Editing Quantum logic gate (section)

=== Pauli gates (''X'',''Y'',''Z'') ===
{{Further|Clifford gates|Pauli group}}
{{Multiple image
| image1            = Qcircuit I.svg
| image2            = Qcircuit NOT.svg
| image3            = Qcircuit Y.svg
| total_width       = 120
| footer            = Quantum gates (from top to bottom): Identity gate, NOT gate, Pauli Y, Pauli Z
| direction         = vertical
| image4            = Qcircuit Z.svg
}}
The Pauli gates <math>(X,Y,Z)</math> are the three [[Pauli matrices]] <math>(\sigma_x,\sigma_y,\sigma_z)</math> and act on a single qubit. The Pauli ''X'', ''Y'' and ''Z'' equate, respectively, to a rotation around the ''x'', ''y'' and ''z'' axes of the [[Bloch sphere]] by <math>\pi</math> radians.{{efn|Note, here a full rotation about the Bloch sphere is <math>2 \pi</math> radians, as opposed to the [[List of quantum logic gates#Rotation operator gates|rotation operator gates]] where a full turn is <math>4 \pi.</math>}}

The Pauli-''X'' gate is the quantum equivalent of the [[NOT gate]] for classical computers with respect to the standard basis {{nowrap|<math>|0\rangle</math>, <math>|1\rangle</math>,}} which distinguishes the ''z'' axis on the [[Bloch sphere]]. It is sometimes called a bit-flip as it maps <math>|0\rangle</math> to <math>|1\rangle</math> and <math>|1\rangle</math> to <math>|0\rangle</math>. Similarly, the Pauli-''Y'' maps <math>|0\rangle</math> to <math>i|1\rangle</math> and <math>|1\rangle</math> to {{nowrap|<math>-i|0\rangle</math>}}. Pauli ''Z'' leaves the basis state <math>|0\rangle</math> unchanged and maps <math>|1\rangle</math> to {{nowrap|<math>-|1\rangle</math>.}} Due to this nature, Pauli ''Z'' is sometimes called phase-flip.

These matrices are usually represented as
:<math> X = \sigma_x =\operatorname{NOT} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} ,</math>
:<math> Y = \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, </math>
:<math> Z = \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.</math>

The Pauli matrices are [[Involutory matrix|involutory]], meaning that the square of a Pauli matrix is the [[identity matrix]].

:<math>I^2 = X^2 = Y^2 = Z^2 = -iXYZ = I</math>

The Pauli matrices also [[Anticommutative property|anti-commute]], for example <math>ZX=iY=-XZ.</math>

The [[matrix exponential]] of a Pauli matrix <math>\sigma_j</math> is a [[List of quantum logic gates#Rotation operator gates|rotation operator]], often written as <math>e^{-i\sigma_j\theta/2}.</math>

{{anchor|squares}}

{{Anchor|CX|CY|CZ|controlled gate|Controlled gates|Controlled|CNOT}}