View source for Quantum computing

{{Short description|Technology that uses quantum mechanics}}
{{Use American English|date=February 2023}}
{{Use dmy dates|date=February 2021}}
[[File:Bloch sphere.svg|thumb|[[Bloch sphere]] representation of a qubit. The state <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle</math> is a point on the surface of the sphere, partway between the poles, <math>|0\rangle</math> and <math>|1\rangle</math>.]]

A '''quantum computer''' is a [[computer]] that exploits [[quantum mechanical]] phenomena. On small scales, physical matter exhibits properties of [[wave-particle duality|both particles and waves]], and quantum computing leverages this behavior using specialized hardware. [[Classical physics]] cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations [[Exponential growth|exponentially]] faster{{efn|As used in this article, "exponentially faster" has a precise [[computational complexity|complexity theoretical]] meaning. Usually, it means that as a function of input size in bits, the best known classical algorithm for a problem requires an [[exponential growth|exponentially growing]] number of steps, while a quantum algorithm uses only a polynomial number of steps.}} than any modern "classical" computer. Theoretically a large-scale quantum computer could [[post-quantum cryptography|break some widely used encryption schemes]] and aid physicists in performing [[quantum simulator|physical simulations]]; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications.

<!-- Basic principles of quantum computing -->
The basic [[unit of information]] in quantum computing, the [[qubit]] (or "quantum bit"), serves the same function as the [[bit]] in classical computing. However, unlike a classical bit, which can be in one of two states (a [[Binary number|binary]]), a qubit can exist in a [[quantum superposition|superposition]] of its two "basis" states, which loosely means that it is in both states simultaneously. When [[measurement in quantum mechanics|measuring]] a qubit, the result is a [[Born rule|probabilistic output]] of a classical bit. If a quantum computer manipulates the qubit in a particular way, [[wave interference]] effects can amplify the desired measurement results. The design of [[quantum algorithms]] involves creating procedures that allow a quantum computer to perform calculations efficiently and quickly.

<!--Physical implementations-->
Quantum computers are not yet practical for real work. Physically engineering high-quality qubits has proven challenging. If a physical qubit is not sufficiently [[isolated system|isolated]] from its environment, it suffers from [[quantum decoherence]], introducing [[noise (signal processing)|noise]] into calculations. National governments have invested heavily in experimental research that aims to develop scalable qubits with longer coherence times and lower error rates. Example implementations include [[superconducting quantum computing|superconductors]] (which isolate an [[electrical current]] by eliminating [[electrical resistance]]) and [[trapped ion quantum computer|ion traps]] (which confine a single [[atom|atomic particle]] using [[electromagnetic fields]]).

<!-- Computability and complexity -->
In principle, a classical computer can solve the same computational problems as a quantum computer, given enough time. Quantum advantage comes in the form of [[time complexity]] rather than [[computability]], and [[quantum complexity theory]] shows that some quantum algorithms are exponentially more efficient than the best known classical algorithms. A large-scale quantum computer could in theory solve computational problems unsolvable by a classical computer in any reasonable amount of time. This concept of extra ability has been called "[[quantum supremacy]]". While such claims have drawn significant attention to the discipline, near-term practical use cases remain limited.

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

For many years, the fields of [[quantum mechanics]] and [[computer science]] formed distinct academic communities.{{sfn|Aaronson|2013|p=132}} [[Modern quantum theory]] developed in the 1920s to explain 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://www.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 |publisher=MIT Press |location=[[Cambridge, Massachusetts]] |pages=3, 46 |language=en-US |oclc=796812982}}</ref> Both disciplines had practical applications during [[World War II]]; computers played a major role in [[World War II cryptography|wartime cryptography]],<ref>{{Cite book |last=Hodges |first=Andrew |title=Alan Turing: The Enigma |publisher=[[Princeton University Press]] |year=2014 |isbn=9780691164724 |location=Princeton, New Jersey |page=xviii |language=en-US}}</ref> and quantum physics was essential for [[nuclear physics]] used in the [[Manhattan Project]].<ref>{{Cite journal |last=Mårtensson-Pendrill |first=Ann-Marie |date=2006-11-01 |title=The Manhattan project—a part of physics history |journal=Physics Education |language=en-US |volume=41 |issue=6 |pages=493–501 |bibcode=2006PhyEd..41..493M |doi=10.1088/0031-9120/41/6/001 |issn=0031-9120 |s2cid=120294023}}</ref>

As [[physicists]] applied quantum mechanical models to computational problems and swapped digital [[bit]]s for [[qubits]], the fields of quantum mechanics and computer science began to converge.
In 1980, [[Paul Benioff]] introduced the [[quantum Turing machine]], which uses quantum theory to describe a simplified computer.<ref name="The computer as a physical system">{{cite journal|last1=Benioff|first1=Paul|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 time|exponential]] increase in overhead when [[simulating quantum dynamics]],<ref>{{Cite journal |last1=Buluta |first1=Iulia |last2=Nori |first2=Franco |date=2009-10-02 |title=Quantum Simulators |journal=Science |language=en |volume=326 |issue=5949 |pages=108–111 |doi=10.1126/science.1177838 |pmid=19797653 |bibcode=2009Sci...326..108B |s2cid=17187000 |issn=0036-8075}}</ref> prompting [[Yuri Manin]] and [[Richard Feynman]] to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation.<ref name="manin1980vychislimoe">{{cite book |author=Manin |first=Yu. I. |url=http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5bdjv-fax%5d.zip |title=Vychislimoe i nevychislimoe |publisher=Soviet Radio |year=1980 |pages=13–15 |language=ru |trans-title=Computable and Noncomputable |access-date=4 March 2013 |archive-url=https://web.archive.org/web/20130510173823/http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5Bdjv%5D.zip |archive-date=10 May 2013 |url-status=dead}}</ref><ref>{{cite journal |last1=Feynman |first1=Richard |title=Simulating Physics with Computers |journal=International Journal of Theoretical Physics |date=June 1982 |volume=21 |issue=6/7 |pages=467–488 |doi=10.1007/BF02650179 |url=https://people.eecs.berkeley.edu/~christos/classics/Feynman.pdf |access-date=28 February 2019 |bibcode=1982IJTP...21..467F |s2cid=124545445 |archive-url=https://web.archive.org/web/20190108115138/https://people.eecs.berkeley.edu/~christos/classics/Feynman.pdf |archive-date=8 January 2019 |url-status=dead}}</ref>{{sfn|Nielsen|Chuang|2010|p=214}}
In a 1984 paper, [[Charles H. Bennett (physicist)|Charles Bennett]] and [[Gilles Brassard]] applied quantum theory to [[cryptography]] protocols and demonstrated that quantum key distribution could enhance [[information security]].<ref name="bb84">{{Cite conference |last1=Bennett |first1=Charles H. |author-link1=Charles H. Bennett (physicist) |last2=Brassard |first2=Gilles |author-link2=Gilles Brassard |date=December 1984 |title=Quantum cryptography: Public key distribution and coin tossing |conference=IEEE International Conference on Computers, Systems & Signal Processing |language=en |location=Bangalore, India |pages=175–179 |arxiv=2003.06557 |doi=10.1016/j.tcs.2014.05.025}}</ref><ref name="personal">{{Cite book |last=Brassard |first=G. |title=IEEE Information Theory Workshop on Theory and Practice in Information-Theoretic Security, 2005 |chapter=Brief history of quantum cryptography: A personal perspective |date=2005 |chapter-url=https://ieeexplore.ieee.org/document/1543949 |location=Awaji Island, Japan |publisher=IEEE |pages=19–23 |doi=10.1109/ITWTPI.2005.1543949 |arxiv=quant-ph/0604072 |isbn=978-0-7803-9491-9|s2cid=16118245 }}</ref>

