quantum computing

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

For many years, the fields of quantum mechanics and computer science formed distinct academic communities.{{sfn|Aaronson|2013|p=132}} Modern quantum theory developed in the 1920s to explain perplexing physical phenomena observed at atomic scales,{{cite book|first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8 |at=§1 |quote=Quantum physics has replaced classical physics as the correct fundamental description of our physical universe. It is used routinely to describe most phenomena that occur at short distances. [...] The era of quantum physics began in earnest in 1925 with the discoveries of Erwin Schrödinger and Werner Heisenberg. The seeds for these discoveries were planted by Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, and others.}}{{cite book|first=Steven |last=Weinberg |author-link=Steven Weinberg |title=Lectures on Quantum Mechanics |year=2015 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-11166-0|chapter=Historical Introduction |pages=1–30}} and digital computers emerged in the following decades to replace human computers for tedious calculations.{{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}} Both disciplines had practical applications during World War II; computers played a major role in wartime cryptography,{{Cite book |last=Hodges |first=Andrew |author-link=Andrew Hodges |title=Alan Turing: The Enigma |publisher=Princeton University Press |year=2014 |isbn=9780691164724 |location=Princeton, New Jersey |page=xviii |language=en-US}} and quantum physics was essential for nuclear physics used in the Manhattan Project.{{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}}

As physicists applied quantum mechanical models to computational problems and swapped digital bits 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.{{cite journal|last1=Benioff|first1=Paul |author-link=Paul Benioff |year=1980|title=The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines|journal=Journal of Statistical Physics|volume=22|issue=5|pages=563–591|bibcode=1980JSP....22..563B|doi=10.1007/bf01011339|s2cid=122949592}}

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

In a 1984 paper, Charles Bennett and Gilles Brassard applied quantum theory to cryptography protocols and demonstrated that quantum key distribution could enhance information security.{{cite book|first1=C. H. |last1=Bennett |first2=G. |last2=Brassard |chapter=Quantum cryptography: Public key distribution and coin tossing |title=Proceedings of the International Conference on Computers, Systems & Signal Processing, Bangalore, India |volume=1 |pages=175–179 |publisher=IEEE |year=1984 |location=New York }} Reprinted as {{cite journal|first1=C. H. |last1=Bennett |first2=G. |last2=Brassard |title=Quantum cryptography: Public key distribution and coin tossing |journal=Theoretical Computer Science |series=Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84 |volume=560 |number=1 |date=4 December 2014 |pages=7–11 |doi=10.1016/j.tcs.2014.05.025 |doi-access=free|arxiv=2003.06557 }}{{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 }}

Quantum algorithms then emerged for solving oracle problems, such as Deutsch's algorithm in 1985,{{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. }} the Bernstein{{en dash}}Vazirani algorithm in 1993,{{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}} and Simon's algorithm in 1994.{{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}}

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 superposition, sometimes referred to as quantum parallelism.{{sfn|Nielsen|Chuang|2010|p=30-32}}

File:Peter Shor 2017 Dirac Medal Award Ceremony.png (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 his 1994 algorithm for breaking the widely used RSA and Diffie{{en dash}}Hellman encryption protocols,{{sfn|Shor|1994}} which drew significant attention to the field of quantum computing. In 1996, Grover's algorithm established a quantum speedup for the widely applicable unstructured search problem.{{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}}{{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,{{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}} validating Feynman's 1982 conjecture.{{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}}

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

In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology,{{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}}{{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}} 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.{{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}}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}}
{{*}}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}}{{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}} However, the validity of this claim is still being actively researched.{{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}}{{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}}

{{See also|Introduction to quantum mechanics}}

Computer engineers 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 from their environment, so any quantum information quickly decoheres.

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

Quantum programs, in contrast, rely on precise control of coherent quantum systems. Physicists describe these systems mathematically using linear algebra. Complex numbers model probability amplitudes, vectors model quantum states, and matrices model the operations that can be performed on these states. Programming a quantum computer is then a matter of composing operations in such a way that the resulting program computes a useful result in theory and is implementable in practice.

