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{{Short description|Basic unit of quantum information}}
{{About|the quantum computing unit}}{{distinguish|Cubit}}{{Fundamental info units}}
[[File:Simple qubits.svg|thumb|upright=0.6|The general definition of a qubit as the quantum state of a two-[[Energy level|level]] quantum system.]]
In [[quantum computing]], a '''qubit''' ({{IPAc-en|ˈ|k|juː|b|ɪ|t}}) or '''quantum bit''' is a basic unit of [[quantum information]]—the quantum version of the classic binary [[bit]] physically realized with a two-state device. A qubit is a [[Two-state quantum system|two-state (or two-level) quantum-mechanical system]], one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the [[Spin (physics)|spin]] of the [[electron]] in which the two levels can be taken as spin up and spin down; or the [[Photon polarization|polarization]] of a single [[photon]] in which the two spin states (left-handed and the right-handed circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent{{clarify|date=February 2025}} [[Quantum superposition|superposition]] of multiple states simultaneously, a property that is fundamental to [[quantum mechanics]] and [[quantum computing]].

==Etymology==
The coining of the term ''qubit'' is attributed to [[Benjamin Schumacher]].<ref name="schumacher1995">
{{cite journal |author=Schumacher |first=B. |author-link=Benjamin Schumacher |year=1995 |title=Quantum coding |journal=[[Physical Review A]] |volume=51 |issue=4 |pages=2738–2747 |bibcode=1995PhRvA..51.2738S |doi=10.1103/PhysRevA.51.2738 |pmid=9911903}}</ref> In the acknowledgments of his 1995 paper, Schumacher states that the term ''qubit'' was created in jest during a conversation with [[William Wootters]].

==Bit versus qubit==
A [[binary digit]], characterized as 0 or 1, is used to represent information in classical computers. When averaged over both of its states (0,1), a binary digit can represent up to one bit of [[information content]], where a [[bit]] is the basic unit of [[information theory|information]]. However, in this article, the word bit is synonymous with a binary digit.

In classical computer technologies, a ''processed'' bit is implemented by one of two levels of low [[Direct Current|direct current]] [[voltage]], and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between two [[logic level]]s must be passed as fast as possible, as electrical voltage cannot change from one level to another instantly.

There are two possible outcomes for the measurement of a qubit—usually taken to have the value "0" and "1", like a bit. However, whereas the state of a bit can only be binary (either 0 or 1), the general state of a qubit according to quantum mechanics can arbitrarily be a [[Quantum superposition|coherent superposition]] of ''all'' computable states simultaneously.<ref name="nielsen2010">{{cite book |last1=Nielsen |first1=Michael A. |title=Quantum Computation and Quantum Information |title-link=Quantum Computation and Quantum Information (book) |last2=Chuang |first2=Isaac L. |date=2010 |publisher=[[Cambridge University Press]] |isbn=978-1-107-00217-3 |page=[https://archive.org/details/quantumcomputati00niel_993/page/n46 13] |language=en-US}}</ref> Moreover, whereas a measurement of a classical bit would not disturb its state, a measurement of a qubit would destroy its coherence and irrevocably disturb the superposition state. It is possible to fully encode one bit in one qubit. However, a qubit can hold more information, e.g., up to two bits using [[superdense coding]].

A bit is always completely in either one of its two states, and a set of {{math|n}} bits (e.g. a [[processor register]] or some bit array) can only hold a single of its {{math|2<sup>n</sup>}} possible states at any time. A quantum state can be in a superposition state, which means that the qubit can have non-zero [[probability amplitude]] in both its states simultaneously (popularly expressed as "it can be in both states simultaneously"). A qubit requires two [[complex number]]s to describe its two probability amplitudes, and these two complex numbers can together be viewed as a 2-dimensional complex [[Vector (mathematics and physics)|vector]], which is called a ''quantum state vector'', or ''superposition state vector.'' Alternatively and equivalently, the value stored in a qubit can be described as a single point in a 2-dimensional [[complex coordinate space]]. Similarly, a set of ''n'' qubits, which is also called a [[quantum register|register]], requires {{math|2<sup>n</sup>}} complex numbers to describe its superposition state vector.<ref name="shor1996">{{cite journal |last1=Shor |first1=Peter |title=Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer∗ |journal=SIAM Journal on Computing |volume=26 |issue=5 |pages=1484–1509 |year=1997 |arxiv=quant-ph/9508027 |bibcode=1995quant.ph..8027S |doi=10.1137/S0097539795293172 |s2cid=2337707}}</ref>{{r|Williams|pages=7–17}}{{r|nielsen2010|pages=13–17}}

