# Rotation-X gate
**$R_x(\theta)$ gate** rotates a qubit by angle $\theta$ about the $x$-axis of the [[bloch-sphere|Bloch sphere]]. It is one of the three [[rotation-gates|rotation gates]] and is defined as the matrix exponential of the [[pauli-gates|Pauli-X]] operator:

$$R_x(\theta) = e^{-i\theta X/2} = \begin{pmatrix}\cos\dfrac{\theta}{2} & -i\sin\dfrac{\theta}{2}\\[6pt] -i\sin\dfrac{\theta}{2} & \cos\dfrac{\theta}{2}\end{pmatrix}$$

Applied to the computational basis states:

$$R_x(\theta)\lvert 0\rangle = \cos\tfrac{\theta}{2}\lvert 0\rangle - i\sin\tfrac{\theta}{2}\lvert 1\rangle \qquad R_x(\theta)\lvert 1\rangle = -i\sin\tfrac{\theta}{2}\lvert 0\rangle + \cos\tfrac{\theta}{2}\lvert 1\rangle$$

The rotation mixes $\lvert 0\rangle$ and $\lvert 1\rangle$ with real cosine and imaginary sine coefficients. At $\theta = \pi/2$ it produces $\tfrac{1}{\sqrt{2}}(\lvert 0\rangle - i\lvert 1\rangle)$, a state midway between the two basis states with an imaginary phase on $\lvert 1\rangle$. At $\theta = \pi$, $R_x(\pi) = -iX$ — a bit flip up to global phase. The eigenstates of $R_x(\theta)$ for any nonzero $\theta$ are $\lvert +\rangle$ and $\lvert -\rangle$ (the $x$-axis poles of the Bloch sphere), since those states are invariant under rotation about $x$.

## List of code implementations

- [[r-gate-qiskit|Rotation gates (Qiskit)]]
- [[r-gate-custatevec|Rotation gates (cuStateVec)]]
- [[r-gate-cudaq|Rotation gates (CUDA-Q)]]