=2006-11-17 |date=2005-12-12}}</ref>

=== Skepticism ===

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]] [[atom]]s that acted like a single atom that could occupy two states.<ref>{{cite journal | url=http://www.nature.com/nature/journal/vaop/ncurrent/pdf/nature08121.pdf |title= Demonstration of two-qubit algorithms with a superconducting quantum processor |accessdate=2009-07-02 |last=DiCarlo |first=L. |last2=Chow |first2=J. M. |last3=Gambetta |first3=J. M. |last4=Bishop |first4=Lev S. |last5=Johnson |first5=B. R. |last6=Schuster |first6=D. I. |last7=Majer |first7=J. |last8=Blais |first8=A. |last9=Frunzio |first9=L. |author10= S. M. Girvin |author11= R. J. Schoelkopf |date=9 July 2009 |journal = [[Nature (journal)|Nature]] | doi=10.1038/nature08121 |pmid=19561592 |volume=460 |issue=7252 |pages=240–4 |bibcode = 2009Natur.460..240D }}</ref><ref>{{cite web | url=http://opa.yale.edu/news/article.aspx?id=6764 |title= Scientists Create First Electronic Quantum Processor |publisher=Yale University |accessdate=2009-07-02 |date=2009-07-02}}</ref>

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 [[Nature (journal)|Nature]] spotlight article summarized current quantum computers as being "For now, [good for] absolutely nothing".<ref name="good-for-nothing">

A team at the [[University of Bristol]], also created a [[silicon]] chip based on [[quantum optics]], able to run [[Shor's algorithm]].<ref>{{cite web | url=http://www.newscientist.com/article/dn17736-codebreaking-quantum-algorithm-run-on-a-silicon-chip.html |title= Code-breaking quantum algorithm runs on a silicon chip |accessdate=2009-10-14 |work=New Scientist |date=2009-09-04}}</ref>

{{Cite journal| journal = Nature | title = Quantum computers: what are they good for? | date = 24 May 2023 | first = Michael | last = Brooks| volume = 617 | issue = 7962 | pages = S1–S3 | doi = 10.1038/d41586-023-01692-9 | pmid = 37225885 | bibcode = 2023Natur.617S...1B | s2cid = 258847001 | doi-access = free }}

Further developments were made in 2010.<ref>{{cite web |title= New Trends in Quantum Computation | url=http://insti.physics.sunysb.edu/itp/conf/simons-qcomputation2/program.html |work=Simons Conference on New Trends in Quantum Computation 2010: Program |publisher=C.N. Yang Institute for Theoretical Physics}}</ref>

</ref> The article elaborated that quantum computers are yet to be more useful or efficient than conventional computers in any case, though it also argued that in the long term such computers are likely to be useful. A 2023 [[Communications of the ACM]] article<ref name = "CACM">{{Cite web | url = https://m-cacm.acm.org/magazines/2023/5/272276-disentangling-hype-from-practicality-on-realistically-achieving-quantum-advantage/fulltext | publisher = Communications of the ACM | date = May 2023 | title = Disentangling Hype from Practicality: On Realistically Achieving Quantum Advantage

Springer publishes a journal (''Quantum Information Processing'') devoted to the subject.<ref>"[http://www.springer.com/new+%26+forthcoming+titles+%28default%29/journal/11128 Quantum Information Processing]". Springer.com. Retrieved on 2011-05-19.</ref>

|author1 = Torsten Hoefler | author2 = Thomas Häner | author3 = Matthias Troyer}}

</ref> found that current quantum computing algorithms are "insufficient for practical quantum advantage without significant improvements across the software/hardware stack". It argues that the most promising candidates for achieving speedup with quantum computers are "small-data problems", for example in chemistry and materials science. However, the article also concludes that a large range of the potential applications it considered, such as machine learning, "will not achieve quantum advantage with current quantum algorithms in the foreseeable future", and it identified I/O constraints that make speedup unlikely for "big data problems, unstructured linear systems, and database search based on Grover's algorithm".

This state of affairs can be traced to several current and long-term considerations.

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.<ref>{{cite web | url=http://www.unsw.edu.au/news/pad/articles/2011/apr/Quantum_teleport_paper.html |title=Quantum teleporter breakthrough |publisher=University of New South Wales |date=2011-04-15 |archiveurl=http://web.archive.org/web/20110418200747/http://www.unsw.edu.au/news/pad/articles/2011/apr/Quantum_teleport_paper.html |archivedate=2011-04-18}}</ref><ref>{{cite web |last=Lai |first=Richard | url=http://www.engadget.com/2011/04/18/first-light-wave-quantum-teleportation-achieved-opens-door-to-u/ |title=First light wave quantum teleportation achieved, opens door to ultra fast data transmission |publisher=Engadget |date=2011-04-18}}</ref>

* Conventional computer hardware and algorithms are not only optimized for practical tasks, but are still improving rapidly, particularly [[GPU]] accelerators.

[[File:DWave 128chip.jpg|thumb|Photograph of a chip constructed by D-Wave Systems Inc., mounted and wire-bonded in a sample holder. The D-Wave processor is designed to use 128 [[superconductivity|superconducting]] logic elements that exhibit controllable and tunable coupling to perform operations.]]

* Current quantum computing hardware generates only a limited amount of [[Quantum entanglement|entanglement]] before getting overwhelmed by noise.

* Quantum algorithms provide speedup over conventional algorithms only for some tasks, and matching these tasks with practical applications proved challenging. Some promising tasks and applications require resources far beyond those available today.<ref>{{Cite web| url = https://m-cacm.acm.org/magazines/2022/12/266916-quantum-computers-and-the-universe/fulltext | publisher = Communications of the ACM

|title = Quantum Computers and the Universe | first = Don | last = Monroe | date = December 2022}}

</ref><ref>{{Cite web| url = https://thequantuminsider.com/2023/06/20/psiquantum-sees-700x-reduction-in-computational-resource-requirements-to-break-elliptic-curve-cryptography-with-a-fault-tolerant-quantum-computer/ | website = The Quanrum Insider | title = PsiQuantum Sees 700x Reduction in Computational Resource Requirements to Break Elliptic Curve Cryptography With a Fault Tolerant Quantum Computer| first = Matt | last = Swayne | date = June 20, 2023 }}

</ref> In particular, processing large amounts of non-quantum data is a challenge for quantum computers.<ref name=CACM/>

* Some promising algorithms have been "dequantized", i.e., their non-quantum analogues with similar complexity have been found.

