View source for Quantum computing

{{Short description|Technology that uses quantum mechanics}}
[[File:IBM Q system (Fraunhofer 2).jpg|thumb|upright=1.2|[[Engineering:IBM Q System One|IBM Q System One]], a quantum computer with 20 [[Superconducting quantum computing|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>]]
A '''quantum computer''' is a [[Computer|computer]] that exploits [[Physics:Quantum mechanical|quantum mechanical]] phenomena.
At small scales, physical matter exhibits properties of both particles and waves, and quantum computing leverages this behavior using specialized hardware.
[[Physics:Classical physics|Classical physics]] cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations exponentially faster than any modern "classical" computer.

In particular, a large-scale quantum computer could [[Post-quantum cryptography|break widely used encryption schemes]] and aid physicists in performing [[Quantum simulator|physical simulations]]; however, the current state of the art is largely experimental and impractical.

<!-- Basic principles of quantum computing -->
The basic unit of information in quantum computing is the [[Qubit|qubit]], similar to the [[Bit|bit]] in traditional digital electronics. Unlike a classical bit, a qubit can exist in a [[Physics:Quantum superposition|superposition]] of its two "basis" states, which loosely means that it is in both states simultaneously. When [[Physics:Measurement in quantum mechanics|measuring]] a qubit, the result is a [[Physics:Born rule|probabilistic output]] of a classical bit. If a quantum computer manipulates the qubit in a particular way, [[Physics:Wave interference|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.

<!--Physical implementations-->
Physically engineering high-quality qubits has proven challenging.
If a physical qubit is not sufficiently [[Physics:Isolated system|isolated]] from its environment, it suffers from [[Physics:Quantum decoherence|quantum decoherence]], introducing [[Noise (signal processing)|noise]] into calculations.
National governments have invested heavily in experimental research that aims to develop scalable qubits with longer coherence times and lower error rates.
Two of the most promising technologies are [[Superconducting quantum computing|superconductors]] (which isolate an electrical current by eliminating [[Physics:Electrical resistance|electrical resistance]]) and [[Trapped ion quantum computer|ion traps]] (which confine a single [[Physics:Atom|atomic particle]] using electromagnetic fields).

<!-- Computability and complexity -->
Any computational problem that can be solved by a classical computer can also be solved by a quantum computer.{{sfn|Nielsen|Chuang|2010|p=29}} Conversely, any problem that can be solved by a quantum computer can also be solved by a classical computer, at least in principle given enough time. In other words, quantum computers obey the [[Church–Turing thesis]]. This means that while quantum computers provide no additional advantages over classical computers in terms of [[Computability|computability]], quantum algorithms for certain problems have significantly lower [[Time complexity|time complexities]] than corresponding known classical algorithms. Notably, quantum computers are believed to be able to solve many problems quickly that no classical computer could solve in any ''feasible'' amount of time—a feat known as "[[Quantum supremacy|quantum supremacy]]." The study of the [[Computational complexity|computational complexity]] of problems with respect to quantum computers is known as [[Quantum complexity theory|quantum complexity theory]].

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

[[File:Mach-Zehnder photons animation.gif|thumb|The [[Physics:Mach–Zehnder interferometer|Mach–Zehnder interferometer]] shows that photons can exhibit wave-like interference.|upright=0.9]]
For many years, the fields of [[Physics:Quantum mechanics|quantum mechanics]] and [[Computer science|computer science]] formed distinct academic communities.{{sfn|Aaronson|2013|p=132}} Modern quantum theory developed in the 1920s to explain the wave{{en dash}}particle duality observed at atomic scales,<ref>{{Cite journal |last=Bhatta |first=Varun S. |date=2020-05-10 |title=Plurality of Wave–Particle Duality |url=https://wwwops.currentscience.ac.in/Volumes/118/09/1365.pdf |journal=Current Science |language=en-US |volume=118 |issue=9 |pages=1365 |doi=10.18520/cs/v118/i9/1365-1374 |issn=0011-3891 |s2cid=216143449}}</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 |location=Cambridge, Massachusetts |pages=3,46 |language=en-US |oclc=796812982}}</ref> Both disciplines had practical applications during [[Social:World War II|World War II]]; computers played a major role in wartime cryptography,<ref>{{Cite book |last=Hodges |first=Andrew |title=Alan Turing: The Enigma |publisher=Princeton University Press |year=2014 |isbn=9780691164724 |page=xviii |language=en-US}}</ref> and quantum physics was essential for the [[Physics:Nuclear physics|nuclear physics]] used in the [[Physics:Manhattan Project|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>

As physicists applied quantum mechanical models to computational problems and swapped digital [[Bit|bit]]s for qubits, the fields of quantum mechanics and computer science began to converge.
In 1980, [[Biography:Paul Benioff|Paul Benioff]] introduced the [[Quantum Turing machine|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|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>
When digital computers became faster, physicists faced an 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 [[Biography:Yuri Manin|Yuri Manin]] and [[Biography:Richard Feynman|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, Yu. I.| title=Vychislimoe i nevychislimoe| trans-title=Computable and Noncomputable| year=1980| publisher=Sov.Radio| pages=13–15| language=ru| url=http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5bdjv-fax%5d.zip| access-date=4 March 2013| url-status=dead| archive-date=10 May 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}}</ref><ref>{{cite journal |last1=Feynman |first1=Richard |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, [[Biography:Charles H. Bennett (physicist)|Charles Bennett]] and [[Biography:Gilles Brassard|Gilles Brassard]] applied quantum theory to [[Cryptography|cryptography]] protocols and demonstrated that quantum key distribution could enhance information security.<ref name="bb84">{{Cite conference |last1=Bennett |first1=Charles H. |last2=Brassard |first2=Gilles |author-link2=Gilles Brassard |date=December 1984 |title=Quantum cryptography: Public key distribution and coin tossing |arxiv=2003.06557 |conference=IEEE International Conference on Computers, Systems & Signal Processing |language=en |location=Bangalore |pages=175–179 |doi=10.1016/j.tcs.2014.05.025}}</ref><ref name="personal">{{Cite journal |last=Brassard |first=G. |date=2005 |title=Brief history of quantum cryptography: a personal perspective |url=https://ieeexplore.ieee.org/document/1543949 |journal=IEEE Information Theory Workshop on Theory and Practice in Information-Theoretic Security, 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>