==Standard representation==
In quantum mechanics, the general [[quantum state]] of a qubit can be represented by a linear superposition of its two [[Orthonormality|orthonormal]] [[Basis (linear algebra)|basis]] states (or basis [[vector space|vector]]s). These vectors are usually denoted as <math>| 0 \rangle = \bigl[\begin{smallmatrix}
1\\
0
\end{smallmatrix}\bigr]</math> and <math>| 1 \rangle = \bigl[\begin{smallmatrix}
0\\
1
\end{smallmatrix}\bigr]</math>. They are written in the conventional [[List of things named after Paul Dirac|Dirac]]—or [[bra–ket notation|"bra–ket"]]—notation; the <math>| 0 \rangle </math> and <math>| 1 \rangle </math> are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states, <math>\{|0\rangle,|1\rangle\}</math>, together called the computational basis, are said to span the two-dimensional [[Hilbert space|linear vector (Hilbert) space]] of the qubit.<ref name="Yanofsky-Mannucci"/>

Qubit basis states can also be combined to form product basis states. A set of qubits taken together is called a [[quantum register]]. For example, two qubits could be represented in a four-dimensional linear vector space spanned by the following product basis states:

<math>| 00 \rangle = \biggl[\begin{smallmatrix}
1\\
0\\
0\\
0
\end{smallmatrix}\biggr]</math>, <math>| 01 \rangle = \biggl[\begin{smallmatrix}
0\\
1\\
0\\
0
\end{smallmatrix}\biggr]</math>, <math>| 10 \rangle = \biggl[\begin{smallmatrix}
0\\
0\\
1\\
0
\end{smallmatrix}\biggr]</math>, and <math>| 11 \rangle = \biggl[\begin{smallmatrix}
0\\
0\\
0\\
1
\end{smallmatrix}\biggr]</math>.

In general, ''n'' qubits are represented by a superposition state vector in 2<sup>''n''</sup> dimensional Hilbert space.<ref name="Yanofsky-Mannucci">{{Cite book|title=Quantum computing for computer scientists|last1=Yanofsky|first1=Noson S.|last2=Mannucci|first2=Mirco|date=2013|publisher=[[Cambridge University Press]]|isbn=978-0-521-87996-5|pages=138–144}}</ref>

==Qubit states==
[[File:Qubit represented by linear polarization of light.png|thumb|right|[[Polarization_(waves)|Polarization of light]] offers a straightforward way to present orthogonal states. With a typical mapping <math>|H\rangle=|0\rangle</math> and <math>|V\rangle=|1\rangle</math>, quantum states <math>(|0\rangle \pm |1\rangle)/\sqrt{2}</math> have a direct physical representation, both easily demonstrable experimentally in a class with [[Polarizer|linear polarizers]] and, for real <math>\alpha</math> and <math>\beta</math>, matching the high-school definition of [[orthogonality]].<ref>{{cite journal|last1=Seskir|first1=Zeki C.|last2=Migdał|first2=Piotr|last3=Weidner|first3=Carrie|last4=Anupam|first4=Aditya|last5=Case|first5=Nicky|last6=Davis|first6=Noah|last7=Decaroli|first7=Chiara|last8=Ercan|first8=İlke|last9=Foti|first9=Caterina|last10=Gora|first10=Paweł|last11=Jankiewicz|first11=Klementyna|last12=La Cour|first12=Brian R.|last13=Malo|first13=Jorge Yago|last14=Maniscalco|first14=Sabrina|last15=Naeemi|first15=Azad|last16=Nita|first16=Laurentiu|last17=Parvin|first17=Nassim|last18=Scafirimuto|first18=Fabio|last19=Sherson|first19=Jacob F.|last20=Surer|first20=Elif|last21=Wootton|first21=James|last22=Yeh|first22=Lia|last23=Zabello|first23=Olga|last24=Chiofalo|first24=Marilù|title=Quantum games and interactive tools for quantum technologies outreach and education|journal=Optical Engineering|volume=61|issue=8|pages=081809|year=2022|arxiv=2202.07756|doi=10.1117/1.OE.61.8.081809|bibcode=2022OptEn..61h1809S }}{{Creative Commons text attribution notice|cc=by4|from this source=yes}}</ref>]]

A pure qubit state is a [[quantum coherence|coherent]] [[quantum superposition|superposition]] of the basis states. This means that a single qubit (<math>\psi</math>) can be described by a [[linear combination]] of <math>|0 \rangle </math> and <math>|1 \rangle </math>:

: <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle  </math>

where <var>α</var> and <var>β</var> are the [[probability amplitude]]s, and are both [[complex number]]s. When we measure this qubit in the standard basis, according to the [[Born rule]], the probability of outcome <math>|0 \rangle </math> with value "0" is <math>| \alpha |^2</math> and the probability of outcome <math>|1 \rangle </math> with value "1" is <math>| \beta |^2</math>. Because the absolute squares of the amplitudes equate to probabilities, it follows that <math>\alpha</math> and <math>\beta</math> must be constrained according to the [[Probability axioms#Second axiom|second axiom of probability theory]] by the equation<ref name="Williams">{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |year=2011 |isbn=978-1-84628-887-6 |pages=9–13}}</ref>

: <math>| \alpha |^2 + | \beta |^2 = 1.</math>

The probability amplitudes, <math>\alpha</math> and <math>\beta</math>, encode more than just the probabilities of the outcomes of a measurement; the ''relative phase'' between <math>\alpha</math> and <math>\beta</math> is for example responsible for [[wave interference|quantum interference]], as seen in the [[double-slit experiment]].