As physicist Charlie Bennett describes the relationship between quantum and classical computers,{{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}}

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

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 $|0\backslash rangle$ and $|1\backslash rangle$ serve as the quantum counterparts of the classical states 0 and 1. However, the quantum states $|0\backslash rangle$ and $|1\backslash rangle$ 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 $|0\backslash rangle$ and $|1\backslash rangle$.{{sfn|Nielsen|Chuang|2010|page=13}}{{sfn|Mermin|2007|p=17}}

A two-dimensional vector mathematically represents a qubit state. Physicists typically use Dirac notation for quantum mechanical linear algebra, writing $|\backslash psi\backslash rangle$ {{gloss|ket psi}} for a vector labeled $\backslash psi$ . Because a qubit is a two-state system, any qubit state takes the form $\backslash alpha|0\backslash rangle+\backslash beta|1\backslash rangle$ , where $|0\backslash rangle$ and $|1\backslash rangle$ are the standard basis states,{{efn|The standard basis is also the computational basis.{{sfn|Mermin|2007|p=18}}}} and $\backslash alpha$ and $\backslash beta$ are the probability amplitudes, which are in general complex numbers.{{sfn|Mermin|2007|p=17}} If either $\backslash alpha$ or $\backslash beta$ is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a quantum state vector acts similarly to a (classical) probability vector, 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 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 $\backslash alpha|0\backslash rangle+\backslash beta|1\backslash rangle$, the state collapses to $|0\backslash rangle$ with probability $|\backslash alpha|^2$, or to $|1\backslash rangle$ with probability $|\backslash beta|^2$.

Any valid qubit state has coefficients $\backslash alpha$ and $\backslash beta$ such that $|\backslash alpha|^2+|\backslash beta|^2\; =\; 1$.

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

Each additional qubit doubles the dimension of the 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 entangled because neither qubit has a state vector of its own.

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

{{See also|Unitarity (physics)}}

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

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

: $X|0\backslash rangle\; =\; |1\backslash rangle$ and $X|1\backslash rangle\; =\; |0\backslash rangle$.

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

|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}.

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

\operatorname{CNOT} :=

\begin{pmatrix}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 0 & 1 \\

0 & 0 & 1 & 0

\end{pmatrix}.

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

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

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}}

{{Further|Quantum programming}}

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

File:Quantum Toffoli Gate Implementation.svg from more primitive gates|upright=1.15]]

A quantum gate array decomposes computation into a sequence of few-qubit quantum gates. A quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits 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 $2^n\; \backslash times\; 2^n$ over $n$ 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 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. Implementation of Boolean functions using the few-qubit quantum gates is presented here.{{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}}

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

An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.{{cite journal|last1=Das|first1=A.|last2=Chakrabarti|first2=B. K.|year=2008|title=Quantum Annealing and Analog Quantum Computation|journal=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}}

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

A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice.{{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}}

A quantum Turing machine is the quantum analog of a Turing machine. All of these models of computation—quantum circuits,{{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}} one-way quantum computation,{{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}} adiabatic quantum computation,{{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}} and topological quantum computation{{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}}—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.

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". According to some researchers, noisy intermediate-scale quantum (NISQ) machines may have specialized uses in the near future, but noise in quantum gates limits their reliability.{{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}}

Scientists at 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.{{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 }}{{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}} The Harvard research team was supported by MIT, QuEra Computing, Caltech, and Princeton University and funded by DARPA's Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program.{{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}}{{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}}

Quantum computing has significant potential applications in the fields of cryptography and cybersecurity. Quantum cryptography, which leverages the principles of quantum mechanics, offers the possibility of secure communication channels that are fundamentally 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. Additionally, quantum random number generators (QRNGs) can produce high-quality randomness, which is essential for secure encryption.

At the same time, quantum computing poses substantial challenges to traditional cryptographic systems. Shor's algorithm, a quantum algorithm for integer factorization, could potentially break widely used public-key encryption schemes like RSA, which rely on the intractability of factoring large numbers. This has prompted a global effort to develop post-quantum cryptography—algorithms designed to resist both classical and quantum attacks. This field remains an active area of research and standardization, aiming to future-proof critical infrastructure against quantum-enabled threats.