* If [[quantum error correction]] is used to scale quantum computers to practical applications, its overhead may undermine speedup offered by many quantum algorithms.<ref name=CACM/>

* Complexity analysis of algorithms sometimes makes abstract assumptions that do not hold in applications. For example, input data may not already be available encoded in quantum states, and "oracle functions" used in Grover's algorithm often have internal structure that can be exploited for faster algorithms.

In particular, building computers with large numbers of qubits may be futile if those qubits are not connected well enough and cannot maintain sufficiently high degree of entanglement for a long time. When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations. Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms (which may be unknown) and show that some quantum algorithms asymptomatically improve upon those bounds.

In 2011, [[D-Wave Systems]] announced the first commercial quantum annealer, the D-Wave One, claiming a 128 qubit processor.<ref>{{cite web |title=Learning to program the D-Wave One |url=http://dwave.wordpress.com/2011/05/11/learning-to-program-the-d-wave-one/ |accessdate=2011-05-11 |work=Hack The Multiverse |publisher=D-Wave}}</ref> On May 25, 2011 [[Lockheed Martin]] agreed to purchase a D-Wave One system.<ref>{{cite web | url=http://www.dwavesys.com/en/pressreleases.html#lm_2011 |title= D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation |accessdate=2011-05-30 |date=2011-05-25 |publisher=D-Wave}}</ref> Lockheed and the University of Southern California (USC) will house the D-Wave One at the newly formed USC Lockheed Martin Quantum Computing Center.<ref>{{cite web | url=http://www.viterbi.usc.edu/news/news/2011/operational-quantum-computing334119.htm |title= Operational Quantum Computing Center Established at USC |accessdate=2011-12-06 |date=2011-10-29 |publisher=University of Southern California}}</ref> 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 (journal)|Nature]], that proved the chips have some quantum properties.<ref>{{cite journal |url=http://www.nature.com/nature/journal/v473/n7346/full/nature10012.html |title=Quantum annealing with manufactured spins |journal=Nature |volume=473 |pages=194–198 |date=12 May 2011 |issue=7346 |doi=10.1038/nature10012}}</ref><ref>{{cite journal |url=http://www.technologyreview.com/news/429429/the-cia-and-jeff-bezos-bet-on-quantum-computing/ |title=The CIA and Jeff Bezos Bet on Quantum Computing |journal=Technology Review |date=October 4, 2012 |first=Tom |last=Simonite}}</ref> Two published papers have suggested that the D-Wave machine's operation can be explained classically, rather than requiring quantum models.<ref>{{cite web |url=http://arxiv.org/abs/1401.7087 |title=How "Quantum" is the D-Wave Machine? |date=2014-05-02 |publisher=[[Arxiv]]}}</ref><ref>{{cite web |url=http://arxiv.org/abs/1304.4595v2 |title=Quantum Annealing With More Than 100 Qbits |date=2013-04-16 |publisher=[[Arxiv]]}}</ref> Later work showed that classical models are insufficient when all available data is considered.<ref>{{cite web |url=http://arxiv.org/abs/1409.3827 |title=Reexamining classical and quantum models for the D-Wave One processor |date=2014-09-12 |publisher=[[Arxiv]]}}</ref> Experts remain divided on the ultimate classification of the D-Wave systems though their quantum behavior was established concretely with a demonstration of entanglement.<ref>{{cite web |url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.021041 |title=Entanglement in a quantum annealing processor |date=2014-05-29 |publisher=prx}}</ref>

[[Bill Unruh]] doubted the practicality of quantum computers in a paper published in 1994.<ref>{{Cite journal |last1=Unruh |first1=Bill |title=Maintaining coherence in Quantum Computers |journal=Physical Review A |volume=51 |issue=2 |pages=992–997 |arxiv=hep-th/9406058 |bibcode=1995PhRvA..51..992U |year=1995 |doi=10.1103/PhysRevA.51.992 |pmid=9911677 |s2cid=13980886}}</ref> [[Paul Davies]] argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the [[holographic principle]].<ref>{{cite arXiv|last1=Davies|first1=Paul|date=6 March 2007 |title=The implications of a holographic universe for quantum information science and the nature of physical law |eprint=quant-ph/0703041}}</ref> Skeptics like [[Gil Kalai]] doubt that quantum supremacy will ever be achieved.<ref>{{cite web |author=Regan |first=K. W. |date=23 April 2016 |title=Quantum Supremacy and Complexity |url=https://rjlipton.wordpress.com/2016/04/22/quantum-supremacy-and-complexity/ |website=Gödel's Lost Letter and P=NP}}</ref><ref>{{cite journal |last1=Kalai |first1=Gil |date=May 2016 |title=The Quantum Computer Puzzle |journal=Notices of the AMS |volume=63 |number=5 |pages=508–516 |url=https://www.ams.org/journals/notices/201605/rnoti-p508.pdf}}</ref><ref>{{cite arXiv |last1=Rinott |first1=Yosef |last2=Shoham |first2=Tomer |last3=Kalai |first3=Gil |date=2021-07-13 |title=Statistical Aspects of the Quantum Supremacy Demonstration |class=quant-ph |eprint=2008.05177}}</ref> Physicist [[Mikhail Dyakonov]] has expressed skepticism of quantum computing as follows:

During the same year, researchers at the [[University of Bristol]] created an all-bulk optics system that ran a version of [[Shor's algorithm]] to successfully factor 21.<ref>{{cite journal |title=Implementation of an iterative quantum order finding algorithm |last=Lopez |first=Enrique Martin |last2=Laing |first2=Anthony |last3=Lawson |first3=Thomas |last4=Alvarez |first4=Roberto| first5= Xiao-Qi |last5=Zhou |last6=O'Brien |first6=Jeremy L. |year=2011 |doi=10.1038/nphoton.2012.259 |journal=Nature Photonics |volume=6 |issue=11 |pages=773–776 | arxiv=1111.4147}}</ref>

:"So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10³⁰⁰... Could we ever learn to control the more than 10³⁰⁰ continuously variable parameters defining the quantum state of such a system? My answer is simple. ''No, never.''"<ref>{{cite web |last1=Dyakonov |first1=Mikhail |title=The Case Against Quantum Computing |url=https://spectrum.ieee.org/the-case-against-quantum-computing |website=IEEE Spectrum |date=15 November 2018 |access-date=3 December 2019}}</ref>

=== Physical realizations ===

In September 2011 researchers proved quantum computers can be made with a [[Von Neumann architecture]] (separation of RAM).<ref>{{cite web|url=http://arxiv.org/abs/1109.3743|title=Quantum computer with Von Neumann architecture|publisher=|accessdate=2014-10-26}}</ref>