===Bloch sphere representation===
[[File:Bloch sphere.svg|thumb|[[Bloch sphere]] representation of a qubit. The [[probability amplitude]]s for the superposition state, <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\,</math> are given by <math> \alpha = \cos\left(\frac{\theta}{2}\right) </math> and <math> \beta = e^{i \varphi} \sin\left(\frac{\theta}{2}\right)</math>]]

It might, at first sight, seem that there should be four [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle\,</math>, as <math>\alpha</math> and <math>\beta</math> are [[complex number]]s with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint {{math|{{!}}''α''{{!}}<sup>2</sup> + {{!}}''β''{{!}}<sup>2</sup> {{=}} 1}}. This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of [[3-sphere#Hopf coordinates|Hopf coordinates]]:
:<math>\begin{align}
\alpha &= e^{i \delta} \cos\frac{\theta}{2}, \\
\beta &= e^{i (\delta + \varphi)} \sin\frac{\theta}{2}.
\end{align}</math>
Additionally, for a single qubit the ''global [[phase factor|phase]]'' of the state <math>e^{i\delta}</math> has no physically observable consequences,{{efn|This is because of the [[Born rule]]. The probability to observe an outcome upon [[Quantum measurement|measurement]] is the [[modulus squared]] of the [[probability amplitude]] for that outcome (or basis state, [[eigenstate]]). The ''global phase'' factor <math>e^{i\delta}</math> is not measurable, because it applies to both basis states, and is on the complex [[unit circle]] so <math>|e^{i\delta}|^2 = 1.</math><br>Note that by removing <math>e^{i\delta}</math> it means that [[quantum state]]s with global phase can not be represented as points on the surface of the Bloch sphere.}} so we can arbitrarily choose {{math|''α''}} to be real (or {{math|''β''}} in the case that {{math|''α''}} is zero), leaving just two degrees of freedom:
:<math>\begin{align}
\alpha &= \cos\frac{\theta}{2}, \\
\beta &= e^{i \varphi} \sin\frac{\theta}{2},
\end{align}</math>
where <math> e^{i \varphi} </math> is the physically significant ''relative phase''.<ref name="Nielsen-Chuang">{{Cite book|title=Quantum Computation and Quantum Information|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac|date=2010|publisher=[[Cambridge University Press]]|isbn=978-1-10700-217-3|location=Cambridge|oclc=43641333|author-link=Michael Nielsen|author-link2=Isaac Chuang|url=https://www.cambridge.org/9781107002173|pages=13–16}}</ref>{{efn|The Pauli Z basis is usually called the ''computational basis'', where the relative phase have no effect on measurement. [[Quantum measurement|Measuring]] instead in the X or Y Pauli basis depends on the relative phase. For example, <math>(|0\rangle + e^{i\pi/2}|1\rangle)/{\sqrt{2}}</math> will (because this state lies on the positive pole of the Y-axis) in the Y-basis always measure to the same value, while in the Z-basis results in equal probability of being measured to <math>|0\rangle</math> or <math>|1\rangle</math>.<br/>Because measurement [[Wave function collapse|collapses]] the quantum state, measuring the state in one basis hides some of the values that would have been measurable the other basis; See the [[uncertainty principle]].}}

The possible quantum states for a single qubit can be visualised using a [[Bloch sphere]] (see picture). Represented on such a [[2-sphere]], a classical bit could only be at the "North Pole" or the "South Pole", in the locations where <math>|0 \rangle</math> and <math>|1 \rangle</math> are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state <math>(|0 \rangle + |1 \rangle)/{\sqrt{2}}</math> would lie on the equator of the sphere at the positive X-axis. In the [[classical limit]], a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles.

The surface of the Bloch sphere is a [[Plane (mathematics)|two-dimensional space]], which represents the observable [[state space (physics)|state space]] of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles <math>\varphi</math> and <math>\theta</math>.

===Mixed state===
{{Main|Density matrix}} A pure state is fully specified by a single ket, <math>|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,\,</math> a coherent superposition, represented by a point on the surface of the Bloch sphere as described above. Coherence is essential for a qubit to be in a superposition state. With interactions, [[quantum noise]] and [[decoherence]], it is possible to put the qubit in a [[Mixed state (physics)|mixed state]], a statistical combination or "incoherent mixture" of different pure states. Mixed states can be represented by points ''inside'' the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles <math>\varphi</math> and <math>\theta </math>, as well as the length <math>r</math> of the vector that represents the mixed state.

[[Quantum error correction]] can be used to maintain the purity of qubits.