[[Quantum algorithm]]s then emerged for solving [[oracle machine|oracle problems]], such as [[Deutsch's algorithm]] in 1985,<ref>{{Cite journal |date=1985-07-08 |title=Quantum theory, the Church–Turing principle and the universal quantum computer |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |language=en |volume=400 |issue=1818 |pages=97–117 |doi=10.1098/rspa.1985.0070 |bibcode=1985RSPSA.400...97D |s2cid=1438116 |issn=0080-4630|last1=Deutsch |first1=D. }}</ref> the [[Bernstein{{en dash}}Vazirani algorithm]] in 1993,<ref>{{Cite book |last1=Bernstein |first1=Ethan |title=Proceedings of the twenty-fifth annual ACM symposium on Theory of computing – STOC '93 |last2=Vazirani |first2=Umesh |date=1993 |publisher=ACM Press |isbn=978-0-89791-591-5 |location=San Diego, California, United States |pages=11–20 |language=en |chapter=Quantum complexity theory |doi=10.1145/167088.167097 |chapter-url=http://portal.acm.org/citation.cfm?doid=167088.167097 |s2cid=676378}}</ref> and [[Simon's algorithm]] in 1994.<ref>{{Cite book |last=Simon |first=D. R. |title=Proceedings 35th Annual Symposium on Foundations of Computer Science |chapter=On the power of quantum computation |date=1994 |chapter-url=https://ieeexplore.ieee.org/document/365701 |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]] with a quantum state in [[quantum superposition|superposition]], sometimes referred to as ''quantum parallelism''.{{sfn|Nielsen|Chuang|2010|p=30-32}}

[[File:Peter Shor 2017 Dirac Medal Award Ceremony.png|thumb|[[Peter Shor]] (pictured here in 2017) showed in 1994 that a scalable quantum computer would be able to break [[RSA encryption]].|upright=0.9]]
[[Peter Shor]] built on these results with [[Shor's algorithm|his 1994 algorithm]] for breaking the widely used [[RSA (cryptosystem)|RSA]] and [[Diffie{{en dash}}Hellman]] encryption protocols,{{sfn|Shor|1994}} which drew significant attention to the field of quantum computing.{{sfn|Grumbling|Horowitz|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, [[Seth Lloyd]] proved that quantum computers could simulate quantum systems without the exponential overhead present in classical simulations,<ref name="273.5278.1073">{{Cite journal |last=Lloyd |first=Seth |date=1996-08-23 |title=Universal Quantum Simulators |journal=Science |volume=273 |issue=5278 |pages=1073–1078 |pmid=8688088 |s2cid=43496899 |bibcode=1996Sci...273.1073L |doi=10.1126/science.273.5278.1073 |issn=0036-8075}}</ref> validating Feynman's 1982 conjecture.<ref>{{Cite journal |last1=Cao |first1=Yudong |last2=Romero |first2=Jonathan |last3=Olson |first3=Jonathan P. |last4=Degroote |first4=Matthias |last5=Johnson |first5=Peter D. |last6=Kieferová |first6=Mária |last7=Kivlichan |first7=Ian D. |last8=Menke |first8=Tim |last9=Peropadre |first9=Borja |last10=Sawaya |first10=Nicolas P. D. |last11=Sim |first11=Sukin |last12=Veis |first12=Libor |last13=Aspuru-Guzik |first13=Alán |date=2019-10-09 |title=Quantum Chemistry in the Age of Quantum Computing |journal=Chemical Reviews |language=en-US |volume=119 |issue=19 |pages=10856–10915 |arxiv=1812.09976 |doi=10.1021/acs.chemrev.8b00803 |issn=0009-2665 |pmid=31469277 |s2cid=119417908 |display-authors=5}}</ref>

Over the years, [[experimental physics|experimentalists]] have constructed small-scale quantum computers using [[trapped ion quantum computer|trapped ions]] and [[superconducting quantum computing|superconductors]].{{sfn|Grumbling|Horowitz|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=[[American Physical Society]] |volume=80 |issue=15 |pages=3408–3411 |bibcode=1998PhRvL..80.3408C |doi=10.1103/PhysRevLett.80.3408}}</ref><ref>{{cite news |last=Holton |first=William Coffeen |title=quantum computer |url=https://www.britannica.com/technology/quantum-computer |access-date=4 Dec 2021 |newspaper=Encyclopedia Britannica |publisher=[[Encyclopædia Britannica]]}}</ref> and subsequent experiments have increased the number of qubits and reduced error rates.{{sfn|Grumbling|Horowitz|2019 |pp=164-169}}

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

In December 2023, physicists, for the first time, reported the entanglement of individual molecules, which may have significant applications in quantum computing.<ref name="PHYS-20231207">{{cite news |author=Staff |title=Physicists 'entangle' individual molecules for the first time, hastening possibilities for quantum computing |url=https://phys.org/news/2023-12-physicists-entangle-individual-molecules-hastening.html |date=7 December 2023 |work=[[Phys.org]] |url-status=live |archiveurl=https://archive.today/20231208152543/https://phys.org/news/2023-12-physicists-entangle-individual-molecules-hastening.html |archivedate=8 December 2023 |accessdate=8 December 2023 }}</ref>

== Quantum information processing ==
{{See also|Introduction to quantum mechanics}}

[[Computer engineer]]s typically describe a [[modern computer]]'s operation in terms of [[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 [[isolated system|isolated]] from their environment, so any [[quantum information]] quickly [[quantum decoherence|decoheres]].
While [[programmers]] may depend on [[probability theory]] when designing a [[randomized algorithm]], quantum mechanical notions like [[quantum superposition|superposition]] and [[quantum interference|interference]] are largely irrelevant for [[program analysis]].

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

As physicist [[Charles H. Bennett (physicist)|Charlie Bennett]] describes the relationship between quantum and classical computers,<ref>{{Cite AV media |url=https://www.youtube.com/live/rslt-LwtDK4&t=4102 |title=Information Is Quantum: How Physics Helped Explain the Nature of Information and What Can Be Done With It |date=2020-07-31 |last=Bennett |first=Charlie |type=Videotape |author-link=Charles H. Bennett (physicist) |time=1:08:22 |via=YouTube}}</ref>
{{Blockquote|text=A classical computer is a quantum computer&nbsp;... so we shouldn't be asking about "where do quantum speedups come from?" We should say, "well, all computers are quantum.&nbsp;... Where do classical slowdowns come from?"}}

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

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

When a qubit is [[quantum measurement|measured]] in the [[standard basis]], the result is a classical bit.
The [[Born rule]] describes the [[norm-squared]] correspondence between amplitudes and probabilities{{mdash}}when measuring a qubit <math>\alpha|0\rangle+\beta|1\rangle</math>, the state [[wave function collapse|collapses]] to <math>|0\rangle</math> with probability <math>|\alpha|^2</math>, or to <math>|1\rangle</math> with probability <math>|\beta|^2</math>.
Any valid qubit state has coefficients <math>\alpha</math> and <math>\beta</math> such that <math>|\alpha|^2+|\beta|^2 = 1</math>.
As an example, measuring the qubit <math>1/\sqrt {2}|0\rangle+1/\sqrt{2}|1\rangle</math> would produce either <math>|0\rangle</math> or <math>|1\rangle</math> with equal probability.