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

Quantum computing also presents broader systemic and geopolitical risks. These include the potential to break current encryption protocols, disrupt financial systems, and accelerate the development of dual-use technologies such as advanced military systems or engineered pathogens. As a result, nations and corporations are actively investing in post-quantum safeguards, and the race for quantum supremacy is increasingly shaping global power dynamics.{{cite web |last=Inderwildi |first=Oliver |title=The Quantum Computing Revolution: From Technological Opportunity to Geopolitical Power Shift |url=https://medium.com/the-geopolitical-economist/the-quantum-computing-revolution-geopolitics-economics-c2380e0167ee |website=The Geopolitical Economist |date=April 14, 2025 |access-date=April 14, 2025}}

{{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.{{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}} When a sender and receiver exchange quantum states, they can guarantee that an adversary does not intercept the message, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change.{{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 }} With appropriate cryptographic protocols, the sender and receiver can thus establish shared private information resistant to eavesdropping.{{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}}

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.{{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 }}{{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}}

Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.{{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}}

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 logarithms, solving Pell's equation, and more generally solving the hidden subgroup problem for abelian finite groups. 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.{{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}} Certain oracle problems like Simon's problem and the Bernstein–Vazirani problem do give provable speedups, though this is in the 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 polynomials, 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. 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}}

{{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 simulation may be an important application of quantum computing.{{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 }} Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.{{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}} In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.{{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}}{{cite journal |author=Kim, Youngseok |display-authors=et al. |title=Evidence for the utility of quantum computing before fault tolerance |date=14 June 2023 |journal=Nature |volume=618 |issue=7965 |pages=500–505 |doi=10.1038/s41586-023-06096-3 |pmid=37316724 |pmc=10266970 |bibcode=2023Natur.618..500K }}

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.{{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 TV |via=YouTube |date=21 November 2018 |access-date=4 February 2021 |first=Andrea |last=Morello |author-link=Andrea Morello }} It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process{{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}} by the mid-2020s{{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}} although some have predicted it will take longer.{{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}}

{{Main|Post-quantum cryptography}}

A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes).{{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 }} 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 cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of post-quantum cryptography.{{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 }}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. 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.{{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}} Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem.{{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}} It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2ⁿ in the classical case,{{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 }} 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).

{{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 $n$ items in a database. This can be solved by Grover's algorithm using $O(\backslash sqrt\{n\})$ queries to the database, quadratically fewer than the $\backslash Omega(n)$ 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 Brassard, Høyer, and Tapp's algorithm for finding collisions in two-to-one functions,{{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}} and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.{{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}}

Problems that can be efficiently addressed with Grover's algorithm have the following properties:{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=Springer |year=2011 |isbn=978-1-84628-887-6 |pages=242–244}}{{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}}

1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. There exists a Boolean function 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{{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 }} 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 cracker that attempts to guess a password. Breaking symmetric ciphers with this algorithm is of interest to government agencies.{{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}}

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.{{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}}

{{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.{{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}}

For example, the HHL Algorithm, named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.{{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}} Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.{{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}}{{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}}{{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 }}

{{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 and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models{{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}} including quantum GANs{{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}} may eventually be developed into ultimate generative chemistry algorithms.

File:A Wafer of the Latest D-Wave Quantum Computers (39188583425).jpg of adiabatic quantum computers]]

{{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.

There are a number of technical challenges in building a large-scale quantum computer.{{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}} Physicist David DiVincenzo has listed these requirements for a practical quantum computer:{{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}}

• 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 computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.{{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}}

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.{{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}}

One of the greatest challenges involved in constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T₂ (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperatures.{{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}} Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator{{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}}) in order to prevent significant decoherence.{{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}} A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.{{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 }}

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.{{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}}

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

As described by the threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often-cited figure for the required error rate in each gate for fault-tolerant computation is 10⁻³, assuming the noise is depolarizing.

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L², where L is the number of binary digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 10⁴ bits without error correction.{{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}} With error correction, the figure would rise to about 10⁷ bits. Computation time is about L² or about 10⁷ steps and at 1{{nbsp}}MHz, about 10 seconds. However, the encoding and error-correction overheads increase the size of a real fault-tolerant quantum computer by several orders of magnitude. Careful estimates{{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 |bibcode=2015PhDT........56A |language=en-US |oclc=923881411}}{{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 }} 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{{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}} 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 anyons, quasi-particles used as threads, and relying on braid theory to form stable logic gates.{{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}}{{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}}

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.{{cite arXiv |last=Preskill |first=John |date=2012-03-26 |title=Quantum computing and the entanglement frontier |eprint=1203.5813 |class=quant-ph}}{{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}}{{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}} The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.{{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}}