==Operations on qubits==
{{Further|DiVincenzo's criteria|Physical and logical qubits}} There are various kinds of physical operations that can be performed on qubits.
* [[Quantum logic gate]]s, building blocks for a [[quantum circuit]] in a [[quantum computing|quantum computer]], operate on a set of qubits (a [[quantum register|register]]); mathematically, the qubits undergo a ([[reversible computing|reversible]]) [[unitary transformation]] described by [[matrix multiplication|multiplying]] the quantum gates [[unitary matrix]] with the [[quantum state]] vector. The result from this multiplication is a new quantum state vector.
* [[Quantum measurement]] is an irreversible operation in which information is gained about the state of a single qubit, and [[quantum coherence|coherence]] is lost. The result of the measurement of a single qubit with the state <math>|\psi\rangle = \alpha |0\rangle + \beta |1\rangle</math> will be either <math>|0\rangle</math> with probability <math>|\alpha|^2</math> or <math>|1\rangle </math> with probability <math>|\beta|^2</math>. Measurement of the state of the qubit alters the magnitudes of <var>α</var> and <var>β</var>. For instance, if the result of the measurement is <math>|1\rangle</math>, <var>α</var> is changed to 0 and <var>β</var> is changed to the phase factor <math>e^{i \phi}</math> no longer experimentally accessible. If measurement is performed on a qubit that is [[quantum entanglement|entangled]], the measurement may [[Wave function collapse|collapse]] the state of the other entangled qubits.
* Initialization or re-initialization to a known value, often <math>|0\rangle</math>. This operation collapses the quantum state (exactly like with measurement). Initialization to <math>|0\rangle</math> may be implemented logically or physically: Logically as a measurement, followed by the application of the [[Quantum logic gate#X gate|Pauli-X gate]] if the result from the measurement was <math>|1\rangle</math>. Physically, for example if it is a [[Superconducting quantum computing|superconducting]] [[phase qubit]], by lowering the energy of the quantum system to its [[ground state]].
* Sending the qubit through a [[quantum channel]] to a remote system or machine (an [[Input/output|I/O]] operation), potentially as part of a [[quantum network]].

==Quantum entanglement==
{{Main|Quantum entanglement|Bell state}} An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit [[quantum entanglement]]; the qubit itself is an exhibition of quantum entanglement. In this case, quantum entanglement is a local or [[quantum nonlocality|nonlocal]] property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.

The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the <math>|\Phi^+\rangle</math> [[Bell state]]:

:<math>\frac{1}{\sqrt{2}} (|00\rangle + |11\rangle).</math>

In this state, called an ''equal superposition'', there are equal probabilities of measuring either product state <math>|00\rangle</math> or <math>|11\rangle</math>, as <math>|1/\sqrt{2}|^2 = 1/2</math>. In other words, there is no way to tell if the first qubit has value "0" or "1" and likewise for the second qubit.

Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either <math>|0\rangle</math> or <math>|1\rangle</math>, i.e., she can now tell if her qubit has value "0" or "1". Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a <math>|0\rangle</math>, Bob must measure the same, as <math>|00\rangle</math> is the only state where Alice's qubit is a <math>|0\rangle</math>. In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value "0" or "1"—a most surprising circumstance that cannot be explained by classical physics.

===Controlled gate to construct the Bell state===
[[Quantum logic gate#Controlled gates|Controlled gates]] act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the [[controlled NOT gate]] (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is <math>|1\rangle</math>, and otherwise leaves it unchanged. With respect to the unentangled product basis <math>\{|00\rangle</math>, <math>|01\rangle</math>, <math>|10\rangle</math>, <math>|11\rangle\}</math>, it maps the basis states as follows:
:<math> | 0 0 \rangle \mapsto | 0 0 \rangle </math>
:<math> | 0 1 \rangle \mapsto | 0 1 \rangle </math>
:<math> | 1 0 \rangle \mapsto | 1 1 \rangle </math>
:<math> | 1 1 \rangle \mapsto | 1 0 \rangle </math>.

A common application of the CNOT gate is to maximally entangle two qubits into the <math>|\Phi^+\rangle</math> [[Bell state]]. To construct <math>|\Phi^+\rangle</math>, the inputs A (control) and B (target) to the CNOT gate are:

<math>\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)_A</math> <math>\otimes</math> <math>|0\rangle_B</math> = <math>\frac{1}{\sqrt{2}}</math> <math>(|00\rangle + |10\rangle)</math>.

After applying CNOT, the output is the <math>|\Phi^+\rangle</math> Bell State: <math>\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)</math>.

===Applications===
The <math>|\Phi^+\rangle</math> Bell state forms part of the setup of the [[superdense coding]], [[quantum teleportation]], and entangled [[quantum cryptography]] algorithms.

Quantum entanglement also allows multiple states (such as the Bell state mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical computer. Many of the successes of quantum computation and communication, such as quantum teleportation and superdense coding, make use of entanglement, suggesting that entanglement is a [[Computational resource|resource]] that is unique to quantum computation.<ref>{{Cite journal |last=Horodecki |first=Ryszard |display-authors=et al |date=2009 |title=Quantum entanglement |journal=Reviews of Modern Physics |volume=81|issue=2 |pages=865–942 |doi=10.1103/RevModPhys.81.865 |arxiv=quant-ph/0702225 |bibcode=2009RvMP...81..865H |s2cid=59577352}}</ref> A major hurdle facing quantum computing, as of 2018, in its quest to surpass classical digital computing, is noise in quantum gates that limits the size of [[quantum circuit]]s that can be executed reliably.<ref name="preskill2018">{{cite journal |last1=Preskill |first1=John |date=2018 |title=Quantum Computing in the NISQ era and beyond |journal=Quantum |volume=2 |pages=79 |arxiv=1801.00862 |doi=10.22331/q-2018-08-06-79 |bibcode=2018Quant...2...79P |s2cid=44098998 }}</ref>

==Quantum register==
A number of qubits taken together is a [[quantum register|qubit register]]. [[Quantum computer]]s perform calculations by manipulating qubits within a register.