Each additional qubit doubles the [[dimension (vector space)|dimension]] of the [[state space (physics)|state space]].{{sfn|Mermin|2007|p=18}}
As an example, the vector {{nowrap|{{sfrac|1|√2}}{{ket|00}} + {{sfrac|1|√2}}{{ket|01}}}} represents a two-qubit state, a [[tensor product]] of the qubit {{ket|0}} with the qubit {{nowrap|{{sfrac|1|√2}}{{ket|0}} + {{sfrac|1|√2}}{{ket|1}}}}.
This vector inhabits a four-dimensional [[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<span class="anchor" id="gate-application"></span> ===
{{See also|Unitarity (physics)}}

The state of this one-qubit [[quantum memory]] can be manipulated by applying [[quantum logic gate]]s, analogous to how classical memory can be manipulated with [[Logic gate|classical logic gates]]. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a [[Matrix (mathematics)|matrix]]
<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>
Mathematically, the application of such a logic gate to a quantum state vector is modelled with [[matrix multiplication]]. Thus
: <math>X|0\rangle = |1\rangle</math> and <math>X|1\rangle = |0\rangle</math>.
The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are
<math display="block">
 |00\rangle := \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix};\quad
 |01\rangle := \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix};\quad
 |10\rangle := \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix};\quad
 |11\rangle := \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.
</math>
The [[Controlled NOT gate|controlled NOT (CNOT)]] gate can then be represented using the following matrix:
<math display="block">
 \operatorname{CNOT} :=
 \begin{pmatrix}
  1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 0 & 1 \\
   0 & 0 & 1 & 0
 \end{pmatrix}.
</math>
As a mathematical consequence of this definition, <math display="inline">\operatorname{CNOT}|00\rangle = |00\rangle</math>, <math display="inline">\operatorname{CNOT}|01\rangle = |01\rangle</math>, <math display="inline">\operatorname{CNOT}|10\rangle = |11\rangle</math>, and <math display="inline">\operatorname{CNOT}|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

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

=== Quantum parallelism ===

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

=== Quantum programming<span class="anchor" id="Models of computation for quantum computing"></span> ===
{{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]] from [[Quantum logic gate|more primitive gates]]|upright=1.15]]

A [[quantum circuit|quantum gate array]] decomposes computation into a sequence of few-qubit [[quantum gate]]s. A quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most [[quantum 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]] of size <math>2^n \times 2^n</math> over <math>n</math> qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a [[Quantum logic gate#Universal quantum gates|universal gate set]], since a computer that can run such circuits is a [[universal quantum computer]]. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the [[Solovay–Kitaev theorem|Solovay-Kitaev theorem]]. Implementation of Boolean functions using the few-qubit quantum gates is presented here.<ref>{{Cite book |last1=Kurgalin |first1=Sergei |title=Concise guide to quantum computing: algorithms, exercises, and implementations |last2=Borzunov |first2=Sergei |date=2021 |publisher=Springer |isbn=978-3-030-65054-4 |series=Texts in computer science |location=Cham}}</ref> 

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

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

==== Adiabatic quantum computing ====

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

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

==== Topological quantum computing ====

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

==== Quantum Turing machine ====

A [[quantum Turing machine]] is the quantum analog of a [[Turing machine]].<ref name="The computer as a physical system"/> All of these models of computation—quantum circuits,<ref>{{Cite book|last=Chi-Chih Yao|first=A.|title=Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science |chapter=Quantum circuit complexity |year=1993|chapter-url=https://ieeexplore.ieee.org/document/366852|pages=352–361|doi=10.1109/SFCS.1993.366852|isbn=0-8186-4370-6|s2cid=195866146}}</ref> [[One-way quantum computer|one-way quantum computation]],<ref>{{Cite journal |last1=Raussendorf |first1=Robert |last2=Browne |first2=Daniel E. |last3=Briegel |first3=Hans J. |date=2003-08-25 |title=Measurement-based quantum computation on cluster states |journal=Physical Review A |volume=68 |issue=2 |pages=022312 |doi=10.1103/PhysRevA.68.022312 |arxiv=quant-ph/0301052 |bibcode=2003PhRvA..68b2312R |s2cid=6197709}}</ref> adiabatic quantum computation,<ref>{{Cite journal |last1=Aharonov |first1=Dorit |last2=van Dam |first2=Wim |last3=Kempe |first3=Julia |last4=Landau |first4=Zeph |last5=Lloyd |first5=Seth |last6=Regev |first6=Oded |date=2008-01-01 |title=Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation |journal=SIAM Review |volume=50 |issue=4 |pages=755–787 |doi=10.1137/080734479 |arxiv=quant-ph/0405098 |bibcode=2008SIAMR..50..755A |s2cid=1503123 |issn=0036-1445}}</ref> and topological quantum computation<ref name="FLW02">{{Cite journal |last1=Freedman |first1=Michael H. |last2=Larsen |first2=Michael |last3=Wang |first3=Zhenghan |date=2002-06-01 |title=A Modular Functor Which is Universal for Quantum Computation |journal=Communications in Mathematical Physics |volume=227 |issue=3 |pages=605–622 |doi=10.1007/s002200200645 |issn=0010-3616 |arxiv=quant-ph/0001108 |bibcode=2002CMaPh.227..605F |s2cid=8990600}}</ref>—have been shown to be equivalent to the quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical.

====Noisy intermediate-scale quantum computing====
The [[threshold theorem]] shows how increasing the number of qubits can mitigate errors,{{sfn|Nielsen|Chuang|2010 |p=481}} yet fully fault-tolerant quantum computing remains "a rather distant dream".<ref name="preskill2018"/> According to some researchers, ''noisy intermediate-scale quantum'' ([[NISQ]]) machines may have specialized uses in the near future, but [[noise (signal processing)|noise]] in quantum gates limits their reliability.<ref name="preskill2018">{{Cite journal |last=Preskill |first=John |date=6 August 2018 |title=Quantum Computing in the NISQ era and beyond |journal=Quantum |volume=2 |page=79 |arxiv=1801.00862 |doi=10.22331/q-2018-08-06-79 |doi-access=free |bibcode=2018Quant...2...79P |s2cid=44098998}}</ref>
Scientists at [[Harvard University|Harvard]] University successfully created "quantum circuits" that correct errors more efficiently than alternative methods, which may potentially remove a major obstacle to practical quantum computers.<ref>{{Cite journal |last1=Bluvstein |first1=Dolev |last2=Evered |first2=Simon J. |last3=Geim |first3=Alexandra A. |last4=Li |first4=Sophie H. |last5=Zhou |first5=Hengyun |last6=Manovitz |first6=Tom |last7=Ebadi |first7=Sepehr |last8=Cain |first8=Madelyn |last9=Kalinowski |first9=Marcin |last10=Hangleiter |first10=Dominik |last11=Ataides |first11=J. Pablo Bonilla |last12=Maskara |first12=Nishad |last13=Cong |first13=Iris |last14=Gao |first14=Xun |last15=Rodriguez |first15=Pedro Sales |date=2023-12-06 |title=Logical quantum processor based on reconfigurable atom arrays |journal=Nature |volume=626 |issue=7997 |language=en |pages=58–65 |doi=10.1038/s41586-023-06927-3 |pmid=38056497 |pmc=10830422 |issn=1476-4687|arxiv=2312.03982 |s2cid=266052773 }}</ref><ref>{{Cite web |last=Freedberg Jr. |first=Sydney J. |date=2023-12-07 |title='Off to the races': DARPA, Harvard breakthrough brings quantum computing years closer |url=https://breakingdefense.sites.breakingmedia.com/2023/12/off-to-the-races-darpa-harvard-breakthrough-brings-quantum-computing-years-closer/ |access-date=2023-12-09 |website=Breaking Defense |language=en-US}}</ref> The Harvard research team was supported by [[Massachusetts Institute of Technology|MIT]], [[QuEra Computing]], [[California Institute of Technology|Caltech]], and [[Princeton University|Princeton]] University and funded by [[DARPA]]'s Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program.<ref>{{Cite web |date=December 6, 2023 |title=DARPA-Funded Research Leads to Quantum Computing Breakthrough |url=https://www.darpa.mil/news-events/2023-12-06 |access-date=January 5, 2024 |website=darpa.mil}}</ref><ref>{{Cite web |last=Choudhury |first=Rizwan |date=2023-12-30 |title=Top 7 innovation stories of 2023 – Interesting Engineering |url=https://interestingengineering.com/lists/top-7-innovation-stories-of-2023-interesting-engineering |access-date=2024-01-06 |website=interestingengineering.com |language=en-US}}</ref>