{{Further|List of proposed quantum registers}}

[[File:IBM Q system (Fraunhofer 2).jpg|thumb|upright=1.2|[[IBM Q System One|Quantum System One]], a quantum computer by [[IBM]] from 2019 with 20 superconducting qubits<ref>{{Cite news |last=Russell |first=John |date=January 10, 2019 |title=IBM Quantum Update: Q System One Launch, New Collaborators, and QC Center Plans |language=en-US |website=HPCwire |url=https://www.hpcwire.com/2019/01/10/ibm-quantum-update-q-system-one-launch-new-collaborators-and-qc-center-plans/ |access-date=2023-01-09}}</ref>]]

In November 2011 researchers factorized 143 using 4 qubits.<ref>{{cite web|url=http://arxiv.org/abs/1111.3726|title=Quantum Factorization of 143 on a Dipolar-Coupling NMR system|publisher=|accessdate=2014-10-26}}</ref>

A practical quantum computer must use a physical system as a programmable quantum register.<ref>{{Cite journal |last1=Tacchino |first1=Francesco |last2=Chiesa |first2=Alessandro |last3=Carretta |first3=Stefano |last4=Gerace |first4=Dario |date=2019-12-19 |title=Quantum Computers as Universal Quantum Simulators: State-of-the-Art and Perspectives |url=https://onlinelibrary.wiley.com/doi/10.1002/qute.201900052 |journal=Advanced Quantum Technologies |language=en |volume=3 |issue=3 |pages=1900052 |doi=10.1002/qute.201900052 |arxiv=1907.03505 |s2cid=195833616 |issn=2511-9044}}</ref> Researchers are exploring several technologies as candidates for reliable qubit implementations.{{sfn|Grumbling|Horowitz|2019|page=127}} [[Superconductors]] and [[trapped ion]]s are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.{{sfn|Grumbling|Horowitz|2019|page=114}}

In February 2012 [[IBM]] scientists said that they had made several breakthroughs in quantum computing with superconducting integrated circuits.<ref>{{cite web|url=http://www.pcmag.com/article2/0,2817,2400930,00.asp|title=IBM Says It's 'On the Cusp' of Building a Quantum Computer|work=PCMAG|accessdate=2014-10-26}}</ref>

For example, [[topological quantum computer]] approaches are being explored for more fault-tolerance computing systems.<ref>{{Cite journal |last1=Nayak |first1=Chetan |last2=Simon |first2=Steven H. |last3=Stern |first3=Ady |last4=Freedman |first4=Michael |last5=Das Sarma |first5=Sankar |date=2008-09-12 |title=Non-Abelian anyons and topological quantum computation |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.80.1083 |journal=Reviews of Modern Physics |volume=80 |issue=3 |pages=1083–1159 |doi=10.1103/RevModPhys.80.1083|arxiv=0707.1889 |bibcode=2008RvMP...80.1083N }}</ref>

The first quantum logic gates were implemented with [[trapped ion]]s and prototype general purpose machines with up to 20 qubits have been realized. However, the technology behind these devices combines complex vacuum equipment, lasers, microwave and radio frequency equipment making full scale processors difficult to integrate with standard computing equipment. Moreover, the trapped ion system itself has engineering challenges to overcome.{{sfn|Grumbling|Horowitz|2019|page=119}}

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.<ref>{{cite web|url=http://www.futurity.org/science-technology/quantum-computer-built-inside-diamond/|title=Quantum computer built inside diamond|work=Futurity|accessdate=2014-10-26}}</ref>

The largest commercial systems are based on [[superconductor]] devices and have scaled to 2000 qubits. However, the error rates for larger machines have been on the order of 5%. Technologically these devices are all cryogenic and scaling to large numbers of qubits requires wafer-scale integration, a serious engineering challenge by itself.{{sfn|Grumbling|Horowitz|2019|page=126}}

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.<ref>{{cite web|title=Australian engineers write quantum computer 'qubit' in global breakthrough|url=http://www.theaustralian.com.au/australian-it/government/australian-engineers-write-quantum-computer-qubit-in-global-breakthrough/story-fn4htb9o-1226477592578|work=The Australian|accessdate=2012-10-03}}</ref>

<ref>{{cite web|title=Breakthrough in bid to create first quantum computer|url=http://newsroom.unsw.edu.au/news/technology/breakthrough-bid-create-first-quantum-computer|publisher=University of New South Wales|accessdate=2012-10-03}}</ref>

== Potential applications ==

In October 2012, [[Nobel Prize]]s were presented to [[David J. Wineland]] and [[Serge Haroche]] for their basic work on understanding the quantum world, which may help make quantum computing possible.<ref name="NYT-20121014">{{cite web |last=Frank |first=Adam |title=Cracking the Quantum Safe |url=http://www.nytimes.com/2012/10/14/opinion/sunday/the-possibilities-of-quantum-information.html |date=October 14, 2012 |work=[[New York Times]] |accessdate=2012-10-14 }}</ref><ref name="NYT-20121009">{{cite web |last=Overbye |first=Dennis |title=A Nobel for Teasing Out the Secret Life of Atoms |url=http://www.nytimes.com/2012/10/10/science/french-and-us-scientists-win-nobel-physics-prize.html |date=October 9, 2012 |work=[[New York Times]] |accessdate=2012-10-14 }}</ref>

With focus on business management's point of view, the potential applications of quantum computing into four major categories are cybersecurity, data analytics and artificial intelligence, optimization and simulation, and data management and searching.<ref>{{cite web |last1=Leong |first1=Kelvin |last2=Sung |first2=Anna |date= November 2022 |title= What Business Managers Should Know About Quantum Computing? |url=http://journalofinterdisciplinarysciences.com/wp-content/uploads/2022/10/3-What-Business-Managers-Should-Know-About-Quantum-Computing.pdf |access-date=13 August 2023 |website=[[Journal of Interdisciplinary Sciences]] |language=en-US}}</ref>

In November 2012, the first [[quantum teleportation]] from one [[Macroscopic scale|macroscopic]] object to another was reported.<ref name="MIT-TR-20121115">{{cite web |subtitle=|title=First Teleportation from One Macroscopic Object to Another: The Physics arXiv Blog |url=http://www.technologyreview.com/view/507531/first-teleportation-from-one-macroscopic-object-to-another/ |date=November 15, 2012 |work=[[MIT Technology Review]] |accessdate=2012-11-17 }}</ref><ref name="Arxiv-20121113">{{cite journal |last=Bao |first=Xiao-Hui |last2=Xu |first2=Xiao-Fan |last3=Li |first3=Che-Ming |last4=Yuan |first4=Zhen-Sheng |last5=Lu |first5=Chao-Yang |last6=Pan |first6=Jian-wei |title=Quantum teleportation between remote atomic-ensemble quantum memories |date=November 13, 2012 |journal=[[arXiv]] |arxiv=1211.2892 }}</ref>

