{title2=$R_y(\theta)$}

= Rotations about the y-axis
{synonym}

The one-qubit <rotation gate>
$$
R_y(\theta)=\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}
$$
satisfies $R_y(2\theta)|0\rangle=\cos\theta|0\rangle+\sin\theta|1\rangle$ and $R_y(\alpha)R_y(\beta)=R_y(\alpha+\beta)$. The latter identity enables a <binary-angle implementation of a quantum variable rotation>.


Rotation about the y-axis

{title2=$R_z(\theta)$}

The one-qubit <rotation gate> is $R_z(\theta)=\operatorname{diag}(e^{-i\theta/2},e^{i\theta/2})$. It implements the two relative phases needed in <simulation of a computable diagonal Hamiltonian>.


Rotation about the z-axis

A one-qubit rotation gate is $R_\alpha(\theta)=e^{-i\theta\sigma_\alpha/2}$, where $\sigma_\alpha$ is a <Pauli matrix>. It rotates the <Bloch vector> by angle $\theta$ about the corresponding axis.


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 Incoming links: Rotation gate