=== Quantum cryptography and cybersecurity ===
[[Image:Laser optique.jpg|thumb|right|Example of a quantum cryptosystem layout]]

Quantum computing has significant potential applications in the fields of cryptography and cybersecurity. Quantum cryptography, which relies on the principles of quantum mechanics, offers the possibility of secure communication channels that are resistant to eavesdropping. Quantum key distribution (QKD) protocols, such as BB84, enable the secure exchange of cryptographic keys between parties, ensuring the confidentiality and integrity of communication. Moreover, quantum random number generators (QRNGs) can produce high-quality random numbers, which are essential for secure encryption.

However, quantum computing also poses challenges to traditional cryptographic systems. [[Shor's algorithm]], a quantum algorithm for integer factorization, could potentially break widely used public-key cryptography schemes like RSA, which rely on the difficulty of factoring large numbers. Post-quantum cryptography, which involves the development of cryptographic algorithms that are resistant to attacks by both classical and quantum computers, is an active area of research aimed at addressing this concern.

Ongoing research in quantum cryptography and post-quantum cryptography is crucial for ensuring the security of communication and data in the face of evolving quantum computing capabilities. Advances in these fields, such as the development of new QKD protocols, the improvement of QRNGs, and the standardization of post-quantum cryptographic algorithms, will play a key role in maintaining the integrity and confidentiality of information in the quantum era.<ref>{{cite journal |last1=Pirandola |first1=S. |last2=Andersen |first2=U. L. |last3=Banchi |first3=L. |last4=Berta |first4=M. |last5=Bunandar |first5=D. |last6=Colbeck |first6=R. |last7=Englund |first7=D. |last8=Gehring |first8=T. |last9=Lupo |first9=C. |last10=Ottaviani |first10=C. |last11=Pereira |first11=J. |last12=Razavi |first12=M. |last13=Shamsul Shaari |first13=J. |last14=Tomamichel |first14=M. |last15=Usenko |first15=V. C. |year=2020 |title=Advances in quantum cryptography |journal=Advances in Optics and Photonics |volume=12 |issue=4 |pages=1012–1236 |doi=10.1364/AOP.361502 |last16=Vallone |first16=G. |last17=Villoresi |first17=P. |last18=Wallden |first18=P.|arxiv=1906.01645 |bibcode=2020AdOP...12.1012P }}</ref>

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

[[Quantum cryptography]] enables new ways to transmit data securely; for example, [[quantum key distribution]] uses entangled quantum states to establish secure [[cryptographic keys]].<ref>{{Cite journal |last1=Pirandola |first1=S. |last2=Andersen |first2=U. L. |last3=Banchi |first3=L. |last4=Berta |first4=M. |last5=Bunandar |first5=D. |last6=Colbeck |first6=R. |last7=Englund |first7=D. |last8=Gehring |first8=T. |last9=Lupo |first9=C. |last10=Ottaviani |first10=C. |last11=Pereira |first11=J. 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 |arxiv=1906.01645 |bibcode=2020AdOP...12.1012P |s2cid=174799187 |issn=1943-8206}}</ref> When a sender and receiver exchange quantum states, they can guarantee that an [[adversary (cryptography)|adversary]] does not intercept the message, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change.<ref>{{Cite journal |last1=Xu |first1=Feihu |last2=Ma |first2=Xiongfeng |last3=Zhang |first3=Qiang |last4=Lo |first4=Hoi-Kwong |last5=Pan |first5=Jian-Wei |date=2020-05-26 |title=Secure quantum key distribution with realistic devices |journal=Reviews of Modern Physics |volume=92 |issue=2 |page=025002{{hyphen}}3 |doi=10.1103/RevModPhys.92.025002|arxiv=1903.09051 |bibcode=2020RvMP...92b5002X |s2cid=210942877 }}</ref> With appropriate [[cryptographic protocols]], the sender and receiver can thus establish shared private information resistant to eavesdropping.<ref name="bb84" /><ref>{{Cite conference |last1=Xu |first1=Guobin |last2=Mao |first2=Jianzhou |last3=Sakk |first3=Eric |last4=Wang |first4=Shuangbao Paul |title=2023 57th Annual Conference on Information Sciences and Systems (CISS) |date=2023-03-22 |chapter=An Overview of Quantum-Safe Approaches: Quantum Key Distribution and Post-Quantum Cryptography |publisher=[[IEEE]] |page=3 |doi=10.1109/CISS56502.2023.10089619 |isbn=978-1-6654-5181-9}}</ref>

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

== Algorithms ==

<!-- Overview of quantum algorithms, particularly abstract routines with no explicit application -->
Progress in finding [[quantum algorithms]] typically focuses on this quantum circuit model, though exceptions like the [[Adiabatic quantum computation|quantum adiabatic algorithm]] exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.<ref name="zoo">{{cite web |author=Jordan |first=Stephen |date=14 October 2022 |title=Quantum Algorithm Zoo |url=http://math.nist.gov/quantum/zoo/ |url-status=live |archive-url=https://web.archive.org/web/20180429014516/https://math.nist.gov/quantum/zoo/ |archive-date=29 April 2018 |orig-year=22 April 2011}}</ref>

