Hadamard Gate

English Translation

Definition¹

The gate defined as follows, known as the Hadamard gate when referring to a $1$qubit, is defined as follows.

$$ H : \mathbb{C}^{2} \to \mathbb{C}^{2} $$

$$ \textstyle H \ket{0} = \frac{1}{\sqrt{2}} \ket{0} + \frac{1}{\sqrt{2}} \ket{1} \\[1em] H \ket{1} = \frac{1}{\sqrt{2}} \ket{0} - \frac{1}{\sqrt{2}} \ket{1} $$

Its matrix representation is as follows.

$$ H = \dfrac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} $$

Description

The Hadamard gate changes a qubit existing only in the states of $\ket{0}$ or $\ket{1}$ into a superposition state where both exist simultaneously. Moreover, because the coefficients of the two states are $\frac{1}{\sqrt{2}}$, the probability of being measured as either $\ket{0}$ or $\ket{1}$ becomes equal.