[[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]]
[[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 journal |last1=Bernstein |first1=Ethan |last2=Vazirani |first2=Umesh |date=1993 |title=Quantum complexity theory |url=http://portal.acm.org/citation.cfm?doid=167088.167097 |journal=Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing – STOC '93 |language=en |location=San Diego, California, United States |publisher=ACM Press |pages=11–20 |doi=10.1145/167088.167097 |isbn=978-0-89791-591-5 |s2cid=676378}}</ref> and Simon's algorithm in 1994.<ref>{{Cite journal |last=Simon |first=D. R. |date=1994 |title=On the power of quantum computation |url=https://ieeexplore.ieee.org/document/365701 |journal=Proceedings 35th Annual Symposium on Foundations of Computer Science |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>
These algorithms did not solve practical problems, but demonstrated mathematically that one could gain more information by querying a [[Black box|black box]] with a quantum state in [[Physics:Quantum superposition|superposition]], sometimes referred to as ''quantum parallelism''.{{sfn|Nielsen|Chuang|2010|p=30-32}}
[[Biography:Peter Shor|Peter Shor]] built on these results with [[Shor's algorithm|his 1994 algorithms]] 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.{{sfn|National Academies of Sciences, Engineering, and Medicine|2019|p=15}}
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, [[Biography:Seth Lloyd|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, [[Physics:Experimental physics|experimentalists]] have constructed small-scale quantum computers using [[Trapped ion quantum computer|trapped ions]] and [[Superconducting quantum computing|superconductors]].{{sfn|National Academies of Sciences, Engineering, and Medicine|2019|pp=164-169}}
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=[[Organization:American Physical Society|American Physical Society]] |volume=80 |issue=15 |pages=3408–3411 |bibcode=1998PhRvL..80.3408C |doi=10.1103/PhysRevLett.80.3408}}</ref><ref>{{cite web |last=Holton |first=William Coffeen |title=quantum computer |url=https://www.britannica.com/technology/quantum-computer |access-date=4 Dec 2021 |website=Time-sharing {{!}} Definition, Advantages, & Facts {{!}} Britannica |publisher=[[Encyclopædia Britannica]]}}</ref> and subsequent experiments have increased the number of qubits and reduced error rates.{{sfn|National Academies of Sciences, Engineering, and Medicine|2019 |pp=164-169}}
In 2019, [[Company:Google AI|Google AI]] and [[Organization:NASA|NASA]] announced that they had achieved [[Quantum supremacy|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=[[Company:Google AI|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> However, the validity of this claim is still being actively researched.<ref>{{cite web |last=Pednault |first=Edwin |date=22 October 2019 |title=On 'Quantum Supremacy' |url=https://www.ibm.com/blogs/research/2019/10/on-quantum-supremacy/ |access-date=9 February 2021 |website=IBM Research Blog}}</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>

The [[Threshold theorem|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 |page=79 |arxiv=1801.00862 |doi=10.22331/q-2018-08-06-79 |doi-access=free |bibcode=2018Quant...2...79P |s2cid=44098998}}</ref>

Investment in quantum computing research has increased in the public and private sectors.<ref>{{cite journal |last1=Gibney |first1=Elizabeth |title=Quantum gold rush: the private funding pouring into quantum start-ups |journal=Nature |date=2 October 2019 |volume=574 |issue=7776 |pages=22–24 |doi=10.1038/d41586-019-02935-4 |doi-access=free |pmid=31578480 |bibcode=2019Natur.574...22G}}</ref><ref>{{Cite news |last=Rodrigo |first=Chris Mills |date=12 February 2020 |title=Trump budget proposal boosts funding for artificial intelligence, quantum computing |language=en-US |work=The Hill |url=https://thehill.com/policy/technology/482402-trump-budget-proposal-boosts-funding-for-artificial-intelligence-quantum |access-date=July 11, 2021}}</ref>
As one consulting firm summarized,<ref>{{cite web |last1=Biondi |first1=Matteo |last2=Heid |first2=Anna |last3=Henke |first3=Nicolaus |last4=Mohr |first4=Niko |last5=Pautasso |first5=Lorenzo |last6=Ostojic |first6=Ivan |last7=Wester |first7=Linde |last8=Zemmel |first8=Rodney |display-authors=5 |date=14 December 2021 |title=Quantum computing use cases are getting real—what you need to know |url=https://www.mckinsey.com/business-functions/mckinsey-digital/our-insights/quantum-computing-use-cases-are-getting-real-what-you-need-to-know |access-date=1 April 2022 |website=McKinsey & Company |language=en-US}}</ref>
{{blockquote|...{{nbsp}}investment dollars are pouring in, and quantum-computing start-ups are proliferating.{{nbsp}}... While quantum computing promises to help businesses solve problems that are beyond the reach and speed of conventional high-performance computers, use cases are largely experimental and hypothetical at this early stage.}}

== Quantum information processing ==

Computer engineers typically describe a modern computer's operation in terms of [[Physics:Classical electrodynamics|classical electrodynamics]].
Within these "classical" computers, some components (such as semiconductors and random number generators) may rely on quantum behavior, but these components are not [[Physics:Isolated system|isolated]] from their environment, so any [[Quantum information|quantum information]] quickly [[Physics:Quantum decoherence|decoheres]].
While programmers may depend on [[Probability theory|probability theory]] when designing a [[Randomized algorithm|randomized algorithm]], quantum mechanical notions like [[Physics:Quantum superposition|superposition]] and interference are largely irrelevant for [[Program analysis|program analysis]].

Quantum programs, in contrast, rely on precise control of coherent quantum systems. Physicists [[Physics:Mathematical formulation of quantum mechanics|describe these systems mathematically]] using [[Linear algebra|linear algebra]]. [[Complex number]]s model [[Physics:Probability amplitude|probability amplitude]]s, [[Vector (mathematics and physics)|vectors]] model [[Physics:Quantum state|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.