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

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

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

=== Simulation of quantum systems ===
{{Main|Quantum simulation}}

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

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

=== Post-quantum cryptography ===
{{Main|Post-quantum cryptography}}

A notable application of quantum computation is for [[cryptanalysis|attacks]] on cryptographic systems that are currently in use. [[Integer factorization]], which underpins the security of [[public key cryptography|public key cryptographic]] systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few [[prime number]]s (e.g., products of two 300-digit primes).<ref>{{cite journal |last=Lenstra |first=Arjen K. |url=http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf |title=Integer Factoring |journal=Designs, Codes and Cryptography |volume=19 |pages=101–128 |year=2000 |doi=10.1023/A:1008397921377 |issue=2/3 |s2cid=9816153 |url-status=dead |archive-url=https://web.archive.org/web/20150410234239/http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf |archive-date=10 April 2015 }}</ref> By comparison, a quantum computer could solve this problem exponentially faster using [[Shor's algorithm]] to find its factors.{{sfn|Nielsen|Chuang|2010|p=216}} This ability would allow a quantum computer to break many of the [[cryptography|cryptographic]] systems in use today, in the sense that there would be a [[polynomial time]] (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular [[Asymmetric Algorithms|public key ciphers]] are based on the difficulty of factoring integers or the [[discrete logarithm]] problem, both of which can be solved by Shor's algorithm. In particular, the [[RSA (algorithm)|RSA]], [[Diffie–Hellman]], and [[elliptic curve Diffie–Hellman]] algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of ''post-quantum cryptography''.<ref name="pqcrypto_survey">{{cite book |doi=10.1007/978-3-540-88702-7_1 |chapter=Introduction to post-quantum cryptography |title=Post-Quantum Cryptography |year=2009 |last1=Bernstein |first1=Daniel J. |pages=1–14 |isbn=978-3-540-88701-0 |place=Berlin, Heidelberg |publisher=Springer|s2cid=61401925 }}</ref><ref>See also [http://pqcrypto.org/ pqcrypto.org], a bibliography maintained by Daniel J. Bernstein and [[Tanja Lange]] on cryptography not known to be broken by quantum computing.</ref> Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the [[McEliece cryptosystem]] based on a problem in [[coding theory]].<ref name="pqcrypto_survey" /><ref>{{cite journal |last1=McEliece |first1=R. J. |title=A Public-Key Cryptosystem Based On Algebraic Coding Theory |journal=DSNPR |date=January 1978 |volume=44 |pages=114–116 |url=http://ipnpr.jpl.nasa.gov/progress_report2/42-44/44N.PDF |bibcode=1978DSNPR..44..114M}}</ref> [[Lattice-based cryptography|Lattice-based cryptosystems]] are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the [[dihedral group|dihedral]] [[hidden subgroup problem]], which would break many lattice based cryptosystems, is a well-studied open problem.<ref>{{cite journal |last1=Kobayashi |first1=H. |last2=Gall |first2=F. L. |year=2006 |title=Dihedral Hidden Subgroup Problem: A Survey |journal=Information and Media Technologies |volume=1 |issue=1 |pages=178–185 |doi=10.2197/ipsjdc.1.470 |doi-access=free}}</ref> It has been proven that applying Grover's algorithm to break a [[Symmetric-key algorithm|symmetric (secret key) algorithm]] by brute force requires time equal to roughly 2<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 ''[[Key size#Effect of quantum computing attacks on key strength|Key size]]'').

=== Search problems<span class="anchor" id="Quantum search"></span> ===
{{Main|Grover's algorithm}}

The most well-known example of a problem that allows for a polynomial quantum speedup is ''unstructured search'', which involves finding a marked item out of a list of <math>n</math> items in a database. This can be solved by [[Grover's algorithm]] using <math>O(\sqrt{n})</math> queries to the database, quadratically fewer than the <math>\Omega(n)</math> queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Many examples of provable quantum speedups for query problems are based on Grover's algorithm, including [[BHT algorithm|Brassard, Høyer, and Tapp's algorithm]] for finding collisions in two-to-one functions,<ref>{{cite encyclopedia |year=2016 |title=Quantum Algorithm for the Collision Problem |encyclopedia=Encyclopedia of Algorithms |publisher=Springer |place=New York, New York |editor-last=Kao |editor-first=Ming-Yang |pages=1662–1664 |language=en |arxiv=quant-ph/9705002 |doi=10.1007/978-1-4939-2864-4_304 |isbn=978-1-4939-2864-4 |last2=Høyer |first2=Peter |last3=Tapp |first3=Alain |last1=Brassard |first1=Gilles |s2cid=3116149}}</ref> and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.<ref>{{Cite journal |last1=Farhi |first1=Edward |last2=Goldstone |first2=Jeffrey |last3=Gutmann |first3=Sam |date=23 December 2008 |title=A Quantum Algorithm for the Hamiltonian NAND Tree |journal=Theory of Computing |language=EN |volume=4 |issue=1 |pages=169–190 |doi=10.4086/toc.2008.v004a008 |s2cid=8258191 |issn=1557-2862 |doi-access=free}}</ref>

Problems that can be efficiently addressed with Grover's algorithm have the following properties:<ref>{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |year=2011 |isbn=978-1-84628-887-6 |pages=242–244}}</ref><ref>{{cite arXiv |last=Grover| first=Lov| author-link=Lov Grover |title=A fast quantum mechanical algorithm for database search |date=29 May 1996| eprint=quant-ph/9605043}}</ref>
#There is no searchable structure in the collection of possible answers,
#The number of possible answers to check is the same as the number of inputs to the algorithm, and
#There exists a Boolean function that evaluates each input and determines whether it is the correct answer.

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

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

[[Quantum annealing]] relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. {{anchor|Computational biology}}Adiabatic optimization may be helpful for solving [[computational biology]] problems.<ref>{{cite journal |last1=Outeiral |first1=Carlos| last2=Strahm |first2=Martin |last3=Morris |first3=Garrett |last4=Benjamin |first4=Simon |last5=Deane |first5=Charlotte |last6=Shi |first6=Jiye |title=The prospects of quantum computing in computational molecular biology |journal=WIREs Computational Molecular Science |year=2021|volume=11|doi=10.1002/wcms.1481 |arxiv=2005.12792 |s2cid=218889377 |doi-access=free}}</ref>

=== Machine learning ===
{{Main|Quantum machine learning}}

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

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

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

== Engineering<span class="anchor" id="Developing physical quantum computers"></span> ==
[[File:A Wafer of the Latest D-Wave Quantum Computers (39188583425).jpg|thumb|A [[wafer (electronics)|wafer]] of [[adiabatic quantum computer]]s]]
{{As of|2023|post=,}} classical computers outperform quantum computers for all real-world applications. While current quantum computers may speed up solutions to particular mathematical problems, they give no computational advantage for practical tasks. Scientists and engineers are exploring multiple technologies for quantum computing hardware and hope to develop scalable quantum architectures, but serious obstacles remain.<ref name="good-for-nothing" /><ref name="CACM" />

