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In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the in which the two levels can be taken as spin up and spin down; or the polarization of a single 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 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. In the acknowledgments of his 1995 paper, Schumacher states that the term qubit was created in jest during a conversation with .


Bit versus qubit
A , 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 is the basic unit of 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 , and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between two 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 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 arbitrary coherent superposition of all computable states simultaneously.

(2025). 9781107002173, Cambridge University Press.
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 bits (e.g. a processor register or some bit array) can only hold a single of its 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 to describe its two probability amplitudes, and these two complex numbers can together be viewed as a 2-dimensional complex 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.

Furthermore, a set of bits can be represented by binary digits, simply by concatenating the representations of each of the bits, whereas a set of qubits, which is also called a , requires complex numbers to describe its superposition state vector.


Standard representation
In quantum mechanics, the general of a qubit can be represented by a linear superposition of its two basis states (or basis ). These vectors are usually denoted as | 0 \rangle = \bigl[\begin{smallmatrix} 1\\ 0 \end{smallmatrix}\bigr] and | 1 \rangle = \bigl[\begin{smallmatrix} 0\\ 1 \end{smallmatrix}\bigr]. They are written in the conventional Dirac—or "bra–ket"—notation; the | 0 \rangle and | 1 \rangle are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states, \, together called the computational basis, are said to span the two-dimensional of the qubit.>

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

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

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

(2025). 9780521879965, Cambridge University Press.


Qubit states
A pure qubit state is a coherent superposition of the basis states. This means that a single qubit (\psi) can be described by a linear combination of
and >1 \rangle :

1 \rangle

where α and β are the probability amplitudes, and are both . When we measure this qubit in the standard basis, according to the , the probability of outcome

^2. Because the absolute squares of the amplitudes equate to probabilities, it follows that \alpha and \beta must be constrained according to the second axiom of probability theory by the equation
(2025). 9781846288876, Springer.

^2 = 1.

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


Bloch sphere representation
It might, at first sight, seem that there should be four degrees of freedom in
1 \rangle\,, as \alpha and \beta are with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint . This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of Hopf coordinates:
\begin{align}
\alpha &= e^{i \delta} \cos\frac{\theta}{2}, \\ \beta &= e^{i (\delta + \varphi)} \sin\frac{\theta}{2}. \end{align} Additionally, for a single qubit, the global of the state e^{i\delta} has no physically observable consequences, so we can arbitrarily choose to be real (or in the case that is zero), leaving just two degrees of freedom:
\begin{align}
\alpha &= \cos\frac{\theta}{2}, \\ \beta &= e^{i \varphi} \sin\frac{\theta}{2}, \end{align}

The possible quantum states for a single qubit can be visualised using a (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

1 \rangle)/{\sqrt{2}} would lie on the equator of the sphere at the positive X-axis. In the , 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 two-dimensional space, which represents the observable state space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles \varphi and \theta.


Mixed state
A pure state is fully specified by a single ket, 
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]] and [[decoherence]], it is possible to put the qubit in a 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 \varphi and \theta , as well as the length r of the vector that represents the mixed state.

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


Operations on qubits
Various kinds of physical operations can be performed on qubits.
     
  • Quantum logic gates, building blocks for a in a quantum computer, operate on a set of qubits (a ); mathematically, the qubits undergo a (reversible) unitary transformation described by multiplying the quantum gates with the 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 coherence is lost. The result of the measurement of a single qubit with the state
. Measurement of the state of the qubit alters the magnitudes of α and β. For instance, if the result of the measurement is >1\rangle, α is changed to 0 and β is changed to 1, while the phase factor e^{i \phi} is no longer experimentally accessible. If measurement is performed on a qubit that is entangled, the measurement may collapse the state of the other entangled qubits.
  • Initialization or re-initialization to a known value, often
  • may be implemented logically or physically: Logically as a measurement, followed by the application of the Pauli-X gate if the result from the measurement was >1\rangle. Physically, for example, if it is a superconducting , by lowering the energy of the quantum system to its .
  • Sending the qubit through a to a remote system or machine (an I/O operation), potentially as part of a .

  • Quantum entanglement
    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 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

    \Phi^+\rangle :

    \frac{1}{\sqrt{2}} (
    11\rangle).

    In this state, called an equal superposition, there are equal probabilities of measuring either product state

    ^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

    is the only state where Alice's qubit is a >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
    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 (CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is
    , >11\rangle\}, it maps the basis states as follows:
    0 0 \rangle
    0 1 \rangle
    1 1 \rangle
    1 0 \rangle .

    A common application of the CNOT gate is to maximally entangle two qubits into the

    . To construct >\Phi^+\rangle, the inputs A (control) and B (target) to the CNOT gate are:

    \frac{1}{\sqrt{2}}(

    10\rangle).

    After applying CNOT, the output is the

    11\rangle).


    Applications
    The
    \Phi^+\rangle 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 resource that is unique to quantum computation. 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 that can be executed reliably.


    Quantum register
    A number of qubits taken together is a . 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. A qubit register that can be measured to N states is identical to an N-level qudit. A rarely usedAs of June 2022 1150 uses versus 31 uses on in the quant-ph category of arxiv.org. for qudit is quNit, since both d and N are frequently used to denote the dimension of a quantum system.

    Qudits are similar to the integer types 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.

    In 2022, researchers at the University of Innsbruck succeeded in developing a universal qudit quantum processor with trapped ions. 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. In 2025, the Innsbruck team managed to simulate two-dimensional lattice gauge theories on their qudit quantum computer.

    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. This was followed by a demonstration of extensible control of superconducting qudits up to d=4 in 2024 based on programmable two-photon interactions.

    Similar to the qubit, the 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 of . Besides the advantage associated with the enlarged computational space, the third qutrit level can be exploited to implement efficient compilation of multi-qubit gates.


    Physical implementations
    Any 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 , 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 T1 lifetime and T2 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 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"
    polarization encodingpolarization of lighthorizontalvertical
    number of photonssingle-photon state
    time-bin encodingtime of arrivalearlylate
    of squeezed lightquadrature-squeezed phase-squeezed state
    electronic spinspinupdown
    numberchargeno electrontwo electron
    nucleus through NMRspinupdown
    neutral atomatomic spinupdown
    trapped atomic spinupdown
    Josephson junctionsuperconducting chargeuncharged superconducting island ()charged superconducting island (, one extra )
    superconducting counterclockwise current
    superconducting first excited state
    singly charged pairelectron localizationchargeelectron on left dotelectron on right dot
    spinspindownup
    gapped topological systemnon-abelian depends on specific depends on specific topological system
    vibrational qubit /vibron>01 \rangle superposition>10 \rangle superposition
    van der Waals heterostructure electron localizationcharge on bottom sheetelectron 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 "memory" qubit. 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. 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. An increased coherence of qubits is being explored by researchers who are testing the limitations of a spin-orbit qubit structure.


    See also


    Notes

    Further reading
    • (2025). 9780521632355, Cambridge University Press.
    • (2025). 9781846288876, Springer.
    • (2025). 9780521879965, Cambridge University Press.
    • 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 (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 , 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 introduction to entanglement, showcasing a Bell state and the measurement of it, is found in Quantum entanglement for babies, by (2017). .

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