===Qudits and qutrits===
The term '''qudit''' denotes the unit of quantum information that can be realized in suitable ''d''-level quantum systems.<ref>{{Cite journal|last1=Nisbet-Jones|first1=Peter B. R.|last2=Dilley|first2=Jerome|last3=Holleczek|first3=Annemarie|last4=Barter|first4=Oliver |last5=Kuhn|first5=Axel |date=2013|title=Photonic qubits, qutrits and ququads accurately prepared and delivered on demand|url=http://stacks.iop.org/1367-2630/15/i=5/a=053007|journal=New Journal of Physics|language=en |volume=15|issue=5|pages=053007|doi=10.1088/1367-2630/15/5/053007|issn=1367-2630|arxiv=1203.5614 |bibcode=2013NJPh...15e3007N|s2cid=110606655}}</ref> A qubit register that can be measured to ''N'' states is identical to an ''N''-level qudit. A rarely used<ref>As of June 2022 [https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=qudit&terms-0-field=all&classification-physics=True&classification-physics_archives=quant-ph&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first 1150 uses] versus [https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=qunit&terms-0-field=all&classification-physics=True&classification-physics_archives=quant-ph&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first 31 uses] on in the quant-ph category of [[arxiv.org]].</ref> [[synonym]] for qudit is '''quNit''',<ref>{{Cite journal |last1=Kaszlikowski |first1=Dagomir |last2=Gnaciński |first2=Piotr |last3=Żukowski |first3=Marek |last4=Miklaszewski |first4=Wieslaw |last5=Zeilinger |first5=Anton |year=2000 |title=Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits |journal=Physical Review Letters |volume=85 |issue=21 |pages=4418–4421 |arxiv=quant-ph/0005028 |bibcode=2000PhRvL..85.4418K |doi=10.1103/PhysRevLett.85.4418 |pmid=11082560 |s2cid=39822693}}</ref> since both ''d'' and ''N'' are frequently used to denote the dimension of a quantum system.

Qudits are similar to the [[Integer (computer science)|integer type]]s in classical computing, and may be mapped to (or realized by) arrays of qubits. Qudits where the ''d''-level system is not an exponent of 2 cannot be mapped to arrays of qubits. It is for example possible to have 5-level qudits.

In 2017, scientists at the [[National Institute of Scientific Research]] constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits.<ref>{{Cite web |last=Choi |first=Charles Q. |date=2017-06-28 |title=Qudits: The Real Future of Quantum Computing? |url=https://spectrum.ieee.org/qudits-the-real-future-of-quantum-computing |access-date=2017-06-29 |website=IEEE Spectrum |language=en-US}}</ref>

In 2022, researchers at the [[University of Innsbruck]] succeeded in developing a universal qudit quantum processor with trapped ions.<ref>{{cite journal |last1=Ringbauer |first1=Martin |last2=Meth |first2=Michael |last3=Postler |first3=Lukas |last4=Stricker |first4=Roman |last5=Blatt |first5=Rainer |last6=Schindler |first6=Philipp |last7=Monz |first7=Thomas |title=A universal qudit quantum processor with trapped ions |journal=Nature Physics |date=21 July 2022 |volume=18 |issue=9 |pages=1053–1057 |doi=10.1038/s41567-022-01658-0 |arxiv=2109.06903 |bibcode=2022NatPh..18.1053R |s2cid=237513730 |url=https://www.nature.com/articles/s41567-022-01658-0 |access-date=21 July 2022 |language=en |issn=1745-2481}}</ref> In the same year, researchers at [[Tsinghua University]]'s Center for Quantum Information implemented the dual-type qubit scheme in trapped ion quantum computers using the same ion species.<ref>{{cite web|first1=Ingrid|last1=Fardelli|url=https://phys.org/news/2022-08-coherently-qubit-ion-species.amp|title=Researchers realize two coherently convertible qubit types using a single ion species|date=August 18, 2022|publisher=[[Phys.org]]}}</ref>