=== Challenges ===

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

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

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

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

One of the greatest challenges involved with constructing quantum computers is controlling or removing [[quantum decoherence]]. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time ''T''<sub>2</sub> (for [[Nuclear magnetic resonance|NMR]] and [[MRI]] technology, also called the ''dephasing time''), typically range between nanoseconds and seconds at low temperature.<ref name="DiVincenzo 1995">{{cite journal |last=DiVincenzo |first=David P. |title=Quantum Computation |journal=Science |year=1995 |volume=270 |issue=5234 |pages=255–261 |doi=10.1126/science.270.5234.255 |bibcode=1995Sci...270..255D |citeseerx=10.1.1.242.2165 |s2cid=220110562}}</ref> Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a [[dilution refrigerator]]<ref>{{Cite journal |doi=10.1016/j.cryogenics.2021.103390| issn=0011-2275 |title=Development of Dilution refrigerators – A review |journal=Cryogenics| volume=121| year=2022| last1=Zu| first1=H.| last2=Dai| first2=W.| last3=de Waele| first3=A.T.A.M.| s2cid=244005391}}</ref>) in order to prevent significant decoherence.<ref>{{cite journal |last1=Jones |first1=Nicola |title=Computing: The quantum company |journal=Nature |date=19 June 2013 |volume=498 |issue=7454 |pages=286–288 |doi=10.1038/498286a|pmid=23783610|bibcode=2013Natur.498..286J|doi-access=free}}</ref> A 2020 study argues that [[ionizing radiation]] such as [[cosmic rays]] can nevertheless cause certain systems to decohere within milliseconds.<ref>{{cite journal |last1=Vepsäläinen |first1=Antti P. |last2=Karamlou |first2=Amir H. |last3=Orrell |first3=John L. |last4=Dogra |first4=Akshunna S. |last5=Loer |first5=Ben |last6=Vasconcelos |first6=Francisca |last7=Kim |first7=David K. |last8=Melville |first8=Alexander J. |last9=Niedzielski |first9=Bethany M. |last10=Yoder |first10=Jonilyn L. |last11=Gustavsson |first11=Simon |last12=Formaggio |first12=Joseph A. |last13=VanDevender |first13=Brent A. |last14=Oliver |first14=William D. |display-authors=5 |title=Impact of ionizing radiation on superconducting qubit coherence |journal=Nature |date=August 2020 |volume=584 |issue=7822 |pages=551–556 |doi=10.1038/s41586-020-2619-8 |pmid=32848227 |url=https://www.nature.com/articles/s41586-020-2619-8 |language=en |issn=1476-4687|arxiv=2001.09190 |bibcode=2020Natur.584..551V |s2cid=210920566 }}</ref>

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

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical [[pulse shaping]]. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

As described by the [[threshold theorem]], if the error rate is small enough, it is thought to be possible to use [[quantum error correction]] to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often-cited figure for the required error rate in each gate for fault-tolerant computation is 10<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 |arxiv=1512.00796 }}</ref> show that at least 3{{nbsp}}million physical qubits would factor 2,048-bit integer in 5 months on a fully error-corrected trapped-ion quantum computer. In terms of the number of physical qubits, to date, this remains the lowest estimate<ref>{{Cite journal |last1=Gidney |first1=Craig |last2=Ekerå |first2=Martin |date=2021-04-15 |title=How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits |journal=Quantum |volume=5 |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]] with [[anyon]]s, [[quasi-particle]]s used as threads, and relying on [[braid theory]] to form stable logic gates.<ref>{{cite journal
 | last1 = Freedman | first1 = Michael H. | author1-link = Michael Freedman
 | last2 = Kitaev | first2 = Alexei | author2-link = Alexei Kitaev
 | last3 = Larsen | first3 = Michael J. | author3-link = Michael J. Larsen
 | 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 ===

Physicist [[John Preskill]] coined the term ''[[quantum supremacy]]'' to describe the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers.<ref>{{cite arXiv |last=Preskill |first=John |date=2012-03-26 |title=Quantum computing and the entanglement frontier |eprint=1203.5813 |class=quant-ph}}</ref><ref>{{Cite journal|last=Preskill |first=John |date=2018-08-06 |title=Quantum Computing in the NISQ era and beyond |journal=Quantum |volume=2 |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 [[Sycamore processor|Sycamore quantum computer]] more than 3,000,000 times faster than they could be done on [[Summit (supercomputer)|Summit]], generally considered the world's fastest computer.<ref name="1910.11333"/><ref>{{cite web |last=Giles |first=Martin |date=September 20, 2019 |title=Google researchers have reportedly achieved 'quantum supremacy' |website=MIT Technology Review |language=en |url=https://www.technologyreview.com/f/614416/google-researchers-have-reportedly-achieved-quantum-supremacy/ |access-date=May 15, 2020}}</ref><ref>{{Cite web |last=Tavares |first=Frank |date=2019-10-23 |title=Google and NASA Achieve Quantum Supremacy |url=http://www.nasa.gov/feature/ames/quantum-supremacy |access-date=2021-11-16 |website=NASA |language=en-US}}</ref> This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,<ref>{{cite arXiv |last1=Pednault |first1=Edwin |last2=Gunnels |first2=John A. |last3=Nannicini |first3=Giacomo |last4=Horesh |first4=Lior |last5=Wisnieff |first5=Robert |date=2019-10-22|title=Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits |class=quant-ph |eprint=1910.09534}}</ref><ref>{{Cite journal |last=Cho |first=Adrian |date=2019-10-23 |title=IBM casts doubt on Google's claims of quantum supremacy |url=https://www.science.org/content/article/ibm-casts-doubt-googles-claims-quantum-supremacy |journal=Science |doi=10.1126/science.aaz6080 |s2cid=211982610 |issn=0036-8075}}</ref> and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Sycamore and classical supercomputers<ref>{{Cite book |last1=Liu |first1=Yong (Alexander) |last2=Liu |first2=Xin (Lucy) |last3=Li |first3=Fang (Nancy) |last4=Fu |first4=Haohuan |last5=Yang |first5=Yuling |last6=Song |first6=Jiawei |last7=Zhao |first7=Pengpeng |last8=Wang |first8=Zhen |last9=Peng |first9=Dajia |last10=Chen |first10=Huarong |last11=Guo |first11=Chu |title=Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis |chapter=Closing the "quantum supremacy" gap |display-authors=5 |date=2021-11-14 |series=SC '21 |location=New York, New York |publisher=Association for Computing Machinery |pages=1–12 |arxiv=2110.14502 |doi=10.1145/3458817.3487399 |isbn=978-1-4503-8442-1 |s2cid=239036985}}</ref><ref>{{Cite journal |last1=Bulmer |first1=Jacob F. F. |last2=Bell |first2=Bryn A. |last3=Chadwick |first3=Rachel S. |last4=Jones |first4=Alex E. |last5=Moise |first5=Diana |last6=Rigazzi |first6=Alessandro |last7=Thorbecke |first7=Jan |last8=Haus |first8=Utz-Uwe |last9=Van Vaerenbergh |first9=Thomas |last10=Patel |first10=Raj B. |last11=Walmsley |first11=Ian A. |display-authors=5 |date=2022-01-28 |title=The boundary for quantum advantage in Gaussian boson sampling |journal=Science Advances |language=en |volume=8 |issue=4 |pages=eabl9236 |doi=10.1126/sciadv.abl9236 |issn=2375-2548 |pmc=8791606 |pmid=35080972 |arxiv=2108.01622 |bibcode=2022SciA....8.9236B}}</ref><ref>{{Cite journal |last=McCormick |first=Katie |date=2022-02-10 |title=Race Not Over Between Classical and Quantum Computers |url=https://physics.aps.org/articles/v15/19 |journal=Physics |language=en |volume=15|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=Science |volume=377 |doi=10.1126/science.ade2364}}</ref><ref>{{Cite web |title=Google's 'quantum supremacy' usurped by researchers using ordinary supercomputer |url=https://techcrunch.com/2022/08/05/googles-quantum-supremacy-usurped-by-researchers-using-ordinary-supercomputer/ |access-date=2022-08-07 |website=TechCrunch |date=5 August 2022 |language=en-US}}</ref>

