Whereas in a classical computer the bit is the unit of information, in a quantum device called a
quantum computer–which is a special type of quantum automaton– this is replaced by a
corresponding concept called qubit.
Definition 0.1. 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 Ψ in a a
two-dimensional vector space ℂ2 over the 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 quantum state space ℂn of n × 1 column vectors with
the inner product (x,y) = x † y is also such a condition, where A † denotes the transpose
conjugate of A. Then a unit vector ψ in ℂn 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.
Definition 0.2. A qudit is defined as the unit of quantum information in a d-level quantum
system ℂd which is contained in the unit vector in a vector space ℂd of dimension d.
Furthermore, one can define as follows a more complex concept than the qudit by allowing for
entanglement of quantum states.
Definition 0.3. A quantum register ℛg 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 [1].
References
[1] Graham P. Collins.,October 17, 2005, Quantum Bug: Qubits might spontaneously
decay in seconds. Scientific American
