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Qubit

From Wikipedia, the free encyclopedia
Basic unit of quantum information
This article is about the quantum computing unit. For other uses, seeQubit (disambiguation).
Not to be confused withCubit.
Units of
information
Information-theoretic
Data storage
Quantum information
The general definition of a qubit as the quantum state of a two-level quantum system

Inquantum computing, aqubit (/ˈkjuːbɪt/) orquantum bit is a basic unit ofquantum information; a binaryqudit – the quantum version of the classicbinarybit physically realized with a two-state device. A qubit is atwo-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include thespin of theelectron in which the two levels can be taken as spin up and spin down; or thepolarization of a singlephoton 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 acoherentsuperposition of multiple states simultaneously, a property that is fundamental toquantum mechanics andquantum computing.

Etymology

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The coining of the termqubit is attributed toBenjamin Schumacher.[1] In the acknowledgments of his 1995 paper, Schumacher states that the termqubit was created in jest during a conversation withWilliam Wootters.

Comparison with classical bits

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Abinary 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 ofinformation content, where abit is the basic unit ofinformation. However, in this article, the word bit is synonymous with a binary digit.

In classical computer technologies, aprocessed bit is implemented by one of two levels of lowdirect currentvoltage, and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between twologic levels must be passed as fast as possible, inasmuch as electrical voltage cannot changeamplitude instantaneously.

There are two possible outcomes for the measurement of a qubit, usually taken to have the values "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 be an arbitrarycoherent superposition ofall computable states simultaneously.[2] 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 usingsuperdense coding.

A bit is always completely in either one of its two states, and a set ofn bits (e.g. aprocessor register or some bit array) can only hold a single of its2n possible states at any time. A quantum state can be in a superposition state, which means that the qubit can have non-zeroprobability amplitude in both its states simultaneously (popularly expressed as "it can be in both states simultaneously"). A qubit requires twocomplex numbers to describe its two probability amplitudes, and these two complex numbers can together be viewed as a 2-dimensional complexvector, which is called aquantum state vector, orsuperposition state vector. Alternatively and equivalently, the value stored in a qubit can be described as a single point in a 2-dimensionalcomplex coordinate space.

Furthermore, a set ofn bits can be represented byn binary digits, simply by concatenating the representations of each of the bits, whereas a set ofn qubits, which is also called aregister, requires2n complex numbers to describe its superposition state vector.[3][4]: 7–17 [2]: 13–17 

Standard representation

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In quantum mechanics, the generalquantum state of a qubit can be represented by a linear superposition of its twoorthonormalbasis states (or basisvectors). These vectors are usually denoted as|0⟩=[10]{\displaystyle |0\rangle ={\bigl [}{\begin{smallmatrix}1\\0\end{smallmatrix}}{\bigr ]}} and|1⟩=[01]{\displaystyle |1\rangle ={\bigl [}{\begin{smallmatrix}0\\1\end{smallmatrix}}{\bigr ]}}. They are written in the conventionalDirac—or"bra–ket"—notation; the|0⟩{\displaystyle |0\rangle } and|1⟩{\displaystyle |1\rangle } are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states,{|0⟩,|1⟩}{\displaystyle \{|0\rangle ,|1\rangle \}}, together called the computational basis, are said to span the two-dimensionallinear vector (Hilbert) space of the qubit.[5]

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

|00⟩=[1000]{\displaystyle |00\rangle ={\biggl [}{\begin{smallmatrix}1\\0\\0\\0\end{smallmatrix}}{\biggr ]}},|01⟩=[0100]{\displaystyle |01\rangle ={\biggl [}{\begin{smallmatrix}0\\1\\0\\0\end{smallmatrix}}{\biggr ]}},|10⟩=[0010]{\displaystyle |10\rangle ={\biggl [}{\begin{smallmatrix}0\\0\\1\\0\end{smallmatrix}}{\biggr ]}}, and|11⟩=[0001]{\displaystyle |11\rangle ={\biggl [}{\begin{smallmatrix}0\\0\\0\\1\end{smallmatrix}}{\biggr ]}}.

