Editing Quantum logic gate (section)

=== Hadamard gate ===
{{further|Hadamard transform#Quantum computing applications|Hadamard matrix}}The Hadamard or Walsh-Hadamard gate, named after [[Jacques Hadamard]] ({{IPA|fr|adamaʁ|lang}}) and [[Joseph L. Walsh]], acts on a single qubit. It maps the basis states <math display="inline">|0\rangle \mapsto \frac{|0\rangle + |1\rangle}{\sqrt{2}}</math> and <math display="inline">|1\rangle \mapsto \frac{|0\rangle - |1\rangle}{\sqrt{2}}</math> (it creates an equal superposition state if given a computational basis state). The two states <math>(|0\rangle + |1\rangle)/\sqrt{2}</math> and <math>(|0\rangle - |1\rangle)/\sqrt{2}</math> are sometimes written <math>|+\rangle</math> and <math>|-\rangle</math> respectively. The Hadamard gate performs a rotation of <math>\pi</math> about the axis <math>(\hat{x}+\hat{z})/\sqrt{2}</math> at the [[Bloch sphere]], and is therefore [[Involutory matrix|involutory]]. It is represented by the [[Hadamard matrix]]: 
[[Image:Hadamard gate.svg|upright=0.8|thumb|Circuit representation of Hadamard gate]]

:<math> H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} .</math>

If the [[Hermitian matrix|Hermitian]] (so <math>H^{\dagger}=H^{-1}=H</math>) Hadamard gate is used to perform a [[Change of basis#Endomorphisms|change of basis]], it flips <math>\hat{x}</math> and <math>\hat{z}</math>. For example, <math>HZH=X</math> and <math>H\sqrt{X}\;H=\sqrt{Z}=S.</math>
{{Anchor|Swap|SWAP|Swap gate}}