PlanetPhysics/Qubit2

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Whereas in a classical [[../SupercomputerArchitercture/|computer]] the bit is the unit of information, in a quantum

device called a quantum computer--which is a special [[../Bijective/|type]] of quantum automaton-- this is replaced by a corresponding [[../PreciseIdea/|concept]] called qubit.

A qubit, qbit or quantum bit is defined as the unit of quantum information which is contained in a quantum [[../QuantumOperatorAlgebra4/|state vector]] for a two-level quantum system consisting of only two [[../CosmologicalConstant/|energy]] levels; mathematically this is expressed as a [[../PureState/|unit vector]] $\mathrm{\Psi }$ in a a [[../CoriolisEffect/|two-dimensional]] [[../NormInducedByInnerProduct/|vector space]] ${ℂ}^2$ over the [[../CosmologicalConstant2/|field]] of complex numbers $ℂ$.

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 [[../QuantumSpinNetworkFunctor2/|quantum state space]] ${ℂ}^n$ of $n\times 1$ column [[../Vectors/|vectors]] with the [[../NormInducedByInnerProduct/|inner product]] $(x,y)=x†y$ is also such a condition, where $A†$ denotes the transpose conjugate of $A$. Then a unit vector $\psi$ in ${ℂ}^n$ 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 [[../Particle/|particles]] ([[../LongRangeCoupling/|Cooper pairs]]) as the superconducting current involves many billions of such coherent electron pairs. In such a many-particle superconducting loop, [[../LongRangeCoupling/|spontaneous symmetry breaking]] 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 [[../TrivialGroupoid/|group]] 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.

A qudit is defined as the unit of quantum information in a $d$-level quantum [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] ${ℂ}^d$ which is contained in the unit vector in a vector space ${ℂ}^d$ of dimension $d$.

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

[[../LongRangeCoupling/|Quantum computers]] 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 .