In general,n qubits are represented by a superposition state vector in 2n dimensional Hilbert space.[5]

Qubit states

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Polarization of light offers a straightforward way to present orthogonal states. With a typical mapping|H⟩=|0⟩{\displaystyle |H\rangle =|0\rangle } and|V⟩=|1⟩{\displaystyle |V\rangle =|1\rangle }, quantum states(|0⟩±|1⟩)/2{\displaystyle (|0\rangle \pm |1\rangle )/{\sqrt {2}}} have a direct physical representation, both easily demonstrable experimentally in a class withlinear polarizers and, for realα{\displaystyle \alpha } andβ{\displaystyle \beta }, matching the high-school definition oforthogonality.[6]

A pure qubit state is acoherentsuperposition of the basis states. This means that a single qubit (ψ{\displaystyle \psi }) can be described by alinear combination of|0⟩{\displaystyle |0\rangle } and|1⟩{\displaystyle |1\rangle }:

|ψ⟩=α|0⟩+β|1⟩{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }

whereα andβ are theprobability amplitudes, and are bothcomplex numbers. When we measure this qubit in the standard basis, according to theBorn rule, the probability of outcome|0⟩{\displaystyle |0\rangle } with value "0" is|α|2{\displaystyle |\alpha |^{2}} and the probability of outcome|1⟩{\displaystyle |1\rangle } with value "1" is|β|2{\displaystyle |\beta |^{2}}. Because the absolute squares of the amplitudes equate to probabilities, it follows thatα{\displaystyle \alpha } andβ{\displaystyle \beta } must be constrained according to thesecond axiom of probability theory by the equation[4]

|α|2+|β|2=1.{\displaystyle |\alpha |^{2}+|\beta |^{2}=1.}

The probability amplitudes,α{\displaystyle \alpha } andβ{\displaystyle \beta }, encode more than just the probabilities of the outcomes of a measurement; therelative phase betweenα{\displaystyle \alpha } andβ{\displaystyle \beta } is for example responsible forquantum interference, as seen in thedouble-slit experiment.

Bloch sphere representation

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Bloch sphere representation of a qubit. Theprobability amplitudes for the superposition state,|ψ⟩=α|0⟩+β|1⟩,{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,} are given byα=cos⁡(θ2){\displaystyle \alpha =\cos \left({\frac {\theta }{2}}\right)} andβ=eiφsin⁡(θ2){\displaystyle \beta =e^{i\varphi }\sin \left({\frac {\theta }{2}}\right)}.

It might, at first sight, seem that there should be fourdegrees of freedom in|ψ⟩=α|0⟩+β|1⟩{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle \,}, asα{\displaystyle \alpha } andβ{\displaystyle \beta } arecomplex numbers with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint|α|2 + |β|2 = 1. This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that ofHopf coordinates:

α=eiδcos⁡θ2,β=ei(δ+φ)sin⁡θ2.{\displaystyle {\begin{aligned}\alpha &=e^{i\delta }\cos {\frac {\theta }{2}},\\\beta &=e^{i(\delta +\varphi )}\sin {\frac {\theta }{2}}.\end{aligned}}}

Additionally, for a single qubit, theglobalphase of the stateeiδ{\displaystyle e^{i\delta }} has no physically observable consequences,[a] so we can arbitrarily chooseα to be real (orβ in the case thatα is zero), leaving just two degrees of freedom:

α=cos⁡θ2,β=eiφsin⁡θ2,{\displaystyle {\begin{aligned}\alpha &=\cos {\frac {\theta }{2}},\\\beta &=e^{i\varphi }\sin {\frac {\theta }{2}},\end{aligned}}}

whereeiφ{\displaystyle e^{i\varphi }} is the physically significantrelative phase.[7][b]

The possible quantum states for a single qubit can be visualised using aBloch sphere (see picture). Represented on such a2-sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where|0⟩{\displaystyle |0\rangle } and|1⟩{\displaystyle |1\rangle } 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(|0⟩+|1⟩)/2{\displaystyle (|0\rangle +|1\rangle )/{\sqrt {2}}} would lie on the equator of the sphere at the positive X-axis. In theclassical 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 atwo-dimensional space, which represents the observablestate space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two anglesφ{\displaystyle \varphi } andθ{\displaystyle \theta }.

Mixed state

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Main article:Density matrix

A pure state is fully specified by a single ket,|ψ⟩=α|0⟩+β|1⟩,{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,} 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 anddecoherence, it is possible to put the qubit in amixed state, a statistical combination or "incoherent mixture" of different pure states. Mixed states can be represented by pointsinside the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the anglesφ{\displaystyle \varphi } andθ{\displaystyle \theta }, as well as the lengthr{\displaystyle r} of the vector that represents the mixed state.