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

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

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

=== Skepticism ===

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 [[Nature (journal)|Nature]] spotlight article summarised current quantum computers as being "For now, [good for] absolutely nothing".<ref name="good-for-nothing">
{{Cite journal| journal = Nature | title = Quantum computers: what are they good for? | date = 24 May 2023 | first = Michael | last = Brooks| volume = 617 | issue = 7962 | pages = S1–S3 | doi = 10.1038/d41586-023-01692-9 | pmid = 37225885 | bibcode = 2023Natur.617S...1B | s2cid = 258847001 | doi-access = free }}
</ref> The article elaborated that quantum computers are yet to be more useful or efficient than conventional computers in any case, though it also argued that in the long term such computers are likely to be useful. A 2023 [[Communications of the ACM]] article<ref name = "CACM">{{Cite web | url = https://m-cacm.acm.org/magazines/2023/5/272276-disentangling-hype-from-practicality-on-realistically-achieving-quantum-advantage/fulltext | publisher = Communications of the ACM | date = May 2023 | title = Disentangling Hype from Practicality: On Realistically Achieving Quantum Advantage | 
author1 = Torsten Hoefler | author2 =  Thomas Häner | author3 = Matthias Troyer}}
</ref> found that current quantum computing algorithms are "insufficient for practical quantum advantage without significant improvements across the software/hardware stack". It argues that the most promising candidates for achieving speedup with quantum computers are "small-data problems", for example in chemistry and materials science. However, the article also concludes that a large range of the potential applications it considered, such as machine learning, "will not achieve quantum advantage with current quantum algorithms in the foreseeable future", and it identified I/O constraints that make speedup unlikely for "big data problems, unstructured linear systems, and database search based on Grover's algorithm".

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

* Conventional computer hardware and algorithms are not only optimized for practical tasks, but are still improving rapidly, particularly [[GPU]] accelerators. 
* Current quantum computing hardware generates only a limited amount of [[Quantum entanglement|entanglement]] before getting overwhelmed by noise.
* [[Quantum algorithm]]s provide speedup over conventional algorithms only for some tasks, and matching these tasks with practical applications proved challenging. Some promising tasks and applications require resources far beyond those available today.<ref>{{Cite web| url = https://m-cacm.acm.org/magazines/2022/12/266916-quantum-computers-and-the-universe/fulltext | publisher = Communications of the ACM |
title = Quantum Computers and the Universe | first = Don | last = Monroe | date = December 2022}}
</ref><ref>{{Cite web| url = https://thequantuminsider.com/2023/06/20/psiquantum-sees-700x-reduction-in-computational-resource-requirements-to-break-elliptic-curve-cryptography-with-a-fault-tolerant-quantum-computer/ | website = The Quanrum Insider | title = PsiQuantum Sees 700x Reduction in Computational Resource Requirements to Break Elliptic Curve Cryptography With a Fault Tolerant Quantum Computer| first = Matt | last = Swayne | date = June 20, 2023 }}
</ref> In particular, processing large amounts of non-quantum data is a challenge for quantum computers.<ref name=CACM/>
* Some promising algorithms have been "dequantized", i.e., their non-quantum analogues with similar complexity have been found.
* If [[quantum error correction]] is used to scale quantum computers to practical applications, its overhead may undermine speedup offered by many quantum algorithms.<ref name=CACM/>
* Complexity analysis of algorithms sometimes makes abstract assumptions that do not hold in applications. For example, input data may not already be available encoded in quantum states, and "oracle functions" used in [[Grover's algorithm]] often have internal structure that can be exploited for faster algorithms.

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

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> [[Paul Davies]] argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the [[holographic principle]].<ref>{{cite arXiv|last1=Davies|first1=Paul|date=6 March 2007 |title=The implications of a holographic universe for quantum information science and the nature of physical law |eprint=quant-ph/0703041}}</ref> Skeptics like [[Gil Kalai]] doubt that quantum supremacy will ever be achieved.<ref>{{cite web |author=Regan |first=K. W. |date=23 April 2016 |title=Quantum Supremacy and Complexity |url=https://rjlipton.wordpress.com/2016/04/22/quantum-supremacy-and-complexity/ |website=Gödel's Lost Letter and P=NP}}</ref><ref>{{cite journal |last1=Kalai |first1=Gil |date=May 2016 |title=The Quantum Computer Puzzle |journal=Notices of the AMS |volume=63 |number=5 |pages=508–516 |url=https://www.ams.org/journals/notices/201605/rnoti-p508.pdf}}</ref><ref>{{cite arXiv |last1=Rinott |first1=Yosef |last2=Shoham |first2=Tomer |last3=Kalai |first3=Gil |date=2021-07-13 |title=Statistical Aspects of the Quantum Supremacy Demonstration |class=quant-ph |eprint=2008.05177}}</ref> Physicist [[Mikhail Dyakonov]] has expressed skepticism of quantum computing as follows:
:"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/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>

=== Physical realizations ===
{{Further|List of proposed quantum registers}}

[[File:IBM Q system (Fraunhofer 2).jpg|thumb|upright=1.2|[[IBM Q System One|Quantum System One]], a quantum computer by [[IBM]] from 2019 with 20 [[Superconducting 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 practical quantum computer must use a physical system as a programmable quantum register.<ref>{{Cite journal |last1=Tacchino |first1=Francesco |last2=Chiesa |first2=Alessandro |last3=Carretta |first3=Stefano |last4=Gerace |first4=Dario |date=2019-12-19 |title=Quantum Computers as Universal Quantum Simulators: State-of-the-Art and Perspectives |url=https://onlinelibrary.wiley.com/doi/10.1002/qute.201900052 |journal=Advanced Quantum Technologies |language=en |volume=3 |issue=3 |pages=1900052 |doi=10.1002/qute.201900052 |arxiv=1907.03505 |s2cid=195833616 |issn=2511-9044}}</ref> Researchers are exploring several technologies as candidates for reliable qubit implementations.{{sfn|Grumbling|Horowitz|2019|page=127}} [[Superconductors]] and [[trapped ion]]s are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.{{sfn|Grumbling|Horowitz|2019|page=114}}

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

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

Research efforts to create stabler qubits for quantum computing include [[topological quantum computer]] approaches. For example, [[Microsoft Azure Quantum| Microsoft]] is working on a computer based on the quantum properties of two-dimensional quasiparticles called [[anyons]].<ref>{{Cite web |last=Mackie |first=Kurt |date=February 8, 2024 |title=Microsoft Quantum Computing Getting DARPA Funding |url=https://rcpmag.com/Articles/2024/02/08/Microsoft-Quantum-Computing-DARPA.aspx |access-date=February 9, 2024 |website=rcpmag.com}}</ref><ref>{{Cite web |last=Gent |first=Edd |date=2023-07-05 |title=Microsoft Wants to Build a Quantum Supercomputer Within a Decade |url=https://singularityhub.com/2023/07/05/microsoft-plans-to-build-a-quantum-supercomputer-within-a-decade/ |access-date=2024-10-18 |website=Singularity Hub |language=en-US}}</ref><ref>{{Cite web |author1=Lucian Armasu |date=2016-11-22 |title=Microsoft Aims To Create World's First Topological Quantum Computer |url=https://www.tomshardware.com/news/microsoft-first-topological-quantum-computer,33068.html |access-date=2024-10-18 |website=Tom's Hardware |language=en}}</ref>

== Potential applications ==

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

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= |pmid=31578480 |bibcode=2019Natur.574...22G|s2cid=203626236 }}</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.}}
== Theory<span class="anchor" id="Relation to computability and complexity theory"></span> ==

=== Computability<span class="anchor" id="Computability theory"></span> ===
{{Further|Computability theory}}

Any [[computational problem]] solvable by a classical computer is also solvable by a quantum computer.{{sfn|Nielsen|Chuang|2010|p=29}} Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using [[quantum mechanics]], which underlies the operation of quantum computers.