Other applications include healthcare (ie. drug discovery), financial modeling, and natural language processing.<ref>{{Cite web |title=10 Quantum Computing Applications & Examples to Know |url=https://builtin.com/hardware/quantum-computing-applications |access-date=2025-06-21 |website=Built In |language=en}}</ref>

In December 2012, the first dedicated quantum computing software company, [[1QBit]] was founded in Vancouver, BC.<ref>{{cite web|title=1QBit Founded|url=http://1qbit.com/press.html|website=1QBit.com|accessdate=2014-06-22}}</ref> 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.<ref>{{cite web|title=1QBit Research|url=http://1qbit.com/research.html|website=1QBit.com|accessdate=2014-06-22}}</ref>

== Theory<span class="anchor" id="Relation to computability and complexity theory"></span> ==

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

=== Computability<span class="anchor" id="Computability theory"></span> ===

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.<ref>{{cite web|title=Launching the Quantum Artificial Intelligence Lab|url=http://googleresearch.blogspot.co.uk/2013/05/launching-quantum-artificial.html|publisher=Research@Google Blog|accessdate=2013-05-16}}</ref>

{{Further|Computability theory}}

Any [[computational problem]] solvable by a classical computer is also solvable by a quantum computer.{{sfn|Nielsen|Chuang|2010|p=29}} 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.

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]].<ref>{{cite news|title=NSA seeks to build quantum computer that could crack most types of encryption|url=http://www.washingtonpost.com/world/national-security/nsa-seeks-to-build-quantum-computer-that-could-crack-most-types-of-encryption/2014/01/02/8fff297e-7195-11e3-8def-a33011492df2_story.html|newspaper=Washington Post|date=January 2, 2014}}</ref>

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. 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 problem]]s like the [[halting problem]], and the existence of quantum computers does not disprove the [[Church–Turing thesis]].{{sfn|Nielsen|Chuang|2010|p=126}}

In 2014, a group of researchers from [[ETH Zürich]], [[University of Southern California|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.<ref>[http://arxiv.org/pdf/1401.2910v1.pdf 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.</ref><ref>{{cite journal |url=http://blogs.scientificamerican.com/observations/2014/06/19/quantum-chaos-after-a-failed-speed-test-the-d-wave-debate-continues/ |title=Quantum Chaos: After a Failed Speed Test, the D-Wave Debate Continues |journal=Scientific American |date=2014-06-19}}</ref>

=== Complexity<span class="anchor" id="Quantum complexity theory"></span> ===

In 2014, researchers at [[University of New South Wales]] used silicon as a protectant shell around [[qubit]]s, making them more accurate, increasing the length of time they will hold information and possibly made quantum computers easier to build.<ref>{{cite journal |url=http://www.computerworld.com/article/2837813/researchers-use-silicon-to-push-quantum-computing-toward-reality.html |title=Researchers use silicon to push quantum computing toward reality |journal=Computer World |date=23 October 2014 |first=Sharon |last=Gaudin}}</ref>

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.<ref>[http://www-03.ibm.com/press/us/en/pressrelease/46725.wss IBM Scientists Achieve Critical Steps to Building First Practical Quantum Computer]</ref>

== Relation to computational complexity theory ==

{{Main|Quantum complexity theory}}

<!-- Power and limits of quantum computers -->

[[File:BQP complexity class diagram.svg|thumb|The suspected relationship of BQP to other problem spaces.<ref>Nielsen, p. 42</ref>]]

While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently [[integer factorization|factor integers]], while this is not believed to be the case for classical computers.

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 algorithm|probabilistic]] algorithms, so BQP on quantum computers is the counterpart of [[Bounded-error probabilistic polynomial|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.<ref>Nielsen, p. 41</ref> 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 ''[[Sharp-P|#P]]'' (or more precisely in the associated class of decision problems ''P^#P''),<ref name=BernVazi>{{cite journal |last1=Bernstein |first1=Ethan |last2=Vazirani |first2=Umesh |doi=10.1137/S0097539796300921 |title=Quantum Complexity Theory |year=1997 |page=1411 |volume=26 |journal=SIAM Journal on Computing |url=http://www.cs.berkeley.edu/~vazirani/bv.ps |issue=5}}</ref> which is a subclass of [[PSPACE]].

<!-- Basic definition of BQP -->

BQP is suspected to be disjoint from [[NP-complete]] and a strict superset of [[P (complexity)|P]], but that is not known. Both [[integer factorization]] and [[discrete logarithm problem|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.<ref name=BernVazi />

The class of [[computational problem|problems]] that can be efficiently solved by a quantum computer with bounded error is called [[BQP]], for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to [[Bounded-error probabilistic polynomial|BPP]] ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time [[probabilistic Turing machine]]s with bounded error.{{sfn|Nielsen|Chuang|2010|p=41}} It is known that <math>\mathsf{BPP\subseteq BQP}</math> and is widely suspected that <math>\mathsf{BQP\subsetneq BPP}</math>, which intuitively would mean that quantum computers are more powerful than classical computers in terms of [[time complexity]].{{sfn|Nielsen|Chuang|2010|p=201}}

<!-- Relation of BQP to basic complexity classes -->

By the same token, it is often believed that quantum computers are generally superior to classical computers, that is, faster or at least as fast for any computing task. However, there is no proof of that, and it might not be true considering the next three observations:

[[File:BQP complexity class diagram.svg|thumb|The suspected relationship of BQP to several classical complexity classes{{sfn|Nielsen|Chuang|2010|p=42}}]]

* A quantum superposition of qubit states is basically a distributed representation of data to be processed. For some computing tasks, it can yield an exponential reduction from ''O''(2^''N'') to ''O''(1) in the amount of work and time needed to complete a computing task for input size ''N''.

