Topological Gates

Definition¹

For $\theta \in \mathbb{R}$, a qubit gate defined as follows, referred to as a phase gate, is the $1$ qubit gate.

$$ R_{\theta} : \mathbb{C}^{2} \to \mathbb{C}^{2} $$

$$ \textstyle R_{\theta} \ket{0} = \ket{0} \\[1em] R_{\theta} \ket{1} = e^{\rm{i}\theta} \ket{1} $$

The matrix representation is as follows.

$$ R_{\theta} = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\theta} \end{bmatrix} $$

Explanation

The phase gate leaves $\ket{0}$ unchanged and transforms $\ket{1}$ into $e^{\rm{i}\theta} \ket{1}$. Therefore, the measurement probability for both states does not change after passing through the phase gate.