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. More formally, any quantum computer can be simulated by a [[Turing machine]]. In other words, quantum computers provide no additional power over classical computers in terms of [[computability]]. This means that quantum computers cannot solve [[undecidable problem]]s like the [[halting problem]], and the existence of quantum computers does not disprove the [[Church–Turing thesis]].{{sfn|Nielsen|Chuang|2010|p=126}}

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

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

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

== Notes ==
{{notelist}}

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

==Sources==
* {{cite book |last=Aaronson |first=Scott |author-link=Scott Aaronson |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 |doi=10.17226/25196 |title=Quantum Computing: Progress and Prospects |date=2019 |editor-first1=Emily |editor-last1=Grumbling |editor-first2=Mark |editor-last2=Horowitz |publisher=The National Academies Press |isbn=978-0-309-47970-7 |location=Washington, DC |s2cid=125635007 |oclc=1091904777}}
* {{cite book |last1=Mermin |first1=N. David |author1-link=N. David Mermin |title=Quantum Computer Science: An Introduction |date=2007 |isbn=978-0-511-34258-5 |oclc=422727925 |doi=10.1017/CBO9780511813870 }}
* {{cite book | author1-link= Michael Nielsen| last1=Nielsen |first1=Michael |author2-link =  Isaac L. Chuang |last2=Chuang |first2=Isaac |title=[[Quantum Computation and Quantum Information]] |year=2010 |edition=10th anniversary |isbn=978-0-511-99277-3 |oclc= 700706156 |doi=10.1017/CBO9780511976667 | s2cid=59717455 }}
* {{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 |author-link=Peter Shor}}

== Further reading ==
{{refbegin|30em}}
===Textbooks===
* {{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 |publisher=MIT Press |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 |publisher=O'Reilly Media, Incorporated |oclc=1111634190 |isbn=978-1-4920-3968-6 }}
* {{cite book |last1=Kaye |first1=Phillip |last2=Laflamme |first2=Raymond |last3=Mosca |first3=Michele |author-link2=Raymond Laflamme |author-link3=Michele Mosca |title=An Introduction to Quantum Computing |year=2007 |publisher=OUP Oxford |oclc=85896383 |isbn=978-0-19-857000-4 }}  
* {{cite book |last1=Kitaev |first1=Alexei Yu. |author-link1=Alexei Kitaev |last2=Shen |first2=Alexander H. |last3=Vyalyi |first3=Mikhail N. |title=Classical and Quantum Computation |year=2002 |publisher=American Mathematical Soc. |oclc=907358694 |isbn=978-0-8218-3229-5 }}
* {{cite book |last1= Kurgalin|first1= Sergei|last2 = Borzunov|first2 = Sergei|date= 2021|title= Concise Guide to Quantum Computing: Algorithms, Exercises, and Implementations|url= https://dx.doi.org/10.1007/978-3-030-65052-0|publisher= Springer|doi= 10.1007/978-3-030-65052-0|isbn= 978-3-030-65052-0}}
* {{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 |last1=Susskind |first1=Leonard |title=Quantum Mechanics: The Theoretical Minimum |last2=Friedman |first2=Art |date=2014 |publisher=[[Basic Books]] |isbn=978-0-465-08061-8 |location=[[New York City|New York]] |author-link=Leonard Susskind}}
* {{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 |access-date=6 February 2022 |archive-date=29 January 2022 |archive-url=https://web.archive.org/web/20220129214631/http://www.thomaswong.net/introduction-to-classical-and-quantum-computing-1e.pdf |url-status=dead }}
* {{cite book |last1=Zeng |first1=Bei |last2=Chen |first2=Xie |last3=Zhou |first3=Duan-Lu |last4=Wen |first4=Xiao-Gang |title=Quantum Information Meets Quantum Matter |year=2019 |oclc=1091358969 |isbn=978-1-4939-9084-9 |doi=10.1007/978-1-4939-9084-9 |arxiv=1508.02595 |s2cid=118528258 }}

===Academic papers===
*{{cite journal | author1-link=Derek Abbott |last1=Abbot |first1=Derek |author2-link= Charles R. Doering |last2=Doering |first2=Charles R. |author3-link= Carlton M. Caves |last3=Caves |first3=Carlton M. |author4-link=Daniel Lidar |last4=Lidar |first4=Daniel M. |author5-link= Howard Brandt|last5=Brandt |first5=Howard E. |author6-link= Alexander R. Hamilton |last6=Hamilton |first6=Alexander R. |author7-link=David K. Ferry |last7=Ferry |first7=David K. |author8-link=Julio Gea-Banacloche |last8=Gea-Banacloche |first8=Julio |author9-link=Sergey M. Bezrukov |last9=Bezrukov |first9=Sergey M. |author10-link=Laszlo B. Kish |first10=Laszlo B. |last10=Kish |display-authors=5 |title=Dreams versus Reality: Plenary Debate Session on Quantum Computing |journal=Quantum Information Processing |year=2003 |volume=2 |issue=6 |pages=449–472 |doi=10.1023/B:QINP.0000042203.24782.9a | arxiv=quant-ph/0310130 |bibcode=2003QuIP....2..449A |hdl=2027.42/45526|s2cid=34885835 }}
*{{cite book |last=Berthiaume |first=Andre |title=Solution Manual for Quantum Mechanics |date=1 December 1998 |chapter=Quantum Computation |s2cid=128255429 |doi=10.1142/9789814541893_0016 |via=Semantic Scholar|pages=233–234 |isbn=978-981-4541-88-6 }}
*{{Cite journal |last1=DiVincenzo |first1=David P. |author1-link=David DiVincenzo |title=The Physical Implementation of Quantum Computation|journal=Fortschritte der Physik |volume=48|issue=9–11|pages=771–783|year=2000 |doi=10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E |arxiv=quant-ph/0002077 |bibcode=2000ForPh..48..771D |s2cid=15439711 }}
*{{cite journal |last=DiVincenzo |first=David P. |title=Quantum Computation |journal=Science |year=1995 |volume=270 |issue=5234 |pages=255–261 |doi= 10.1126/science.270.5234.255 |bibcode = 1995Sci...270..255D |citeseerx=10.1.1.242.2165 |s2cid=220110562 }} Table 1 lists switching and dephasing times for various systems.
*{{cite journal |last=Feynman |first=Richard |author-link=Richard Feynman |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=[[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 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-inline|Quantum computer}}
*{{Wikiversity inline|Quantum computing}}
* [[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 [[Michael Nielsen]]
 
;Lectures
* [https://www.youtube.com/playlist?list=PL1826E60FD05B44E4 Quantum computing for the determined] – 22 video lectures by [[Michael Nielsen]]
* [http://www.quiprocone.org/Protected/DD_lectures.htm Video Lectures] by [[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]

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

{{Authority control}}

[[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:Computer-related introductions in 1980]]
[[Category:Supercomputers]]