Also in 2022, researchers at the [[University of California, Berkeley]] developed a technique to dynamically control the cross-Kerr interactions between fixed-frequency qutrits, achieving high two-qutrit gate fidelities.<ref name="Goss Morvan Marinelli Mitchell 2022 p. ">{{cite journal | last1=Goss | first1=Noah | last2=Morvan | first2=Alexis | last3=Marinelli | first3=Brian | last4=Mitchell | first4=Bradley K. | last5=Nguyen | first5=Long B. | last6=Naik | first6=Ravi K. | last7=Chen | first7=Larry | last8=Jünger | first8=Christian | last9=Kreikebaum | first9=John Mark | last10=Santiago | first10=David I. | last11=Wallman | first11=Joel J. | last12=Siddiqi | first12=Irfan | title=High-fidelity qutrit entangling gates for superconducting circuits | journal=Nature Communications | publisher=Springer Science and Business Media LLC | volume=13 | issue=1 | date=2022-12-05 | issn=2041-1723 | doi=10.1038/s41467-022-34851-z | page=7481| pmid=36470858 | pmc=9722686 | arxiv=2206.07216 | bibcode=2022NatCo..13.7481G }}</ref> This was followed by a demonstration of extensible control of superconducting qudits up to <math>d=4</math> in 2024 based on programmable two-photon interactions.<ref name="Nguyen Goss Siva Kim 2023 q896">{{cite journal | last1=Nguyen | first1=Long B. | last2=Goss | first2=Noah | last3=Siva | first3=Karthik | last4=Kim | first4=Yosep | last5=Younis | first5=Ed | last6=Qing | first6=Bingcheng | last7=Hashim | first7=Akel | last8=Santiago | first8=David I. | last9=Siddiqi | first9=Irfan | title=Empowering high-dimensional quantum computing by traversing the dual bosonic ladder | journal=Nature Communications | publisher=Springer Science and Business Media LLC | volume=15 | issue=1 | date=2024-08-19 | issn=2041-1723 | doi=10.1038/s41467-024-51434-2 | page=7117| pmid=39160166 | pmc=11333499 | arxiv=2312.17741 | bibcode=2024NatCo..15.7117N }}</ref>

Similar to the qubit, the [[qutrit]] is the unit of quantum information that can be realized in suitable 3-level quantum systems. This is analogous to the unit of classical information [[trit (computing)|trit]] of [[ternary computer]]s.<ref>{{Cite web |last=Irving |first=Michael |date=2022-10-14 |title="64-dimensional quantum space" drastically boosts quantum computing |url=https://newatlas.com/telecommunications/qudits-64-dimensional-quantum-space/ |access-date=2022-10-14 |website=New Atlas |language=en-US}}</ref> Besides the advantage associated with the enlarged computational space, the third qutrit level can be exploited to implement efficient compilation of multi-qubit gates.<ref name="Nguyen Goss Siva Kim 2023 q896"></ref><ref>{{cite journal |last1=Nguyen |first1=L. B. |last2=Kim |first2=Y. |last3=Hashim |first3=A. |last4=Goss |first4=N. |last5=Marinelli |first5=B. |last6=Bhandari |first6=B. |last7=Das |first7=D. |last8=Naik |first8=R. K. |last9=Kreikebaum |first9=J. M. |last10=Jordan |first10=A. |last11=Santiago |first11=D. I. |last12=Siddiqi |first12=I. |date=16 January 2024 |title=Programmable Heisenberg interactions between Floquet qubits |journal=Nature Physics |volume=20 |issue=1 |pages=240–246 |arxiv=2211.10383 |bibcode=2024NatPh..20..240N |doi=10.1038/s41567-023-02326-7 |doi-access=free}}</ref><ref name="Chu He Zhou Yuan 2022 pp. 126–131">{{cite journal | last1=Chu | first1=Ji | last2=He | first2=Xiaoyu | last3=Zhou | first3=Yuxuan | last4=Yuan | first4=Jiahao | last5=Zhang | first5=Libo | last6=Guo | first6=Qihao | last7=Hai | first7=Yongju | last8=Han | first8=Zhikun | last9=Hu | first9=Chang-Kang | last10=Huang | first10=Wenhui | last11=Jia | first11=Hao | last12=Jiao | first12=Dawei | last13=Li | first13=Sai | last14=Liu | first14=Yang | last15=Ni | first15=Zhongchu | last16=Nie | first16=Lifu | last17=Pan | first17=Xianchuang | last18=Qiu | first18=Jiawei | last19=Wei | first19=Weiwei | last20=Nuerbolati | first20=Wuerkaixi | last21=Yang | first21=Zusheng | last22=Zhang | first22=Jiajian | last23=Zhang | first23=Zhida | last24=Zou | first24=Wanjing | last25=Chen | first25=Yuanzhen | last26=Deng | first26=Xiaowei | last27=Deng | first27=Xiuhao | last28=Hu | first28=Ling | last29=Li | first29=Jian | last30=Liu | first30=Song | last31=Lu | first31=Yao | last32=Niu | first32=Jingjing | last33=Tan | first33=Dian | last34=Xu | first34=Yuan | last35=Yan | first35=Tongxing | last36=Zhong | first36=Youpeng | last37=Yan | first37=Fei | last38=Sun | first38=Xiaoming | last39=Yu | first39=Dapeng | title=Scalable algorithm simplification using quantum AND logic | journal=Nature Physics | publisher=Springer Science and Business Media LLC | volume=19 | issue=1 | date=2022-11-14 | issn=1745-2473 | doi=10.1038/s41567-022-01813-7 | pages=126–131| arxiv=2112.14922 }}</ref>

==Physical implementations==
Any [[two-state quantum system|two-level quantum-mechanical system]] can be used as a qubit. Multilevel systems can be used as well, if they possess two states that can be effectively decoupled from the rest (e.g., the ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate two-level systems to various degrees have been successfully realized. Similarly to a classical bit, where the state of a transistor in a processor, the magnetization of a surface in a [[hard disk]], and the presence of current in a cable can all be used to represent bits in the same computer, an eventual quantum computer is likely to use various combinations of qubits in its design.