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

Operations on qubits

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Further information:DiVincenzo's criteria andPhysical and logical qubits

Various kinds of physical operations can be performed on qubits.

Quantum entanglement

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Main articles:Quantum entanglement andBell state

An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibitquantum entanglement; the qubit itself is an exhibition of quantum entanglement. In this case, quantum entanglement is a local ornonlocal 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|Φ+⟩{\displaystyle |\Phi ^{+}\rangle }Bell state:

12(|00⟩+|11⟩).{\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).}

In this state, called anequal superposition, there are equal probabilities of measuring either product state|00⟩{\displaystyle |00\rangle } or|11⟩{\displaystyle |11\rangle }, as|1/2|2=1/2{\displaystyle |1/{\sqrt {2}}|^{2}=1/2}. 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|0⟩{\displaystyle |0\rangle } or|1⟩{\displaystyle |1\rangle }, 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|0⟩{\displaystyle |0\rangle }, Bob must measure the same, as|00⟩{\displaystyle |00\rangle } is the only state where Alice's qubit is a|0⟩{\displaystyle |0\rangle }. 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

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Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, thecontrolled NOT gate (CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is|1⟩{\displaystyle |1\rangle }, and otherwise leaves it unchanged. With respect to the unentangled product basis{|00⟩{\displaystyle \{|00\rangle },|01⟩{\displaystyle |01\rangle },|10⟩{\displaystyle |10\rangle },|11⟩}{\displaystyle |11\rangle \}}, it maps the basis states as follows:

|00⟩↦|00⟩{\displaystyle |00\rangle \mapsto |00\rangle }
|01⟩↦|01⟩{\displaystyle |01\rangle \mapsto |01\rangle }
|10⟩↦|11⟩{\displaystyle |10\rangle \mapsto |11\rangle }
|11⟩↦|10⟩{\displaystyle |11\rangle \mapsto |10\rangle }.

A common application of the CNOT gate is to maximally entangle two qubits into the|Φ+⟩{\displaystyle |\Phi ^{+}\rangle }Bell state. To construct|Φ+⟩{\displaystyle |\Phi ^{+}\rangle }, the inputs A (control) and B (target) to the CNOT gate are:

12(|0⟩+|1⟩)A{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}}⊗{\displaystyle \otimes }|0⟩B{\displaystyle |0\rangle _{B}} =12{\displaystyle {\frac {1}{\sqrt {2}}}}(|00⟩+|10⟩){\displaystyle (|00\rangle +|10\rangle )}.

After applying CNOT, the output is the|Φ+⟩{\displaystyle |\Phi ^{+}\rangle } Bell State:12(|00⟩+|11⟩){\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}.

Applications

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The|Φ+⟩{\displaystyle |\Phi ^{+}\rangle } Bell state forms part of the setup of thesuperdense coding,quantum teleportation, and entangledquantum 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 aresource that is unique to quantum computation.[8] 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 ofquantum circuits that can be executed reliably.[9]

Quantum register

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A number of qubits taken together is aqubit register.Quantum computers perform calculations by manipulating qubits within a register.

Qudits and qutrits

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The termqudit denotes the unit of quantum information that can be realized in suitabled-level quantum systems.[10]

Qudits are similar to theinteger types in classical computing, and may be mapped to (or realized by) arrays of qubits. Qudits where thed-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 theNational Institute of Scientific Research constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits.[11]

In 2022, researchers at theUniversity of Innsbruck succeeded in developing a universal qudit quantum processor with trapped ions.[12] In the same year, researchers atTsinghua University's Center for Quantum Information implemented the dual-type qubit scheme in trapped ion quantum computers using the same ion species.[13] In 2025, the Innsbruck team managed to simulate two-dimensionallattice gauge theories on their qudit quantum computer.[14]

Also in 2022, researchers at theUniversity of California, Berkeley developed a technique to dynamically control the cross-Kerr interactions between fixed-frequency qutrits, achieving high two-qutrit gate fidelities.[15] This was followed by a demonstration of extensible control of superconducting qudits up tod=4{\displaystyle d=4} in 2024 based on programmable two-photon interactions.[16]

Similar to the qubit, thequtrit is the unit of quantum information that can be realized in suitable 3-level quantum systems. This is analogous to the unit of classical informationtrit ofternary computers.[17] Besides the advantage associated with the enlarged computational space, the third qutrit level can be exploited to implement efficient compilation of multi-qubit gates.[16][18][19]

Physical implementation

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Anytwo-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 ahard 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-calledT1 lifetime andT2 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 forquantum computing because gate times and fidelities need to be considered, too.