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.{{sfn|Nielsen|Chuang|2010}} This model is a [[Complex number|complex]] [[Linear algebra|linear-algebraic]] generalization of [[Boolean circuit|boolean circuit]]s.{{efn|The [[Engineering:Logic gate|classical logic gates]] such as [[Logical conjunction|AND]], [[Logical disjunction|OR]], [[Negation|NOT]], etc., that act on classical bits can be written as matrices, and used in the exact same way as [[Quantum logic gate|quantum logic gate]]s, as presented in this article. The same rules for [[Quantum logic gate#Serially wired gates|series]] and [[Quantum logic gate#Parallel gates|parallel]] quantum circuits can then also be used, and also [[Quantum logic gate#Unitary inversion of gates|inversion]] if the classical circuit is [[Reversible computing#Logical reversibility|reversible]].<br/>The equations used for describing NOT and CNOT (below) are the same for both the classical and quantum case (since they are not applied to superposition states).<br/>Unlike quantum gates, classical gates are often not [[Unitary matrix|unitary matrices]]. For example <math>\operatorname{OR} := \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \end{pmatrix}</math> and <math>\operatorname{AND} := \begin{pmatrix} 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}</math> which are not unitary.<br/>In the classical case, the matrix entries can only be 0s and 1s, while for quantum computers this is generalized to complex numbers.<ref name="qc-for-cs">{{Cite book|title=Quantum computing for computer scientists|last1=Yanofsky|first1=Noson S.|last2=Mannucci|first2=Mirco|date=2013|publisher={{wipe|Cambridge University Press}}|isbn=978-0-521-87996-5|pages=144–147, 158–169}}</ref>}}

=== Quantum information ===

The ''qubit'' serves as the basic unit of [[Quantum information|quantum information]].
It represents a two-state system, just like a classical bit, except that it can exist in a ''superposition'' of its two states.
In one sense, a superposition is like a probability distribution over the two values.
However, a quantum computation can be influenced by both values at once, inexplicable by either state individually.
In this sense, a "superposed" qubit stores both values simultaneously.

A two-dimensional [[Vector (mathematics and physics)|vector]] mathematically represents a qubit state. Physicists typically use Dirac notation for quantum mechanical [[Linear algebra|linear algebra]], writing {{ket|''&psi;''}} {{gloss|ket psi}} for a vector labeled ''&psi;''. Because a qubit is a two-state system, any qubit state takes the form {{nowrap|''α''{{ket|0}} + ''&beta;''{{ket|1}}}}, where {{ket|0}} and {{ket|1}} are the standard ''basis states'',{{efn|The [[Standard basis|standard basis]] is also the "computational basis".{{sfn|Mermin|2007|p=18}}}} and ''α'' and ''&beta;'' are the ''[[Physics:Probability amplitude|probability amplitude]]s''. If either ''α'' or ''&beta;'' 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|probability vector]], with one key difference: unlike probabilities, probability {{em|amplitudes}} are not necessarily positive numbers. Negative amplitudes allow for destructive wave interference.{{efn|In general, probability amplitudes are complex numbers.}}

When a qubit is [[Physics:Quantum measurement|measured]] in the [[Standard basis|standard basis]], the result is a classical bit.
The [[Physics:Born rule|Born rule]] describes the norm-squared correspondence between amplitudes and probabilities{{mdash}}when measuring a qubit {{nowrap|''α''{{ket|0}} + ''&beta;''{{ket|1}}}}, the state [[Physics:Wave function collapse|collapses]] to {{ket|0}} with probability {{abs|''α''}}<sup>2</sup>, or to {{ket|1}} with probability {{abs|''&beta;''}}<sup>2</sup>.
Any valid qubit state has coefficients ''α'' and ''&beta;'' such that {{nowrap|1={{abs|''α''}}<sup>2</sup> + {{abs|''&beta;''}}<sup>2</sup> = 1}}.
As an example, measuring the qubit {{nowrap|{{sfrac|1|√2}}{{ket|0}} + {{sfrac|1|√2}}{{ket|1}}}} would produce either {{ket|0}} or {{ket|1}} with equal probability.

Each additional qubit doubles the [[Dimension (vector space)|dimension]] of the [[Physics:State space|state space]].
As an example, the vector {{nowrap|{{sfrac|1|√2}}{{ket|00}} + {{sfrac|1|√2}}{{ket|01}}}} represents a two-qubit state, a [[Tensor product|tensor product]] of the qubit {{ket|0}} with the qubit {{nowrap|{{sfrac|1|√2}}{{ket|0}} + {{sfrac|1|√2}}{{ket|1}}}}.
This vector inhabits a four-dimensional [[Vector space|vector space]] spanned by the basis vectors {{ket|00}}, {{ket|01}}, {{ket|10}}, and {{ket|11}}.
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 their probability amplitudes are correlated.
In general, the vector space for an ''n''-qubit system is 2<sup>''n''</sup>-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2<sup>100</sup> classical values.

=== Unitary operators ===

{{Anchor|gate-application}}The state of this one-qubit quantum memory can be manipulated by applying [[Quantum logic gate|quantum logic gate]]s, analogous to how classical memory can be manipulated with [[Engineering: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]]
<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|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 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, 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 circuit|quantum circuit]]s depict a network consisting only of quantum logic gates and no measurements.

=== Quantum parallelism ===

''Quantum parallelism'' refers to the ability of quantum computers to evaluate 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.{{sfn|Nielsen|Chuang|2010|p=30-32}}

=== Quantum programming {{anchor|Models of computation for quantum computing}} ===
{{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.

==== Gate array {{anchor|Quantum circuit|Definition}} ====
[[File:Quantum Toffoli Gate Implementation.svg|thumb|A quantum circuit diagram implementing a [[Toffoli gate|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 gates. 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 circuit|quantum circuit]]s depict a network consisting only of quantum logic gates and no measurements.

Any quantum computation (which is, in the above formalism, any [[Unitary matrix|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]].

==== 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 gates applied to a highly entangled initial state (a [[Cluster state|cluster state]]), using a technique called [[Quantum gate teleportation|quantum gate teleportation]].

==== Adiabatic quantum computing ====

An [[Physics:Adiabatic quantum computation|adiabatic quantum computer]], based on [[Quantum annealing|quantum annealing]], decomposes computation into a slow continuous transformation of an initial [[Physics: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=[[Physics: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>

==== Topological quantum computing ====

A [[Topological quantum computer|topological quantum computer]] decomposes computation into the braiding of [[Anyon|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|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 journal|last=Chi-Chih Yao|first=A.|year=1993|title=Quantum circuit complexity|url=https://ieeexplore.ieee.org/document/366852|journal=Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science|pages=352–361|doi=10.1109/SFCS.1993.366852|isbn=0-8186-4370-6|s2cid=195866146}}</ref> 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.

