Talk:PlanetPhysics/Qubit2

Original TeX Content from PlanetPhysics Archive

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\begin{document}

 Whereas in a classical \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/SupercomputerArchitercture.html} the bit is the unit of information, in a quantum
device called a \htmladdnormallink{quantum computer}{http://en.wikipedia.org/wiki/File:Quantum_computer.jpg}--which is a special \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of quantum automaton-- this is replaced by a corresponding \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} called qubit.

\begin{definition}
A {\em qubit, qbit} or {\em quantum bit} is defined as the unit of quantum information which is contained in a quantum \htmladdnormallink{state vector}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} for a two-level quantum system consisting of only two \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} levels; mathematically this is expressed as a \htmladdnormallink{unit vector}{http://planetphysics.us/encyclopedia/PureState.html} $\Psi$ in a a \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $\mathbb{C}^2$ over the \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} of complex numbers $\mathbb{C}$.
\end{definition}

In order to have any practical use such a qubit must also meet several conditions, such as: it has to be measurable, undergo controlled unitary transformations, have a long coherence time, be capable of initialization,
and so on. Scalability to a \htmladdnormallink{quantum state space}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} $\mathbb{C}^n$ of $n \times 1$ column \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} with the \htmladdnormallink{inner product}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $(x,y) = x\dagger{}y$ is also such a condition, where $A\dagger{}$ denotes the transpose conjugate of $A$. Then a unit vector $\psi$ in $\mathbb{C}^n$ denotes a quantum state. As an example, in a superconducting flux qubit an electric current can be imagined to circulate simultaneously in a stable (or coherent) loop both clockwise and counterclockwise. A qubit in such a superposition is in a highly symmetrical quantum state. Superconducting qubits involve large numbers of \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} (\htmladdnormallink{Cooper pairs}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html}) as the superconducting current involves many billions of such coherent electron pairs. In such a many-particle superconducting loop, \htmladdnormallink{spontaneous symmetry breaking}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html} tends to determine the qubit to end up in a definite state, by `breaking up the superposition'. On the other hand, an ion suspended in a magnetic trap or a single electron in a quantum dot on a chip
do not exhibit this phenomenon. In August 2005, a \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of physicists at the National Institute of Standards and Technology (NIST) suceeded in preparing single-ion qubits with a coherence time longer than 10 seconds.


\begin{definition}
A {\em qudit} is defined as the unit of quantum information in a $d$-level quantum \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} $\mathbb{C}^d$ which is contained in the unit vector in a vector space $\mathbb{C}^d$ of dimension $d$.
\end{definition}

Furthermore, one can define as follows a more complex concept than the qudit by allowing for entanglement of quantum states.
\begin{definition}
A {\em quantum register} $\mathcal{R}_g$ consists, or is determined by, a number $r$ of entangled qubits.
\end{definition}

\htmladdnormallink{Quantum computers}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html} could then perform calculations by manipulating qubits within a quantum register. However, the requirement for long coherence times may be a major obstacle to building quantum computers \cite{Collins2k5}.

\begin{thebibliography}{9}
\bibitem{Collins2k5}
Graham P. Collins.,October 17, 2005, Quantum Bug: Qubits might spontaneously decay in seconds. {\em Scientific American}
\end{thebibliography} 

\end{document}