Hadamard Gate

Last updated Jun 23, 2022

• Type atom
• Status needs revision
• On quantum computing

The Hadamard Gate turns $\ket 0$ into $\ket +$, and $\ket 1$ into $\ket -$: $$\displaylines{
H\ket0 = \frac{1}{\sqrt 2}(\ket 0 + \ket 1), \newline
H\ket1 = \frac{1}{\sqrt 2}(\ket 0 - \ket 1), \newline
H\ket+ = \ket 0, \newline
H\ket- = \ket 1, \newline
H\ket i = \ket {-i}, \newline
H\ket{-i} = \ket i.
}$$
It is a rotation of $180^\circ$ about the $x+z$-axis. Applying $H$ twice rotates by $360^\circ$, which does nothing. So, $H^2 = I$. ¹
It’s Matrix representation is: $$H = \frac{1}{\sqrt 2} \begin{pmatrix}1 & 1 \\ 1 & -1 \end{pmatrix}.$$