Different applications likequantum sensing,quantum computing andquantum 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.

Physical supportNameInformation support|0⟩{\displaystyle |0\rangle }|1⟩{\displaystyle |1\rangle }
photonpolarizationencodingpolarization of lighthorizontalvertical
number of photonsFock statevacuumsingle-photon state
time-bin encodingtime of arrivalearlylate
coherent state oflightsqueezed lightquadratureamplitude-squeezedstatephase-squeezed state
electronselectronic spinspinupdown
electronnumberchargeno electrontwo electron
nucleusnuclear spinaddressedthroughNMRspinupdown
neutral atomatomicenergy levelspinupdown
trappedionatomicenergy levelspinupdown
Josephson junctionsuperconductingcharge qubitchargeunchargedsuperconducting island (Q = 0)charged superconducting island (Q = 2e, one extraCooper pair)
superconductingflux qubitcurrentclockwisecurrentcounterclockwise current
superconductingphase qubitenergyground statefirst excited state
singly chargedquantum dot pairelectron localizationchargeelectron on left dotelectron on right dot
quantum dotdotspinspindownup
gapped topological systemnon-abeliananyonsbraiding of excitationsdepends on specifictopologicalsystemdepends on specific topological system
vibrational qubit[20]vibrationalstatesphonon/vibron|01⟩{\displaystyle |01\rangle }superposition|10⟩{\displaystyle |10\rangle } superposition
van der Waals heterostructure[21]electron localizationchargeelectron on bottom sheetelectron on top sheet

Qubit storage

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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 anuclear spin "memory" qubit.[22] This event can be considered the first relatively consistent quantum data storage, a vital step towards the development ofquantum 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.[23] 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.[24] An increased coherence of qubits is being explored by researchers who are testing the limitations of aGehole spin-orbit qubit structure.[25]

See also

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Wikimedia Commons has media related toQubit.

Notes

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  1. ^This is because of theBorn rule. The probability of observing an outcome uponmeasurement is themodulus squared of theprobability amplitude for that outcome (or basis state,eigenstate). Theglobal phase factoreiδ{\displaystyle e^{i\delta }} is not measurable, because it applies to both basis states, and is on the complexunit circle so|eiδ|2=1.{\displaystyle |e^{i\delta }|^{2}=1.}
    Note that by removingeiδ{\displaystyle e^{i\delta }} it means thatquantum states with global phase can not be represented as points on the surface of the Bloch sphere.
  2. ^The Pauli Z basis is usually called thecomputational basis, where the relative phase have no effect on measurement.Measuring instead in the X or Y Pauli basis depends on the relative phase. For example,(|0⟩+eiπ/2|1⟩)/2{\displaystyle (|0\rangle +e^{i\pi /2}|1\rangle )/{\sqrt {2}}} 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|0⟩{\displaystyle |0\rangle } or|1⟩{\displaystyle |1\rangle }.
    Because measurementcollapses the quantum state, measuring the state in one basis hides some of the values that would have been measurable in the other basis; See theuncertainty principle.

References

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  23. ^Kamyar Saeedi; et al. (2013). "Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28".Science.342 (6160):830–833.arXiv:2303.17734.Bibcode:2013Sci...342..830S.doi:10.1126/science.1239584.PMID 24233718.S2CID 42906250.
  24. ^Náfrádi, Bálint; Choucair, Mohammad; Dinse, Klaus-Pete; Forró, László (July 18, 2016)."Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres".Nature Communications.7 12232.arXiv:1611.07690.Bibcode:2016NatCo...712232N.doi:10.1038/ncomms12232.PMC 4960311.PMID 27426851.
  25. ^Wang, Zhanning; Marcellina, Elizabeth; Hamilton, A. R.; Cullen, James H.; Rogge, Sven; Salfi, Joe; Culcer, Dimitrie (April 1, 2021)."Qubits composed of holes could be the trick to build faster, larger quantum computers".npj Quantum Information.7 (1).arXiv:1911.11143.doi:10.1038/s41534-021-00386-2.S2CID 232486360.

Further reading

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General
Theorems
Quantum
communication
Quantum cryptography
Quantum algorithms
Quantum
complexity theory
Quantum
processor benchmarks
Quantum
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Quantum
error correction
Physical
implementations
Quantum optics
Ultracold atoms
Spin-based
Superconducting
Quantum
programming
International
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