All physical implementations are affected by noise. The so-called ''T''<sub>1</sub> lifetime and ''T''<sub>2</sub> dephasing time are a time to characterize the physical implementation and represent their sensitivity to noise. A higher time does not necessarily mean that one or the other qubit is better suited for [[quantum computing]] because gate times and fidelities need to be considered, too.

Different applications like [[quantum sensing]], [[quantum computing]] and [[quantum communication]] use different implementations of qubits to suit their application.

The following is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only.

{| class="wikitable" align="center"
|-
! scope="col" | Physical support
! scope="col" | Name
! scope="col" | Information support
! scope="col" style="background: white;" | <math>|0 \rangle</math>
! scope="col" style="background: white;" | <math>|1 \rangle</math>
|-
| rowspan=3 |[[photon]]
| [[Polarization (waves)|polarization]] [[Encoding (memory)|encoding]]
| [[polarization of light]]
| horizontal
| vertical
|-
| [[Amount of substance|number of photons]]
| [[Fock state]]
| [[vacuum]]
| single-photon state
|-
| [[time-bin encoding]]
| [[time of arrival]]
| early
| late
|-
| [[coherent state]] of [[light]]
| [[Squeezed coherent state|squeezed light]]
| [[Optical phase space|quadrature]]
| [[amplitude]]-[[Squeezed coherent state|squeezed]] [[Quantum state|state]]
| phase-squeezed state
|-
| rowspan=2|[[electron]]s
| [[Spin quantum number|electronic spin]]
| [[Spin (physics)|spin]]
| up
| down
|-
| [[electron]] [[Amount of substance|number]]
| [[Charge (physics)|charge]]
| no electron
| two electron
|-
| [[Nuclear magnetic resonance quantum computer|nucleus]]
| [[nuclear spin]] [[Address|addressed]] [[Preposition and postposition|through]] [[Nuclear magnetic resonance|NMR]]
| [[Spin (physics)|spin]]
| up
| down
|-
| [[Neutral atom quantum computer|neutral atom]]
| atomic [[energy level]]
| [[Spin (physics)|spin]]
| up
| down
|-
| trapped [[ion]]
| atomic [[energy level]]
| [[Spin (physics)|spin]]
| up
| down
|-
| rowspan=3| [[Josephson junction]]
| [[Superconductivity|superconducting]] [[charge qubit]]
| [[Charge (physics)|charge]]
| uncharged [[Superconductivity|superconducting]] island ({{nowrap|1=''Q'' = 0}})
| charged superconducting island ({{nowrap|1=''Q'' = 2''e''}}, one extra [[Cooper pair]])
|-
| [[Superconductivity|superconducting]] [[flux qubit]]
| [[Current source|current]]
| [[clockwise]] [[Current source|current]]
| counterclockwise current
|-
| [[Superconductivity|superconducting]] [[phase qubit]]
| [[energy]]
| [[ground state]]
| first excited state
|-
| singly charged [[quantum dot]] pair
| [[Electron localization function|electron localization]]
| [[Charge (physics)|charge]]
| electron on left dot
| electron on right dot
|-
| [[quantum dot]]
| [[Dot product|dot]] [[Spin (physics)|spin]]
| [[Spin (physics)|spin]]
| down
| up
|-
| [[Topological order|gapped topological system]]
| [[Non-abelian group|non-abelian]] [[anyon]]s
| [[Braid group|braiding of excitations]]
| depends on specific [[Topology|topological]] [[system]]
| depends on specific topological system
|-
| vibrational qubit<ref>{{cite journal |last1=Berrios |first1=Eduardo |last2=Gruebele |first2=Martin |last3=Shyshlov |first3=Dmytro |last4=Wang |first4=Lei |last5=Babikov |first5=Dmitri |year=2012 |title=High fidelity quantum gates with vibrational qubits |journal=Journal of Chemical Physics |volume=116 |issue=46 |pages=11347–11354 |bibcode=2012JPCA..11611347B |doi=10.1021/jp3055729 |pmid=22803619}}</ref>
| [[Vibrational bond|vibrational]] [[Quantum state|states]]
| [[phonon]]/[[Vibronic spectroscopy|vibron]]
| <math>|01 \rangle</math> [[Superposition principle|superposition]]
| <math>|10 \rangle</math> superposition
|-
| [[van der Waals heterostructure]]<ref>
{{cite journal |author=Lucatto |first=B. |display-authors=etal |year=2019 |title=Charge qubit in van der Waals heterostructures |journal=Physical Review B |volume=100 |issue=12 |pages=121406 |arxiv=1904.10785 |bibcode=2019PhRvB.100l1406L |doi=10.1103/PhysRevB.100.121406 |s2cid=129945636}}</ref>
| [[Electron localization function|electron localization]]
| [[Charge (physics)|charge]]
| [[electron]] on bottom sheet
| electron on top sheet
|}