The exact relationship of BQP to [[P (complexity)|P]], [[NP (complexity)|NP]], and [[PSPACE (complexity)|PSPACE]] is not known. However, it is known that <math>\mathsf{P\subseteq BQP \subseteq PSPACE}</math>; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer 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 known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that <math>\mathsf{NP\nsubseteq BQP}</math>; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of [[NP-complete]] problems (if an NP-complete problem were in BQP, then it would follow from [[NP-hard]]ness that all problems in NP are in BQP).<ref name=BernVazi>{{cite journal |last1=Bernstein |first1=Ethan |last2=Vazirani |first2=Umesh |doi=10.1137/S0097539796300921 |title=Quantum Complexity Theory |year=1997 |pages=1411–1473 |volume=26 |journal=SIAM Journal on Computing |url=http://www.cs.berkeley.edu/~vazirani/bv.ps |issue=5|citeseerx=10.1.1.144.7852 }}</ref>

* In classical computing, various kinds of distributed representations have been devised to reduce work and time for various computing tasks. This reduction is typically from ''O''(2^''N'') to ''O''(''N''²), but for at least one computing task, [[Transparallel processing|hyperstrings]] are distributed representations that enable an exponential reduction from ''O''(2^''N'') to ''O''(1).<ref>van der Helm, P. A. (2004). [http://www.pnas.org/cgi/reprint/101/30/10862.pdf Transparallel processing by hyperstrings.] ''Proceedings of the National Academy of Sciences USA, 101,'' 10862—10867. [http://dx.doi.org/10.1073/pnas.0403402101 doi:10.1073/pnas.0403402101].</ref><ref>van der Helm, P. A. (2015). Transparallel mind: Classical computing with quantum power. ''Artificial Intelligence Review'' (Online First). [http://dx.doi.org/10.1007/s10462-015-9429-7 doi:10.1007/s10462-015-9429-7].</ref>

* In the end, it always is the computing task at hand that determines if the data to be processed can be gathered in a distributed representation by which the task can be solved efficiently.

Hence, the interdependence between computing tasks and distributed representations suggests not only that quantum superpositions may be more efficient than classical distributed representations for some computing tasks, but also that the inverse may hold for other computing tasks.

== See also ==

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.<ref name=Ozhigov1>{{cite journal |last1=Ozhigov |first1=Yuri |title=Quantum Computers Speed Up Classical with Probability Zero |year=1999 |pages=1707–1714 |volume=10 |journal=Chaos Solitons Fractals |arxiv=quant-ph/9803064 |bibcode = 1998quant.ph..3064O |doi=10.1016/S0960-0779(98)00226-4 |issue=10 }}</ref> A similar fact takes place for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.<ref name=Ozhigov2>{{cite journal |last1=Ozhigov |first1=Yuri |title=Lower Bounds of Quantum Search for Extreme Point |year=1999 |pages=2165–2172 |volume=A455 |journal=Proceedings of the London Royal Society |arxiv=quant-ph/9806001 |bibcode = 1999RSPSA.455.2165O |doi = 10.1098/rspa.1999.0397 |issue=1986 }}</ref>

{{Commons category|Quantum computing}}

<!-- New links in alphabetical order please -->

{{cols|colwidth=21em}}

*{{annotated link|D-Wave Systems}}

*{{annotated link|Electronic quantum holography}}

*{{annotated link|Glossary of quantum computing}}

*{{annotated link|IARPA}}

*{{annotated link|India's quantum computer}}

*{{annotated link|IonQ}}

*{{annotated link|List of emerging technologies}}

*[[List of quantum computing journals]]

*{{annotated link|List of quantum processors}}

*{{annotated link|Magic state distillation}}

*{{annotated link|Metacomputing}}

*{{annotated link|Natural computing}}

*{{annotated link|Optical computing}}

*{{annotated link|Quantum bus}}

*{{annotated link|Quantum cognition}}

*{{annotated link|Quantum sensor}}

*{{annotated link|Quantum volume}}

*{{annotated link|Quantum weirdness}}

*{{annotated link|Rigetti Computing}}

*{{annotated link|Supercomputer}}

*{{annotated link|Theoretical computer science}}

*{{annotated link|Unconventional computing}}

*{{annotated link|Valleytronics}}

{{colend}}

== Notes ==

Although quantum computers may be faster than classical computers for some problems, those described above can't solve any problems that classical computers can't solve, given enough time and memory (however, those amounts might be practically infeasible). A [[Turing machine]] can simulate <!-- add mention about [[Quantum Virtual Machines]] which can simulate quantum computer on classical one -->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]].<ref>Nielsen, p. 126</ref> 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.<ref>[[Scott Aaronson]], ''[http://arxiv.org/abs/quant-ph/0502072 NP-complete Problems and Physical Reality]'', ACM [[SIGACT]] News, Vol. 36, No. 1. (March 2005), pp. 30–52, section 7 "Quantum Gravity": "[…] to anyone who wants

{{notelist}}

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)</ref>

== See also ==

*[[Chemical computer]]

*[[DNA computer]]

*[[Electronic quantum holography]]

*[[List of emerging technologies]]

*[[Natural computing]]

*[[Normal mode]]

*[[Photonic computing]]

*[[Post-quantum cryptography]]

*[[Quantum annealing]]

*[[Quantum bus]]

*[[Quantum cognition]]

*[[Quantum gate]]

*[[Quantum threshold theorem]]

*[[Soliton]]

*[[Timeline of quantum computing]]

*[[Topological quantum computer]]

*[[Valleytronics]]

== References ==

{{Reflist|30em}}

== Bibliography Sources==

* {{cite book | author1-linklast=Aaronson Michael Nielsen| author1first=Nielsen, MichaelScott |author2author-link = Isaac L. Chuang |author2=Chuang,Scott IsaacAaronson |title=Quantum ComputationComputing andSince QuantumDemocritus Information|date=2013 |publisher=Cambridge University Press |___location=Cambridge |year=2000 |isbn=978-0-521-6350319956-98 |oclc= 174527496829706638 |urldoi=http://books10.google.com1017/books?id=aai-P4V9GJ8C&printsec=frontcoverCBO9780511979309 }}

* {{cite book |doi=10.17226/25196 |title=Quantum Computing: Progress and Prospects |date=2019 |editor-first1=Emily |editor-last1=Grumbling |editor-first2=Mark |editor-last2=Horowitz |publisher=The National Academies Press |isbn=978-0-309-47970-7 |___location=Washington, DC |s2cid=125635007 |oclc=1091904777}}

<!-- These need to be inlined -->

* {{cite book |last1=Mermin |first1=N. David |author1-link=N. David Mermin |title=Quantum Computer Science: An Introduction |date=2007 |isbn=978-0-511-34258-5 |oclc=422727925 |doi=10.1017/CBO9780511813870 }}

