RX rotation gate

Definition

The $R_X$ gate rotates a single qubit about the $X$ axis of the Bloch sphere by an angle $\theta$:

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

This follows from the operator-exponential identity $e^{-i\theta X/2} = \cos(\theta/2)\,I - i\sin(\theta/2)\,X$, which holds because $X^2 = I$, so the Taylor series splits into even and odd powers of $X$ that resum to cosine and sine respectively.

Action on basis states

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

At $\theta = \pi/2$ this produces an equal superposition with a relative phase of $-i$ rather than the $+$ that $H$ produces — the two gates land on different points of the Bloch-sphere equator.

Key identities

• $R_X(\pi) = -iX$: a full $\pi$ rotation reproduces the Pauli-X flip up to the global phase $-i$, unobservable in isolation but relevant inside controlled operations.
• $R_X(\theta_1)R_X(\theta_2) = R_X(\theta_1 + \theta_2)$: rotations about the same axis compose additively, so $R_X$ forms a one-parameter subgroup of $SU(2)$.
• $R_X(\theta) = H\,R_Z(\theta)\,H$: since $HXH = Z$, conjugating $R_Z$ by Hadamard swaps the rotation axis from $Z$ to $X$ — the standard way a $Z$-rotation-native compiler synthesizes $R_X$.
• Together with $R_Z$, $R_X$ generates arbitrary single-qubit unitaries via Euler decomposition, e.g. $R_Z(\alpha)R_X(\beta)R_Z(\gamma)$ up to a global phase.