PlanetPhysics/Qubit2

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Whereas in a classical computer the bit is the unit of information, in a quantum device called a --which is a special type of quantum automaton-- this is replaced by a corresponding concept called qubit.

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

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 quantum state space $$\mathbb{C}^n$$ of $$n \times 1$$ column vectors with the inner product $$(x,y) = x\dagger{}y$$ 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 particles (Cooper pairs) as the superconducting current involves many billions of such coherent electron pairs. In such a many-particle superconducting loop, 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 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 system $$\mathbb{C}^d$$ which is contained in the unit vector in a vector space $$\mathbb{C}^d$$ 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 $$\mathcal{R}_g$$ consists, or is determined by, a number $$r$$ of entangled qubits.

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.