*{{cite journal | author1-link=Derek Abbott |last1=Abbot |first1=Derek |author2-link= Charles R. Doering |last2=Doering |first2=Charles R. |author3-link= Carlton M. Caves |last3=Caves |first3=Carlton M. |author4-link=Daniel Lidar |last4=Lidar |first4=Daniel M. |author5-link= Howard Brandt|last5=Brandt |first5=Howard E. |author6-link= Alexander R. Hamilton |last6=Hamilton |first6=Alexander R. |author7-link=David K. Ferry |last7=Ferry |first7=David K. |author8-link=Julio Gea-Banacloche |last8=Gea-Banacloche |first8=Julio |author9-link=Sergey M. Bezrukov |last9=Bezrukov |first9=Sergey M. |author10-link=Laszlo B. Kish |first10=Laszlo B. |last10=Kish | title=Dreams versus Reality: Plenary Debate Session on Quantum Computing |journal=Quantum Information Processing |year=2003 |volume=2 |issue=6 |pages=449–472 |doi=10.1023/B:QINP.0000042203.24782.9a | arxiv=quant-ph/0310130 |id={{hdl|2027.42/45526}}}}

* {{cite book | author1-link= Michael Nielsen| last1=Nielsen |first1=Michael |author2-link = Isaac L. Chuang |last2=Chuang |first2=Isaac |title=[[Quantum Computation and Quantum Information]] |year=2010 |edition=10th anniversary |isbn=978-0-511-99277-3 |oclc= 700706156 |doi=10.1017/CBO9780511976667 | s2cid=59717455 }}

*DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". ''Experimental Proposals for Quantum Computation''. {{arxiv|quant-ph/0002077}}

* {{citeCite journalconference |last=DiVincenzoShor |first=DavidPeter PW. |date=1994 |title=Algorithms for Quantum Computation: Discrete Logarithms and Factoring |journalconference=[[Symposium on Foundations of Computer Science]] |year___location=1995[[Santa |volume=270Fe, New Mexico]] |issuepublisher=5234[[IEEE]] |pages=255–261124{{en dash}}134 |doi= 10.11261109/scienceSFCS.2701994.5234.255365700 |bibcodeisbn=978-0-8186-6580-6 |author-link=Peter 1995Sci...270..255D Shor}} Table 1 lists switching and dephasing times for various systems.

*{{cite journal |last=Feynman |first=Richard |authorlink=Richard Feynman |title=Simulating physics with computers |journal=International Journal of Theoretical Physics |volume=21 |page=467 |year=1982 |doi = 10.1007/BF02650179 |bibcode = 1982IJTP...21..467F | issue=6–7 }}

== Further reading ==

*{{cite book |last=Jaeger |first=Gregg |title=Quantum Information: An Overview |publisher=Springer |___location=Berlin |year=2006 |isbn=0-387-35725-4 |oclc=255569451}}

{{refbegin|30em}}

*{{cite book |last=Singer |first=Stephanie Frank |title=Linearity, Symmetry, and Prediction in the Hydrogen Atom |publisher=Springer |___location=New York |year=2005 |isbn=0-387-24637-1 |oclc= 253709076}}

*{{cite book |last=Benenti |first=Giuliano |title=Principles of Quantum Computation and Information Volume 1 | publisher=World Scientific |___location=New Jersey |year=2004 |isbn=981-238-830-3 |oclc= 179950736}}

===Textbooks===

*Lomonaco, Sam. [http://www.csee.umbc.edu/~lomonaco/Lectures.html#OxfordLectures Four Lectures on Quantum Computing given at Oxford University in July 2006]

* {{cite book |last1=Benenti |first1=Giuliano |last2=Casati |first2=Giulio |last3=Rossini |first3=Davide |last4=Strini |first4=Giuliano |title=Principles of Quantum Computation and Information: A Comprehensive Textbook |edition=2nd |year=2019 |isbn=978-981-3237-23-0 |oclc=1084428655 |doi=10.1142/10909 |s2cid=62280636 }}

*C. Adami, N.J. Cerf. (1998). "Quantum computation with linear optics". {{arxiv|quant-ph/9806048v1}}.

* {{cite book |last=Bernhardt |first=Chris |year=2019 |title=Quantum Computing for Everyone |publisher=MIT Press |oclc=1082867954 |isbn=978-0-262-35091-4 }}

*{{cite book

* {{cite book |editor-last1=Exman |editor-first1=Iaakov |editor-last2=Pérez-Castillo |editor-first2=Ricardo |editor-last3=Piattini |editor-first3=Mario |editor-last4=Felderer |editor-first4=Michael |title=Quantum Software: Aspects of Theory and System Design |year=2024 |url=https://link.springer.com/book/10.1007/978-3-031-64136-7 |isbn=978-3-031-64136-7 |publisher=[[Springer Nature]]|doi=10.1007/978-3-031-64136-7 }}

|last=Stolze |first=Joachim

* {{cite book |last=Hidary |first=Jack D. |year=2021 |edition=2nd |title=Quantum Computing: An Applied Approach |oclc=1272953643 |isbn=978-3-03-083274-2 |doi=10.1007/978-3-030-83274-2 |s2cid=238223274 }}

|last2=Suter |first2=Dieter

* {{cite book |editor-last1=Hiroshi |editor-first1=Imai |editor-last2=Masahito |editor-first2=Hayashi |year=2006 |title = Quantum Computation and Information: From Theory to Experiment |series=Topics in Applied Physics |volume=102 |isbn=978-3-540-33133-9 |doi=10.1007/3-540-33133-6 }}

|year = 2004

* {{cite book |last1=Hughes |first1=Ciaran |last2=Isaacson |first2=Joshua |last3=Perry |first3=Anastasia |last4=Sun |first4=Ranbel F. |last5=Turner |first5=Jessica |title=Quantum Computing for the Quantum Curious |isbn=978-3-03-061601-4 |oclc=1244536372 |doi=10.1007/978-3-030-61601-4 |year=2021 |s2cid=242566636 |url=https://link.springer.com/book/10.1007/978-3-030-61601-4 }}

|title = Quantum Computing

* {{cite book |last=Jaeger |first=Gregg |title=Quantum Information: An Overview |year=2007 |isbn=978-0-387-36944-0 |oclc=186509710 |doi=10.1007/978-0-387-36944-0 }}

|publisher = Wiley-VCH

* {{cite book |last1=Johnston |first1=Eric R. |last2=Harrigan |first2=Nic |last3=Gimeno-Segovia |first3=Mercedes |title=Programming Quantum Computers: Essential Algorithms and Code Samples |year=2019 |publisher=O'Reilly Media, Incorporated |oclc=1111634190 |isbn=978-1-4920-3968-6 }}

|isbn = 3-527-40438-4

* {{cite book |last1=Kaye |first1=Phillip |last2=Laflamme |first2=Raymond |last3=Mosca |first3=Michele |author-link2=Raymond Laflamme |author-link3=Michele Mosca |title=An Introduction to Quantum Computing |year=2007 |publisher=OUP Oxford |oclc=85896383 |isbn=978-0-19-857000-4 }}

