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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Quantum information and computing}}
A '''qubit''' (quantum bit) is the fundamental unit of quantum information. It is realized by a two-level quantum system and forms the quantum analogue of the classical bit.<ref name="NielsenChuang2010">{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |isbn=978-1-107-00217-3}}</ref>
[[File:Bloch_sphere.png|thumb|400px|Representation of a qubit state on the Bloch sphere.]]

== Definition ==

A qubit is described by a state vector in a two-dimensional complex Hilbert space with orthonormal basis states <math>|0\rangle</math> and <math>|1\rangle</math>.<ref name="YanofskyMannucci2013">{{cite book |last1=Yanofsky |first1=Noson S. |last2=Mannucci |first2=Mirco A. |title=Quantum Computing for Computer Scientists |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-87996-5 |pages=138–144}}</ref><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|article-number=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 general qubit state is

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

where <math>\alpha, \beta \in \mathbb{C}</math> satisfy the normalization condition

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

The coefficients <math>\alpha</math> and <math>\beta</math> are probability amplitudes.<ref name="Williams2011">{{cite book |last=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=Springer |year=2011 |isbn=978-1-84628-887-6 |pages=9–13}}</ref>

== Comparison with a classical bit ==

A classical bit can take only one of two values, 0 or 1. A qubit, however, can exist in a coherent superposition of both basis states.<ref name="NielsenChuang2010" />

Upon measurement:

* <math>|0\rangle</math> is obtained with probability <math>|\alpha|^2</math>
* <math>|1\rangle</math> is obtained with probability <math>|\beta|^2</math>

Unlike a classical bit, measurement generally disturbs the qubit state and destroys quantum coherence.<ref name="NielsenChuang2010" />

== Bloch sphere representation ==

Any pure qubit state can be written as

<math>
|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle.
</math>

This allows a geometric representation on the Bloch sphere, where <math>\theta</math> and <math>\phi</math> specify the state.<ref name="NielsenChuang2010" />

Pure states lie on the surface of the Bloch sphere, while the global phase has no observable physical effect.<ref name="NielsenChuang2010" />

== Mixed states ==

A qubit may also be in a mixed state, described by a density matrix

<math>
\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i |.
</math>

Mixed states arise from statistical uncertainty or from interaction with an environment, and correspond to points inside the Bloch sphere.<ref name="NielsenChuang2010" />

== Quantum operations ==

Quantum states evolve according to unitary transformations:

<math>
|\psi\rangle \rightarrow U |\psi\rangle,
</math>

where <math>U</math> is a unitary operator.<ref name="NielsenChuang2010" />

In quantum computing, these transformations are implemented as '''quantum gates'''. Examples include:

* Pauli gates (<math>X, Y, Z</math>)
* Hadamard gate
* Controlled-NOT (CNOT) gate

These operations enable interference, superposition control, and the creation of entanglement.

== Physical realizations ==

Qubits can be implemented in various physical systems, including:

* electron spin
* photon polarization
* trapped ions
* superconducting circuits
* quantum dots

Different implementations are used depending on the application in quantum computing, communication, or sensing.<ref name="NielsenChuang2010" /><ref>{{cite book |last=Preskill |first=John |title=Lecture Notes for Physics 229: Quantum Information and Computation |year=1998}}</ref>

== Quantum registers ==

A collection of qubits forms a '''quantum register'''. For <math>n</math> qubits, the state space has dimension <math>2^n</math>, allowing complex superpositions and correlations.<ref name="YanofskyMannucci2013" />

== Physical significance ==

The qubit:

* is the basic carrier of quantum information
* enables superposition and interference
* forms the foundation of quantum computation and communication

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

= References =
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum qubit|1}}

Physics:Quantum Qubit - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template