== Communication ==
{{Further|Quantum information science}}

[[Quantum cryptography]] enables new ways to transmit data securely; for example, [[Quantum key distribution|quantum key distribution]] uses entangled quantum states to establish secure cryptographic keys.<ref>{{Cite journal |last=Pirandola |first=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. L. |last12=Razavi |first12=M. |last13=Shamsul Shaari |first13=J. |last14=Tomamichel |first14=M. |last15=Usenko |first15=V. C. |date=2020-12-14 |title=Advances in quantum cryptography |journal=Advances in Optics and Photonics |language=en |volume=12 |issue=4 |page=1017 |doi=10.1364/AOP.361502 |issn=1943-8206}}</ref> When a sender and receiver exchange quantum states, they can guarantee that the message is not intercepted, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change.<ref>{{Cite journal |last=Xu |first=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}}</ref> With the right cryptographic protocols, the sender and receiver can thus establish shared private information resistant to eavesdropping.<ref name="bb84" /><ref>{{Cite conference |last=Xu |first=Guobin |last2=Mao |first2=Jianzhou |last3=Sakk |first3=Eric |last4=Wang |first4=Shuangbao Paul |date=2023-03-22 |title=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>

== 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 [[Physics: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|discrete logarithm]]s, solving [[Pell's equation]], and more generally solving the [[Hidden subgroup problem|hidden subgroup problem]] for [[Abelian group|abelian]] finite groups.<ref name="zoo"/> These algorithms depend on the primitive of the [[Quantum Fourier transform|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=[[Organization:Association for Computing Machinery|Association for Computing Machinery]] |pages=333–342 |doi=10.1145/1993636.1993682 |isbn=978-1-4503-0691-1 |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|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|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}}

=== 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 cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few [[Prime number|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 public key ciphers are based on the difficulty of factoring integers or the [[Discrete logarithm|discrete logarithm]] problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

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 [[Biography:Tanja Lange|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|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|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<sup>''n''/2</sup> invocations of the underlying cryptographic algorithm, compared with roughly 2<sup>''n''</sup> 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 ''[[Social:Key size#Effect of quantum computing attacks on key strength|Key size]]'').

=== Search problems {{anchor|Quantum search}} ===
{{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 |last1=Brassard |first1=Gilles |title=Quantum Algorithm for the Collision Problem |year=2016 |encyclopedia=Encyclopedia of Algorithms |pages=1662–1664 |editor-last=Kao |editor-first=Ming-Yang |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-1-4939-2864-4_304 |isbn=978-1-4939-2864-4 |last2=Høyer |first2=Peter |last3=Tapp |first3=Alain |arxiv=quant-ph/9705002 |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=[[Company:Springer Science+Business Media|Springer]] |year=2011 |isbn=978-1-84628-887-6 |pages=242–244}}</ref><ref>{{cite arXiv |last=Grover| first=Lov| 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 [[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>

=== Simulation of quantum systems ===

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 [[Physics:Collider|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={{wipe|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.is/XBFmz |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 |volume=618 |issue=7965 |pages=500–505 |doi=10.1038/s41586-023-06096-3 |pmid=37316724 |pmc=10266970 }}</ref>

About 2% of the annual global energy output is used for [[Biology:Nitrogen fixation|nitrogen fixation]] to produce [[Chemistry:Ammonia|ammonia]] for the [[Chemistry:Haber process|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://ghostarchive.org/varchive/youtube/20211211/7susESgnDv8| archive-date=2021-12-11 |url-status=live |title=Lunch & Learn: Quantum Computing |publisher=[[Finance: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>

=== Quantum annealing {{anchor|Quantum annealing and adiabatic optimization}} ===

[[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. {{anchor|Computational biology}}Adiabatic optimization may be helpful for solving [[Biology:Computational biology|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|machine learning]] tasks.<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><ref name="preskill2018"/>

For example, the [[Quantum algorithm for linear systems of equations|quantum algorithm for linear systems of equations]], or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.<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><ref name="preskill2018"/> 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 |page=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 |pages=117628 |doi=10.1016/j.apenergy.2021.117628 |issn=0306-2619 |doi-access=free}}</ref>

{{anchor|Computer-aided drug design and generative chemistry}} 
Deep generative chemistry models emerge as powerful tools to expedite [[Medicine:Drug discovery|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 [[Engineering:Wafer (electronics)|wafer]] of adiabatic quantum computers]]

=== 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/computing/hardware/the-case-against-quantum-computing |title=The Case Against Quantum Computing |journal=IEEE Spectrum |date=15 November 2018}}</ref> Physicist 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 [[Physics:Decoherence|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 [[Company:Google|Google]] and [[Company:IBM|IBM]], need [[Physics:Helium-3|helium-3]], a [[Physics:Nuclear physics|nuclear]] research byproduct, and special [[Physics:Superconducting|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=[[Engineering:Nature Electronics|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 {{anchor|Quantum decoherence}} ====

One of the greatest challenges involved with constructing quantum computers is controlling or removing [[Physics:Quantum decoherence|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''<sub>2</sub> (for [[Physics: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">{{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 [[Physics:Dilution refrigerator|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.| bibcode=2022Cryo..121....1Z| 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 [[Physics:Ionizing radiation|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|pulse shaping]]. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

As described in the [[Threshold theorem|threshold theorem]], if the error rate is small enough, it is thought to be possible to use [[Quantum error correction|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<sup>−3</sup>, 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''<sup>2</sup>, where ''L'' is the number of 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 10<sup>4</sup> 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 10<sup>7</sup> bits. Computation time is about ''L''<sup>2</sup> or about 10<sup>7</sup> 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 |url=http://worldcat.org/oclc/923881411 |title=Architecture Framework for Trapped-ion Quantum Computer based on Performance Simulation Tool |date=2015 |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 }}</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 |pages=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|topological quantum computer]] with [[Anyon|anyon]]s, quasi-particles used as threads, and relying on [[Braid theory|braid theory]] to form stable logic gates.<ref>{{cite journal
 | last4 = Wang | first4 = Zhenghan
 | arxiv = quant-ph/0101025
 | doi = 10.1090/S0273-0979-02-00964-3
 | issue = 1
 | journal = Bulletin of the American Mathematical Society
 | mr = 1943131
 | pages = 31–38
 | title = Topological quantum computation
 | volume = 40
 | year = 2003}}</ref><ref>{{cite journal |last=Monroe |first=Don |url=https://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>

=== Quantum supremacy ===

''[[Quantum supremacy]]'' is a term coined by [[Biography:John Preskill|John Preskill]] referring to 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 |pages=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>

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 [[Engineering:Sycamore processor|Sycamore quantum computer]] more than 3,000,000 times faster than they could be done on 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 journal |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 |display-authors=5 |date=2021-11-14 |title=Closing the "quantum supremacy" gap: achieving real-time simulation of a random quantum circuit using a new Sunway supercomputer |journal=Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis |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|page=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= |doi=10.1126/science.ade2364}}</ref><ref>{{Cite web |title=Google's 'quantum supremacy' usurped by researchers using ordinary supercomputer |url=https://social.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>

In December 2020, a group at 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 |doi-access=free}}</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>

=== Skepticism ===

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, as of 2023 quantum computers are "good for ... nothing", per Nature spotlight article.<ref>
{{Cite web| url = https://www.nature.com/articles/d41586-023-01692-9 | website = Nature | title = Quantum computers: what are they good for? | date = 24 May 2023 | first = Michael | last = Brooks}}
</ref> In other words, conventional computers are currently more efficient for practical uses and applications. This state of affairs can be traced to several current and long-term considerations. 