}}

* {{cite book |last1=Kitaev |first1=Alexei Yu. |author-link1=Alexei Kitaev |last2=Shen |first2=Alexander H. |last3=Vyalyi |first3=Mikhail N. |title=Classical and Quantum Computation |year=2002 |publisher=American Mathematical Soc. |oclc=907358694 |isbn=978-0-8218-3229-5 }}

*{{cite web

* {{cite book |last1= Kurgalin|first1= Sergei|last2 = Borzunov|first2 = Sergei|date= 2021|title= Concise Guide to Quantum Computing: Algorithms, Exercises, and Implementations|url= https://dx.doi.org/10.1007/978-3-030-65052-0|publisher= Springer|doi= 10.1007/978-3-030-65052-0|isbn= 978-3-030-65052-0}}

|last=Mitchell |first=Ian

* {{cite book |last1=Stolze |first1=Joachim |last2=Suter |first2=Dieter |year=2004 |title =Quantum Computing: A Short Course from Theory to Experiment |isbn =978-3-527-61776-0 |oclc=212140089 |doi=10.1002/9783527617760 }}

|year = 1998

* {{Cite book |last1=Susskind |first1=Leonard |title=Quantum Mechanics: The Theoretical Minimum |last2=Friedman |first2=Art |date=2014 |publisher=[[Basic Books]] |isbn=978-0-465-08061-8 |___location=[[New York City|New York]] |author-link=Leonard Susskind}}

|title = Computing Power into the 21st Century: Moore's Law and Beyond

* {{cite book |last=Wichert |first=Andreas |year=2020 |title=Principles of Quantum Artificial Intelligence: Quantum Problem Solving and Machine Learning |edition=2nd |doi=10.1142/11938 |isbn=978-981-12-2431-7 |s2cid=225498497 |oclc=1178715016 }}

|url = http://citeseer.ist.psu.edu/mitchell98computing.html

* {{cite book |last=Wong |first=Thomas |title=Introduction to Classical and Quantum Computing |publisher=Rooted Grove |year=2022 |isbn=979-8-9855931-0-5 |oclc=1308951401 |url=http://www.thomaswong.net/introduction-to-classical-and-quantum-computing-1e.pdf |access-date=6 February 2022 |archive-date=29 January 2022 |archive-url=https://web.archive.org/web/20220129214631/http://www.thomaswong.net/introduction-to-classical-and-quantum-computing-1e.pdf |url-status=dead }}

}}

* {{cite book |last1=Zeng |first1=Bei |last2=Chen |first2=Xie |last3=Zhou |first3=Duan-Lu |last4=Wen |first4=Xiao-Gang |title=Quantum Information Meets Quantum Matter |year=2019 |oclc=1091358969 |isbn=978-1-4939-9084-9 |doi=10.1007/978-1-4939-9084-9 |arxiv=1508.02595 |s2cid=118528258 }}

*{{cite web

|last=Landauer |first=Rolf | authorlink=Rolf Landauer

===Academic papers===

|year = 1961

*{{cite journal | author1-link=Derek Abbott |last1=Abbot |first1=Derek |author2-link= Charles R. Doering |last2=Doering |first2=Charles R. |author3-link= Carlton M. Caves |last3=Caves |first3=Carlton M. |author4-link=Daniel Lidar |last4=Lidar |first4=Daniel M. |author5-link= Howard Brandt|last5=Brandt |first5=Howard E. |author6-link= Alexander R. Hamilton |last6=Hamilton |first6=Alexander R. |author7-link=David K. Ferry |last7=Ferry |first7=David K. |author8-link=Julio Gea-Banacloche |last8=Gea-Banacloche |first8=Julio |author9-link=Sergey M. Bezrukov |last9=Bezrukov |first9=Sergey M. |author10-link=Laszlo B. Kish |first10=Laszlo B. |last10=Kish |display-authors=5 |title=Dreams versus Reality: Plenary Debate Session on Quantum Computing |journal=Quantum Information Processing |year=2003 |volume=2 |issue=6 |pages=449–472 |doi=10.1023/B:QINP.0000042203.24782.9a | arxiv=quant-ph/0310130 |bibcode=2003QuIP....2..449A |hdl=2027.42/45526|s2cid=34885835 }}

|title = Irreversibility and heat generation in the computing process

*{{cite book |last=Berthiaume |first=Andre |title=Solution Manual for Quantum Mechanics |date=1 December 1998 |chapter=Quantum Computation |s2cid=128255429 |doi=10.1142/9789814541893_0016 |via=Semantic Scholar|pages=233–234 |isbn=978-981-4541-88-6 }}

|url = http://www.research.ibm.com/journal/rd/053/ibmrd0503C.pdf

*{{Cite journal |last1=DiVincenzo |first1=David P. |author1-link=David DiVincenzo |title=The Physical Implementation of Quantum Computation|journal=Fortschritte der Physik |volume=48|issue=9–11|pages=771–783|year=2000 |doi=10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E |arxiv=quant-ph/0002077 |bibcode=2000ForPh..48..771D |s2cid=15439711 }}

}}

*{{cite journal |last=DiVincenzo |first=David P. |title=Quantum Computation |journal=Science |year=1995 |volume=270 |issue=5234 |pages=255–261 |doi= 10.1126/science.270.5234.255 |bibcode = 1995Sci...270..255D |citeseerx=10.1.1.242.2165 |s2cid=220110562 }} Table 1 lists switching and dephasing times for various systems.

