Clifford gates


In quantum computing and quantum information theory, the Clifford gates are the elements of the Clifford group, a set of mathematical transformations which effect permutations of the Pauli operators. The notion was introduced by Daniel Gottesman and is named after the mathematician William Kingdon Clifford.

Clifford group

The Pauli matrices,
provide a basis for the density operators of a single qubit, as well as for the unitaries that can be applied to them. For the -qubit case, one can construct a group, known as the Pauli group, according to
The Clifford group is defined as the group of unitaries that normalize the Pauli group: The Clifford gates are then defined as elements in the Clifford group.
Some authors choose to define the Clifford group as the quotient group. For 1, 2, and 3, this group contains 24, 11,520, and 92,897,280 elements, respectively.
Quantum circuits constructed from Clifford gates can be efficiently simulated with a classical computer, a result commonly known as the Gottesman–Knill theorem.