* Conventional computer hardware and algorithms are not only optimized for practical tasks, but are still improving rapidly. 
* Current quantum computing hardware generates only a limited amount of [[Quantum entanglement|entanglement]] before getting overwhelmed by noise and does not rule out practical simulation on conventional computers, possibly except for contrived cases.
* [[Quantum algorithm|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. In particular, processing large amounts of non-quantum data is a challenge for quantum computers.
* If [[Quantum error correction|quantum error correction]] is used to scale quantum computers to practical applications, its overhead may undermine speedup offered by many quantum 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 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.

Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons.

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> [[Biography:Paul Davies|Paul Davies]] argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the [[Physics:Holographic principle|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 [[Biography:Gil Kalai|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 [[Biography:Mikhail Dyakonov|Mikhail Dyakonov]] has expressed skepticism of quantum computing as follows:
:"So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10<sup>300</sup>... Could we ever learn to control the more than 10<sup>300</sup> 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/computing/hardware/the-case-against-quantum-computing |website=IEEE Spectrum |date=15 November 2018 |access-date=3 December 2019}}</ref><ref>{{cite book |last1=Dyakonov |first1=Mikhail |title=Will We Ever Have a Quantum Computer? |date=24 March 2020 |url=https://www.springer.com/gp/book/9783030420185 |publisher=Springer |isbn=9783030420185 |access-date=22 May 2020}}{{page needed|date=May 2020}}</ref>

