RY rotation gate

Definition

$$R_Y(\theta) = e^{-i\theta Y/2} = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

Unlike $R_X$ or $R_Z$, every entry of $R_Y(\theta)$ is a real number for real $\theta$: the $-i$ prefactor from the exponential exactly cancels the $\pm i$ entries of $Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$, leaving a real $2\times 2$ rotation matrix — the same form as a classical rotation matrix in the $(|0\rangle, |1\rangle)$ plane.

Action on basis states

$$R_Y(\theta)|0\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle, \qquad R_Y(\theta)|1\rangle = -\sin(\theta/2)|0\rangle + \cos(\theta/2)|1\rangle$$

At $\theta = \pi/2$, $R_Y(\pi/2)|0\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ — the same state Hadamard produces on $|0\rangle$, but $R_Y(\pi/2)|1\rangle = \tfrac{1}{\sqrt2}(-|0\rangle + |1\rangle)$ differs from $H|1\rangle$ by an overall sign.

Key identities

• $R_Y(\pi) = -iY$: a full $\pi$ rotation reproduces the Pauli-Y flip up to global phase $-i$.
• $X R_Y(\theta) X = R_Y(-\theta)$: since $XYX = -Y$, conjugating by $X$ reverses the rotation direction.
• Because its output amplitudes are real, $R_Y(\theta)$ is the standard primitive for amplitude encoding — loading a classical probability vector $(\cos(\theta/2), \sin(\theta/2))$ into a qubit — and for the mixer layers of variational circuits such as hardware-efficient ansätze.
• $R_Y(\theta) = R_Z(-\pi/2)\,R_X(\theta)\,R_Z(\pi/2)$: an axis change from $X$ to $Y$ via conjugation by a $Z$ rotation, mirroring the $H$-conjugation identity that relates $R_X$ and $R_Z$.