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 quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world's fastest computer.{{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}}{{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}} This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,{{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}}{{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}} 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{{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}}{{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}}{{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 }} and even beating it.{{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}}{{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}}{{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}}

In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer, Jiuzhang, to demonstrate quantum supremacy.{{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=}}{{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}}{{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}} 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.{{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}}

Claims of quantum supremacy have generated hype around quantum computing,{{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 }} but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.{{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}}

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.{{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 }}

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 Nature spotlight article summarized current quantum computers as being "For now, [good for] absolutely 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 }}

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{{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}}

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 entanglement before getting overwhelmed by noise.
• Quantum algorithms provide speedup over conventional algorithms only for some tasks, and matching these tasks with practical applications proved challenging. Some promising tasks and applications require resources far beyond those available today.{{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}}

{{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 }}

In particular, processing large amounts of non-quantum data is a challenge for quantum computers.

• 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.
• 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.

Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.{{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}} Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.{{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}} Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.{{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}}{{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}}{{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}} 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 10³⁰⁰... Could we ever learn to control the more than 10³⁰⁰ continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never."{{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}}

{{Further|List of proposed quantum registers}}

File:IBM Q system (Fraunhofer 2).jpg, a quantum computer by IBM from 2019 with 20 superconducting qubits{{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}}]]

A practical quantum computer must use a physical system as a programmable quantum register.{{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}} Researchers are exploring several technologies as candidates for reliable qubit implementations.{{sfn|Grumbling|Horowitz|2019|page=127}} Superconductors and trapped ions are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.{{sfn|Grumbling|Horowitz|2019|page=114}}

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

The first quantum logic gates were implemented with trapped ions 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}}

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.{{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}}

{{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 problems like the halting problem, and the existence of quantum computers does not disprove the Church–Turing thesis.{{sfn|Nielsen|Chuang|2010|p=126}}

{{Main|Quantum complexity theory}}

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 factor integers, while this is not believed to be the case for classical computers.

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". 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 BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.{{sfn|Nielsen|Chuang|2010|p=41}} It is known that $\backslash mathsf\{BPP\backslash subseteq\; BQP\}$ and is widely suspected that $\backslash mathsf\{BQP\backslash subsetneq\; BPP\}$, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.{{sfn|Nielsen|Chuang|2010|p=201}}

File:BQP complexity class diagram.svg

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that $\backslash mathsf\{P\backslash subseteq\; BQP\; \backslash subseteq\; PSPACE\}$; 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 $\backslash mathsf\{NP\backslash nsubseteq\; BQP\}$; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).{{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 }}

{{Commons category|Quantum computing}}

{{cols|colwidth=21em}}

• {{annotated link|D-Wave Systems}}
• {{annotated link|Electronic quantum holography}}
• {{annotated link|Glossary of quantum computing}}
• {{annotated link|IARPA}}
• {{annotated link|India's quantum computer}}
• {{annotated link|IonQ}}
• {{annotated link|IQM}}
• {{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 sensor}}
• {{annotated link|Quantum volume}}
• {{annotated link|Quantum weirdness}}
• {{annotated link|Rigetti Computing}}
• {{annotated link|Supercomputer}}
• {{annotated link|Theoretical computer science}}
• {{annotated link|Unconventional computing}}
• {{annotated link|Valleytronics}}

{{colend}}

{{notelist}}

{{Reflist|30em}}

• {{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}}

{{refbegin|30em}}

• {{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 |editor-last1=Exman |editor-first1=Iaakov |editor-last2=Pérez-Castillo |editor-first2=Ricardo |editor-last3=Piattini |editor-first3=Mario |editor-last4=Felderer |editor-first4=Michael |title=Quantum Software: Aspects of Theory and System Design |year=2024 |url=https://link.springer.com/book/10.1007/978-3-031-64136-7 |isbn=978-3-031-64136-7 |publisher=Springer Nature|doi=10.1007/978-3-031-64136-7 }}
• {{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/book/10.1007/978-3-030-61601-4 }}
• {{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 |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 }}

• {{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=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}}

• {{Commons-inline|Quantum computer}}
• {{Wikiversity inline|Quantum computing}}
• Stanford Encyclopedia of Philosophy: "[https://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://introtoquantum.org Introduction to Quantum Computing for Business by Koen Groenland]

;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
• 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