=== Candidates for physical realizations ===

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

*[[Superconducting quantum computing]]<ref name="ClarkeWilhelm2008">{{cite journal |last1=Clarke |first1=John |last2=Wilhelm |first2=Frank K. |title=Superconducting quantum bits |journal=Nature |date=18 June 2008 |volume=453 |issue=7198 |pages=1031–1042 |doi=10.1038/nature07128 |pmid=18563154 |bibcode=2008Natur.453.1031C |s2cid=125213662 |url=https://www.semanticscholar.org/paper/7ee1053ce63f33a62f2ea555547c514ce5f21054}}</ref><ref>{{cite arXiv |last1=Kaminsky |first1=William M. |last2=Lloyd |first2=Seth |last3=Orlando |first3=Terry P. |title=Scalable Superconducting Architecture for Adiabatic Quantum Computation |eprint=quant-ph/0403090 |date=12 March 2004}} {{bibcode|2004quant.ph..3090K}}</ref> (qubit implemented by the state of nonlinear resonant superconducting circuits containing Josephson junctions)
*[[Trapped ion quantum computer]] (qubit implemented by the internal state of trapped ions)
*Neutral atoms in [[Physics:Optical lattice|optical lattice]]s (qubit implemented by internal states of neutral atoms trapped in an optical lattice)<ref>{{Cite journal |last1=Khazali |first1=Mohammadsadegh |last2=Mølmer |first2=Klaus |date=11 June 2020 |title=Fast Multiqubit Gates by Adiabatic Evolution in Interacting Excited-State Manifolds of Rydberg Atoms and Superconducting Circuits |journal=Physical Review X |volume=10 |issue=2 |page=021054 |doi=10.1103/PhysRevX.10.021054 |arxiv=2006.07035 |bibcode=2020PhRvX..10b1054K |doi-access=free}}</ref><ref>{{Cite journal |last1=Henriet |first1=Loic |last2=Beguin |first2=Lucas |last3=Signoles |first3=Adrien |last4=Lahaye |first4=Thierry |last5=Browaeys |first5=Antoine |last6=Reymond |first6=Georges-Olivier |last7=Jurczak |first7=Christophe |date=22 June 2020 |title=Quantum computing with neutral atoms |journal=Quantum |volume=4 |page=327 |doi=10.22331/q-2020-09-21-327 |arxiv=2006.12326 |bibcode=2020Quant...4..327H |s2cid=219966169}}</ref>
*[[Physics:Quantum dot|Quantum dot]] computer, spin-based (e.g. the Loss-DiVincenzo quantum computer<ref>{{cite journal |last1=Imamog¯lu |first1=A. |last2=Awschalom |first2=D. D. |last3=Burkard |first3=G. |last4=DiVincenzo |first4=D. P. |last5=Loss |first5=D. |last6=Sherwin |first6=M. |last7=Small |first7=A. |title=Quantum Information Processing Using Quantum Dot Spins and Cavity QED |journal=Physical Review Letters |date=15 November 1999 |volume=83 |issue=20 |pages=4204–4207 |doi=10.1103/PhysRevLett.83.4204 |s2cid=18324734 |bibcode=1999PhRvL..83.4204I |arxiv=quant-ph/9904096}}</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 |last1=Fedichkin |first1=L. |last2=Yanchenko |first2=M. |last3=Valiev |first3=K. A. |title=Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot |journal=Quantum Computers and Computing |date=June 2000 |volume=1 |page=58 |arxiv=quant-ph/0006097 |bibcode=2000quant.ph..6097F}}</ref>
* Quantum computing using engineered [[Physics:Quantum well|quantum well]]s, which could in principle enable the construction of a quantum computer that operates at room temperature<ref>{{cite journal |last1=Ivády |first1=Viktor |last2=Davidsson |first2=Joel |last3=Delegan |first3=Nazar |last4=Falk |first4=Abram L. |last5=Klimov |first5=Paul V. |last6=Whiteley |first6=Samuel J. |last7=Hruszkewycz |first7=Stephan O. |last8=Holt |first8=Martin V. |last9=Heremans |first9=F. Joseph |last10=Son |first10=Nguyen Tien |last11=Awschalom |first11=David D. |last12=Abrikosov |first12=Igor A. |last13=Gali |first13=Adam |display-authors=5 |title=Stabilization of point-defect spin qubits by quantum wells |journal=Nature Communications |date=6 December 2019 |volume=10 |issue=1 |page=5607 |doi=10.1038/s41467-019-13495-6 |pmid=31811137 |pmc=6898666 |arxiv=1905.11801 |bibcode=2019NatCo..10.5607I }}</ref><ref>{{cite news |title=Scientists Discover New Way to Get Quantum Computing to Work at Room Temperature |url=https://interestingengineering.com/scientists-discover-new-way-to-get-quantum-computing-to-work-at-room-temperature |work=interestingengineering.com |date=24 April 2020 }}</ref>
*Coupled [[Physics:Quantum wire|quantum wire]] (qubit implemented by a pair of quantum wires coupled by a [[Physics:Quantum point contact|quantum point contact]])<ref>{{cite journal |last1=Bertoni |first1=A. |last2=Bordone |first2=P. |last3=Brunetti |first3=R. |last4=Jacoboni |first4=C. |last5=Reggiani |first5=S. |title=Quantum Logic Gates based on Coherent Electron Transport in Quantum Wires |journal=Physical Review Letters |date=19 June 2000 |volume=84 |issue=25 |pages=5912–5915 |doi=10.1103/PhysRevLett.84.5912 |hdl=11380/303796 |hdl-access=free |pmid=10991086 |bibcode=2000PhRvL..84.5912B}}</ref><ref>{{cite journal |last1=Ionicioiu |first1=Radu |last2=Amaratunga |first2=Gehan |last3=Udrea |first3=Florin |title=Quantum Computation with Ballistic Electrons |journal=International Journal of Modern Physics B |date=20 January 2001 |volume=15 |issue=2 |pages=125–133 |doi=10.1142/S0217979201003521 |arxiv=quant-ph/0011051 |bibcode=2001IJMPB..15..125I |citeseerx=10.1.1.251.9617 |s2cid=119389613 }}</ref><ref>{{cite journal |last1=Ramamoorthy |first1=A |last2=Bird |first2=J. P. |last3=Reno |first3=J. L. |date=11 July 2007 |title=Using split-gate structures to explore the implementation of a coupled-electron-waveguide qubit scheme |journal=Journal of Physics: Condensed Matter |volume=19 |issue=27 |page=276205 |bibcode=2007JPCM...19A6205R |doi=10.1088/0953-8984/19/27/276205 |s2cid=121222743}}</ref>
*[[Physics:Nuclear magnetic resonance quantum computer|Nuclear magnetic resonance quantum computer]] (NMRQC) implemented with the [[Physics:Nuclear magnetic resonance|nuclear magnetic resonance]] of molecules in solution, where qubits are provided by [[Physics:Nuclear spin|nuclear spin]]s within the dissolved molecule and probed with radio waves
*Solid-state NMR [[Kane quantum computer]] (qubit realized by the nuclear spin state of [[Chemistry:Phosphorus|phosphorus]] [[Chemistry:Electron donor|donors]] in [[Chemistry:Silicon|silicon]])
*Vibrational quantum computer (qubits realized by vibrational superpositions in cold [[Physics:Molecule|molecule]]s)<ref>{{cite journal |author1=Berrios |first=Eduardo |last2=Gruebele |first2=Martin |last3=Shyshlov |first3=Dmytro |last4=Wang |first4=Lei |last5=Babikov |first5=Dmitri |year=2012 |title=High fidelity quantum gates with vibrational qubits |journal=Journal of Chemical Physics |volume=116 |issue=46 |pages=11347–11354 |bibcode=2012JPCA..11611347B |doi=10.1021/jp3055729 |pmid=22803619}}</ref>
*Electrons-on-helium quantum computer (qubit is the electron spin)
*[[Cavity quantum electrodynamics]] (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
*[[Physics:Single-molecule magnet|Molecular magnet]]<ref>{{cite journal |last1=Leuenberger |first1=Michael N. |last2=Loss |first2=Daniel |title=Quantum computing in molecular magnets |journal=Nature |date=April 2001 |volume=410 |issue=6830 |pages=789–793 |doi=10.1038/35071024 |pmid=11298441 |arxiv=cond-mat/0011415 |bibcode=2001Natur.410..789L |s2cid=4373008 }}</ref> (qubit given by spin states)
*[[Chemistry:Fullerene|Fullerene]]-based [[Physics:Electron paramagnetic resonance|ESR]] quantum computer (qubit based on the electronic spin of [[Chemistry:Endohedral fullerene|atoms or molecules encased in fullerenes]])<ref>{{cite journal |last1=Harneit |first1=Wolfgang |title=Fullerene-based electron-spin quantum computer |journal= Physical Review A|date=27 February 2002 |volume=65 |issue=3 |page=032322 |doi=10.1103/PhysRevA.65.032322 |bibcode=2002PhRvA..65c2322H |url=https://www.researchgate.net/publication/257976907}}</ref>
*Nonlinear optical quantum computer (qubits realized by processing states of different [[Physics:Normal mode|modes]] of light through both linear and [[Physics:Nonlinear optics|nonlinear]] elements)<ref name="qc1988">{{cite conference |first1=K. |last1=Igeta |first2=Y. |last2=Yamamoto |title=Quantum mechanical computers with single atom and photon fields |conference=International Quantum Electronics Conference |year=1988 |url=https://www.osapublishing.org/abstract.cfm?uri=IQEC-1988-TuI4}}</ref><ref name="chuang1995">{{cite journal |last1=Chuang |first1=I. L. |last2=Yamamoto |first2=Y. |year=1995 |title=Simple quantum computer |journal=Physical Review A |volume=52 |issue=5 |pages=3489–3496 |arxiv=quant-ph/9505011 |bibcode=1995PhRvA..52.3489C |doi=10.1103/PhysRevA.52.3489 |pmid=9912648 |s2cid=30735516}}</ref>
*[[Linear optical quantum computing|Linear optical quantum computer]] (LOQC) (qubits realized by processing states of different [[Physics:Normal mode|modes]] of light through linear elements e.g. mirrors, [[Physics:Beam splitter|beam splitter]]s and [[Engineering: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 |s2cid=4362012 |url=https://www.semanticscholar.org/paper/054b680165a7325569ca6e63028ca9cee7f3ac9a }}</ref> Quantum microprocessor based on laser photonics at room temperature made possible.<ref>{{Cite web |date=2023-05-06 |title=Indian scientist among those who made building blocks of quantum computer |url=https://www.deccanherald.com/business/technology/indian-scientist-among-those-who-made-building-blocks-of-quantum-computer-1216384.html |access-date=2023-05-07 |website=Deccan Herald |language=en}}</ref><ref>{{Cite web |date=2022-08-07 |title=Traditional hardware can match Google's quantum computer performance: Researchers |url=https://www.deccanherald.com/science-and-environment/traditional-hardware-can-match-googles-quantum-computer-performance-researchers-1134055.html |access-date=2023-05-07 |website=Deccan Herald |language=en}}</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 |s2cid = 122596827
|url = https://www.semanticscholar.org/paper/a7598ca24265e5537f14dc61b7c3a1d5b5953162
}}</ref><ref>{{cite journal |last1=Dutt |first1=M. V. G. |last2=Childress |first2=L. |last3=Jiang |first3=L. |last4=Togan |first4=E. |last5=Maze |first5=J. |last6=Jelezko |first6=F. |last7=Zibrov |first7=A. S. |last8=Hemmer |first8=P. R. |last9=Lukin |first9=M. D. |display-authors=5 |title=Quantum Register Based on Individual Electronic and Nuclear Spin Qubits in Diamond |journal=Science |date=1 June 2007 |volume=316 |issue=5829 |pages=1312–1316 |doi=10.1126/science.1139831 |pmid=17540898 |bibcode=2007Sci...316.....D |s2cid=20697722 }}</ref><ref>{{cite web |author=Baron |first=David |date=June 7, 2007 |title=At room temperature, carbon-13 nuclei in diamond create stable, controllable quantum register |url=https://news.harvard.edu/gazette/story/2007/06/single-spinning-nuclei-in-diamond-offer-a-stable-quantum-computing-building-block/ |publisher=The Harvard Gazette, FAS Communications}}</ref><ref name="Neumannetal2008">{{cite journal
|journal = Science
|date = 6 June 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 = 5
|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. |s2cid = 8892596
}}</ref> (qubit realized by the electronic or nuclear spin of [[Physics:Nitrogen-vacancy center|nitrogen-vacancy center]]s in diamond)
*[[Physics:Bose–Einstein condensate|Bose-Einstein condensate]]-based quantum computer<ref>{{cite journal |last1=Anderlini |first1=Marco |last2=Lee |first2=Patricia J. |last3=Brown |first3=Benjamin L. |last4=Sebby-Strabley |first4=Jennifer |last5=Phillips |first5=William D. |last6=Porto |first6=J. V. |title=Controlled exchange interaction between pairs of neutral atoms in an optical lattice |journal=Nature |date=July 2007 |volume=448 |issue=7152 |pages=452–456 |doi=10.1038/nature06011 |pmid=17653187 |arxiv=0708.2073 |bibcode=2007Natur.448..452A |s2cid=4410355 }}</ref><ref>{{cite journal|title=Thousands of Atoms Swap 'Spins' with Partners in Quantum Square Dance|url=https://www.nist.gov/news-events/news/2007/07/thousands-atoms-swap-spins-partners-quantum-square-dance|journal=[[Organization:National Institute of Standards and Technology|NIST]]|date=January 8, 2018}}</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 computer<ref name="Ohlsson2002">{{cite journal
|journal = Opt. Commun.
|date = 1 January 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 = 23 September 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 |s2cid = 41374015
}}</ref> (qubit realized by the internal electronic state of [[Chemistry:Dopant|dopant]]s in [[Engineering:Optical fiber|optical fiber]]s)
*Metallic-like carbon nanospheres-based quantum computer<ref name="Nafradi2016">{{cite journal |last1=Náfrádi |first1=Bálint |last2=Choucair |first2=Mohammad |last3=Dinse |first3=Klaus-Peter |last4=Forró |first4=László |title=Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres |journal=Nature Communications |date=18 July 2016 |volume=7 |issue=1 |page=12232 |doi=10.1038/ncomms12232 |pmid=27426851 |pmc=4960311 |arxiv=1611.07690 |bibcode=2016NatCo...712232N }}</ref>