==Qubit storage==
In 2008 a team of scientists from the U.K. and U.S. reported the first relatively long (1.75 seconds) and coherent transfer of a superposition state in an electron spin "processing" qubit to a [[nuclear spin]] "memory" qubit.<ref>{{cite journal |author=Morton |first=J. J. L. |display-authors=etal |year=2008 |title=Solid-state quantum memory using the <sup>31</sup>P nuclear spin |journal=[[Nature (journal)|Nature]] |volume=455 |issue=7216 |pages=1085–1088 |arxiv=0803.2021 |bibcode=2008Natur.455.1085M |doi=10.1038/nature07295 |s2cid=4389416}}</ref> This event can be considered the first relatively consistent quantum data storage, a vital step towards the development of [[quantum computing]]. In 2013, a modification of similar systems (using charged rather than neutral donors) has dramatically extended this time, to 3 hours at very low temperatures and 39 minutes at room temperature.<ref>{{cite journal |author=Kamyar Saeedi |year=2013|title=Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28|volume=342|pages=830–833|doi=10.1126/science.1239584|issue=6160|journal=[[Science (journal)|Science]]|bibcode = 2013Sci...342..830S |display-authors=etal |pmid=24233718|arxiv=2303.17734 |s2cid=42906250}}</ref> Room temperature preparation of a qubit based on electron spins instead of nuclear spin was also demonstrated by a team of scientists from Switzerland and Australia.<ref>{{cite journal|last1=Náfrádi|first1=Bálint|last2=Choucair|first2=Mohammad|last3=Dinse|first3=Klaus-Pete|last4=Forró|first4=László|title=Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres|journal=Nature Communications|date=July 18, 2016|volume=7|page=12232|doi=10.1038/ncomms12232|pmid=27426851|pmc=4960311|arxiv=1611.07690|bibcode=2016NatCo...712232N}}</ref> An increased coherence of qubits is being explored by researchers who are testing the limitations of a [[Germanium|Ge]] [[Electron hole|hole]] spin-orbit qubit structure.<ref>{{Cite journal|first1=Zhanning |last1=Wang |first2=Elizabeth |last2=Marcellina |first3=A. R. |last3=Hamilton |first4=James H. |last4=Cullen |first5=Sven |last5=Rogge |first6=Joe |last6=Salfi |first7=Dimitrie |last7=Culcer|date=April 1, 2021|title=Qubits composed of holes could be the trick to build faster, larger quantum computers|doi=10.1038/s41534-021-00386-2|arxiv=1911.11143|journal=[[npj Quantum Information]]|volume=7|issue=1|s2cid=232486360|url=https://phys.org/news/2021-04-qubits-holes-faster-larger-quantum.html}}</ref>

==See also==
{{Commons category|Qubit}}
* [[Ancilla bit]]
* [[Electron-on-helium qubit]]
* [[Physical and logical qubits]]
* [[Unitary group]]
* [[W state]] and [[Greenberger–Horne–Zeilinger state]] (GHZ state)

==Notes==
{{notelist}}

==References==
{{Reflist}}

==Further reading==
* {{Cite book |title=Quantum Computation and Quantum Information |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac |date=2000 |publisher=[[Cambridge University Press]] |isbn=0521632358 |location=Cambridge, England |oclc=43641333 |author-link=Michael Nielsen |author-link2=Isaac Chuang}}
* {{cite book |author=Williams |first=Colin P. |year=2011 |title=Explorations in Quantum Computing |publisher=[[Springers Science+Business Media|Springer]] |isbn=978-1-84628-887-6}}
* {{Cite book|title=Quantum computing for computer scientists|last1=Yanofsky|first1=Noson S.|last2=Mannucci|first2=Mirco|date=2013|publisher=[[Cambridge University Press]]|isbn=978-0-521-87996-5}}
* A treatment of two-level quantum systems, decades before the term "qubit" was coined, is found in the third volume of ''[[The Feynman Lectures on Physics]]'' [https://feynmanlectures.caltech.edu/III_toc.html (2013 ebook edition)], in chapters 9–11.
* A non-traditional motivation of the qubit aimed at non-physicists is found in ''[[Quantum Computing Since Democritus]]'', by [[Scott Aaronson]], Cambridge University Press (2013).
* An introduction to qubits for non-specialists, by the person who coined the word, is found in Lecture 21 of ''The science of information: from language to black holes'', by Professor [[Benjamin Schumacher]], [[The Great Courses]], The Teaching Company (4 DVDs, 2015).
* A [[picture book]] introduction to entanglement, showcasing a Bell state and the measurement of it, is found in ''Quantum entanglement for babies'', by [[Chris Ferrie]] (2017). {{ISBN|9781492670261}}.

{{quantum computing}}
{{Authority control}}

[[Category:Quantum computing]]
[[Category:Quantum states]]
[[Category:Teleportation]]
[[Category:Units of information]]
[[Category:Australian inventions]]