*{{cite book

*{{cite journal |last=Jeutner |first=Valentin |title=The Quantum Imperative: Addressing the Legal Dimension of Quantum Computers |journal=Morals & Machines |volume=1 |pages=52–59 |year=2021 |doi=10.5771/2747-5174-2021-1-52 |issue=1 |s2cid=236664155 |url=https://lup.lub.lu.se/record/e034e7b7-d17c-4863-9cee-7e654f97225b |doi-access=free }}

|last=Moore |first=Gordon E. | authorlink=Gordon E. Moore

*{{Cite journal |last1=Krantz |first1=P. |last2=Kjaergaard |first2=M. |last3=Yan |first3=F. |last4=Orlando |first4=T. P. |last5=Gustavsson |first5=S. |last6=Oliver |first6=W. D. |date=2019-06-17 |title=A Quantum Engineer's Guide to Superconducting Qubits |journal=[[Applied Physics Reviews]] |language=en |volume=6 |issue=2 |pages=021318 |doi=10.1063/1.5089550 |arxiv=1904.06560 |bibcode=2019ApPRv...6b1318K |s2cid=119104251 |issn=1931-9401}}

|year = 1965

*{{cite web |last=Mitchell |first=Ian |year=1998 |title=Computing Power into the 21st Century: Moore's Law and Beyond |url=http://citeseer.ist.psu.edu/mitchell98computing.html }}

|title = Cramming more components onto integrated circuits

*{{cite web |last = Simon |first = Daniel R. |year = 1994 |title = On the Power of Quantum Computation |publisher = Institute of Electrical and Electronics Engineers Computer Society Press |url = http://citeseer.ist.psu.edu/simon94power.html }}

|journal = Electronics Magazine

{{Refend}}

}}

*{{cite book

|last=Keyes |first=R. W.

|year = 1988

|title = Miniaturization of electronics and its limits

|journal=IBM Journal of Research and Development

}}

*{{cite web

|authorlink = Michael Nielsen|last=Nielsen |first=M. A.

|last2=Knill |first2=E. | author3-link=Raymond Laflamme|last3=Laflamme |first3=R.

|title = Complete Quantum Teleportation By Nuclear Magnetic Resonance

|url = http://citeseer.ist.psu.edu/595490.html

}}

*{{cite book

|last=Vandersypen |first=Lieven M.K.

|last2=Yannoni |first2=Constantino S. |last3=Chuang |first3=Isaac L.

|year = 2000

|title = Liquid state NMR Quantum Computing

}}

*{{cite book

|last=Hiroshi |first=Imai

|last2=Masahito |first2=Hayashi

|year = 2006

|title = Quantum Computation and Information

|publisher = Springer

|isbn = 3-540-33132-8

|___location = Berlin

}}

*{{cite web

|last=Berthiaume |first=Andre

|year = 1997

|title = Quantum Computation

|url = http://citeseer.ist.psu.edu/article/berthiaume97quantum.html

}}

*{{cite web

|last=Simon |first=Daniel R.

|year = 1994

|title = On the Power of Quantum Computation

|publisher = Institute of Electrical and Electronic Engineers Computer Society Press

|url = http://citeseer.ist.psu.edu/simon94power.html

}}

*{{cite web

|title = Seminar Post Quantum Cryptology

|publisher = Chair for communication security at the Ruhr-University Bochum

|url = http://www.crypto.rub.de/its_seminar_ss08.html

}}

*{{cite web

|last=Sanders |first=Laura

|year = 2009

|title = First programmable quantum computer created

|url = http://www.sciencenews.org/view/generic/id/49951/title/First_programmable_quantum_computer_created

}}

*{{cite web

|title = New trends in quantum computation

|url = http://insti.physics.sunysb.edu/itp/conf/simons-qcomputation2/

}}

== External links ==

*{{Commons-inline|Quantum computer}}

*{{Wikiversity inline|Quantum computing}}

*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/qt-quantcomp/ Quantum Computing]" by Amit Hagar.

* [[Stanford Encyclopedia of Philosophy]]: "[https://plato.stanford.edu/entries/qt-quantcomp/ Quantum Computing]" by Amit Hagar and Michael E. Cuffaro

*[http://www.quantiki.org/ Quantiki] – Wiki and portal with free-content related to quantum information science.

* {{springer|title=Quantum computation, theory of|id=p/q130020}}

*[http://www.scottaaronson.com/blog/ Scott Aaronson's blog]<!--Comes highly recommended by Tim Gowers-->, which features informative and critical commentary on developments in the field<!--and which delivers regular smackdowns of D-Wave rubbish-->

* [https://introtoquantum.org Introduction to Quantum Computing for Business by Koen Groenland]

*[http://www.inc.com/will-bourne/d-waves-dream-machine.html D-Wave thinks it has built the world's first commercial quantum computer. Mother Nature has other ideas], in the January 2014 issue of ''[[Inc. (magazine)|Inc. magazine]]''

* Schneider, J., & Smalley, I. (2024, August 5). ''What Is Quantum Computing? | IBM''. https://www.ibm.com/think/topics/quantum-computing

*[http://arxiv.org/pdf/1310.1339.pdf Quantum Annealing and Computation: A Brief Documentary Note], A. Ghosh and S. Mukherjee

*[http://www.lps.umd.edu/Quantum%20Computing%20Group/QuantumComputingIndex.html Maryland University Laboratory for Physical Sciences]: conducts researches for the quantum computer-based project led by the NSA, named 'Penetrating Hard Target'.

*[http://arnetminer.org/event/quantumComputingHistory Visualized history of quantum computing]

*[http://arxiv.org/pdf/0801.2193.pdf Quantum Annealing and Analog Quantum Computation by Arnab Das and BK Chakrabarti ]

;Lectures

* [https://courseswww.edxyoutube.orgcom/courses/BerkeleyX/CS-191xplaylist?list=PL1826E60FD05B44E4 Quantum Mechanicscomputing andfor Quantumthe Computationdetermined] – [[edX]]22 video courselectures by [[UmeshMichael VaziraniNielsen]]

* [http://www.quiprocone.org/Protected/DD_lectures.htm Video Lectures] by [[David Deutsch]]

*[http://www.youtube.com/playlist?list=PL1826E60FD05B44E4 Quantum computing for the determined] – 22 video lectures by [[Michael Nielsen]]

* Lomonaco, Sam. [http://www.csee.umbc.edu/~lomonaco/Lectures.html#OxfordLectures Four Lectures on Quantum Computing given at Oxford University in July 2006]

*[http://www.quiprocone.org/Protected/DD_lectures.htm Video Lectures] by [[David Deutsch]]

*[http://www.quantware.ups-tlse.fr/IHP2006/ Lectures at the Institut Henri Poincaré (slides and videos)]

*[http://nanohub.org/resources/4778 Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008)]

*{{YouTube|dWcT_qrBN_w|Quantum Computing research by Mikko Möttönen at Aalto University (video)}}

{{CPU technologies}}

{{Quantum computing}}

{{Emergingemerging technologies|quantum=yes|other=yes}}

{{Quantum mechanics topics}}

{{Computer science}}

{{Authority control}}

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[[Category:Quantum computing]]

[[Category:Models of computation]]

[[Category:Quantum cryptography]]

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