The large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.<ref>{{Cite web |last=Naveh |first=Yehuda |title=Council Post: Quantum Software Development Is Still In Its Infancy |url=https://www.forbes.com/sites/forbestechcouncil/2021/06/23/quantum-software-development-is-still-in-its-infancy/ |access-date=2022-08-21 |website=Forbes |language=en}}</ref>

== Theory {{anchor|Relation to computability and complexity theory}} ==

=== Computability {{anchor|Computability theory}} ===
{{Further|Computability theory}}

Any [[Computational problem|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 [[Physics:Quantum mechanics|quantum mechanics]], which underlies the operation of quantum computers.

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|computability]]. This means that quantum computers cannot solve [[Undecidable problem|undecidable problem]]s like the [[Halting problem|halting problem]], and the existence of quantum computers does not disprove the [[Church–Turing thesis]].{{sfn|Nielsen|Chuang|2010|p=126}}

=== Complexity<span class="anchor" id="Quantum complexity theory"></span> ===
{{Main|Quantum complexity theory}}

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

<!-- Basic definition of BQP -->
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|quantum Turing machine]] with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time [[Probabilistic Turing machine|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|time complexity]].{{sfn|Nielsen|Chuang|2010|p=201}}

<!-- Relation of BQP to basic complexity classes -->
[[File:BQP complexity class diagram.svg|thumb|The suspected relationship of BQP to several classical complexity classes{{sfn|Nielsen|Chuang|2010|p=42}}]]
The exact relationship of BQP to [[P (complexity)|P]], [[NP (complexity)|NP]], and 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|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-hardness 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>

<!-- Speculation about more advanced types of quantum computers -->
It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on [[Physics:De Broglie–Bohm theory|Bohmian Mechanics]] could implement a search of an {{mvar|N}}-item database in at most <math>O(\sqrt[3]{N})</math> steps, a slight speedup over [[Grover's algorithm]], which runs in <math>O(\sqrt{N})</math> steps. Note, however, that neither search method would allow quantum computers to solve NP-complete problems in polynomial time.<ref name="auto">{{cite web |title=Quantum Computing and Hidden Variables |last=Aaronson |first=Scott |url=http://www.scottaaronson.com/papers/qchvpra.pdf}}</ref> Theories of [[Physics:Quantum gravity|quantum gravity]], such as [[Physics:M-theory|M-theory]] and [[Physics:Loop quantum gravity|loop quantum gravity]], may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the [[Physics:Problem of time|problem of time]]; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.<ref>{{Cite journal |first=Scott |last=Aaronson |title=NP-complete Problems and Physical Reality |journal=ACM SIGACT News |volume=2005 |arxiv=quant-ph/0502072 |year=2005 |bibcode=2005quant.ph..2072A |author-link=Scott Aaronson}} See section 7 "Quantum Gravity": "[...] to anyone who wants a test or benchmark for a favorite quantum gravity theory,[author's footnote: That is, one without all the bother of making numerical predictions and comparing them to observation] let me humbly propose the following: ''can you define Quantum Gravity Polynomial-Time?'' [...] until we can say what it means for a 'user' to specify an 'input' and 'later' receive an 'output'—''there is no such thing as computation, not even theoretically.''" (emphasis in original)</ref><ref name=":0">{{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 |access-date=30 May 2011 |date=25 May 2011 |publisher=D-Wave |archive-date=22 December 2020 |archive-url=https://web.archive.org/web/20201222041457/https://www.dwavesys.com/en/pressreleases.html#lm_2011 |url-status=dead }}</ref>

== See also ==
<!-- New links in alphabetical order please -->
{{cols|colwidth=21em}}
*[[Company:D-Wave Systems|D-Wave Systems]]
*[[Engineering:Electronic quantum holography|Electronic quantum holography]]
*[[Glossary of quantum computing]]
*IARPA
*[[Social:List of emerging technologies|List of emerging technologies]]
*[[List of quantum processors]]
*[[Magic state distillation]]
*[[Natural computing]]
*[[Optical computing]]
*[[Quantum bus]]
*[[Quantum cognition]]
*[[Quantum volume]]
*[[Physics:Quantum weirdness|Quantum weirdness]]
*[[Company:Rigetti Computing|Rigetti Computing]]
*[[Supercomputer]]
*[[Theoretical computer science]]
*[[Unconventional computing]]
*[[Physics:Valleytronics|Valleytronics]]
{{colend}}

== Notes ==
{{notelist}}

== References ==
{{Reflist|30em}}

== Further reading ==
{{Refbegin|2}}

===Textbooks===
* {{cite book |last=Aaronson |first=Scott |title=Quantum Computing Since Democritus |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-19956-8 |oclc=829706638 |doi=10.1017/CBO9780511979309 }}
* {{cite book |last=Akama |first=Seiki |year=2014 |title=Elements of Quantum Computing: History, Theories and Engineering Applications |publisher=Springer |isbn=978-3-319-08284-4 |oclc=884786739 |doi=10.1007/978-3-319-08284-4 }}
* {{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 }}
* {{cite book |last=Bernhardt |first=Chris |year=2019 |title=Quantum Computing for Everyone |oclc=1082867954 |isbn=978-0-262-35091-4 }}
* {{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 }}
* {{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 }}
* {{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/content/pdf/10.1007/978-3-030-61601-4.pdf }}
* {{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 }}
* {{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 |oclc=1111634190 |isbn=978-1-4920-3968-6 }}
* {{cite book |last1=Kaye |first1=Phillip |last2=Laflamme |first2=Raymond |last3=Mosca |first3=Michele |author-link3=Michele Mosca |title=An Introduction to Quantum Computing |year=2007 |oclc=85896383 |isbn=978-0-19-857000-4 }}  
* {{cite book |last1=Kitaev |first1=Alexei Yu. |last2=Shen |first2=Alexander H. |last3=Vyalyi |first3=Mikhail N. |title=Classical and Quantum Computation |year=2002 |oclc=907358694 |isbn=978-0-8218-3229-5 }}
* {{cite book |last1=Mermin |first1=N. David |title=Quantum Computer Science: An Introduction |date=2007 |isbn=978-0-511-34258-5 |oclc=422727925 |doi=10.1017/CBO9780511813870 }}
* {{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 |author=((National Academies of Sciences, Engineering, and Medicine)) |isbn=978-0-309-47970-7 |location=Washington, DC |s2cid=125635007 |oclc=1091904777}}
* {{cite book |  last1=Nielsen |first1=Michael |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 }}
* {{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 }}
* {{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 }}
* {{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 }}
* {{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 }}

===Academic papers===
*{{cite journal | last1=Abbot |first1=Derek |last2=Doering |first2=Charles R. |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 |hdl=2027.42/45526|s2cid=34885835 }}
*{{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 |website=Semantic Scholar|pages=233–234 |isbn=978-981-4541-88-6 }}
*{{Cite journal |last1=DiVincenzo |first1=David P. |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 journal |last=Feynman |first=Richard |title=Simulating physics with computers |journal=International Journal of Theoretical Physics |volume=21 |issue=6–7 |pages=467–488 |year=1982 |doi=10.1007/BF02650179 |bibcode=1982IJTP...21..467F |citeseerx=10.1.1.45.9310 |s2cid=124545445 }}
*{{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 }}
*{{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=[[Physics:Applied Physics Reviews|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}}
*{{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}}
*{{Cite conference |last=Shor |first=Peter W. |date=1994 |title=Algorithms for Quantum Computation: Discrete Logarithms and Factoring |conference=Symposium on Foundations of Computer Science |location=Santa Fe, New Mexico |publisher=IEEE |pages=124{{en dash}}134 |doi=10.1109/SFCS.1994.365700 |isbn=978-0-8186-6580-6 |}}
*{{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}}
{{Refend}}

== External links ==
{{Commons|Quantum computer}}

* [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/qt-quantcomp/ Quantum Computing]" by Amit Hagar and Michael E. Cuffaro.
* {{springer|title=Quantum computation, theory of|id=p/q130020}}
* [https://quantum.country/qcvc Quantum computing for the very curious] by Andy Matuschak and [[Biography:Michael Nielsen|Michael Nielsen]]
 
;Lectures
* [https://www.youtube.com/playlist?list=PL1826E60FD05B44E4 Quantum computing for the determined] – 22 video lectures by [[Biography:Michael Nielsen|Michael Nielsen]]
* [http://www.quiprocone.org/Protected/DD_lectures.htm Video Lectures] by [[Biography:David Deutsch|David Deutsch]]
* [https://web.archive.org/web/20160303183533/http://www.quantware.ups-tlse.fr/IHP2006/ Lectures at the Institut Henri Poincaré (slides and videos)]
* [https://web.archive.org/web/20130901004919/http://nanohub.org/resources/4778 Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008)]
* Lomonaco, Sam. [http://www.csee.umbc.edu/~lomonaco/Lectures.html#OxfordLectures Four Lectures on Quantum Computing given at Oxford University in July 2006]

{{Quantum computing}}
{{emerging technologies|quantum=yes|other=yes}}
{{Computer science}}
{{Quantum mechanics topics}}


[[Category:Quantum computing]]
[[Category:Models of computation]]
[[Category:Quantum cryptography]]
[[Category:Information theory]]
[[Category:Computational complexity theory]]
[[Category:Classes of computers]]
[[Category:Theoretical computer science]]
[[Category:Open problems]]
[[Category:Emerging technologies]]
[[Category:Supercomputers]]

{{Sourceattribution|Quantum computing}}