# RX Gate (Rotation around X)

**RX** rotates a qubit around the x-axis by angle $\theta$.

Matrix:

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

Action: $R_X(\theta)|\psi\rangle$ rotates $|\psi\rangle$ by $\theta$ radians around x-axis on Bloch sphere.

Special cases:

- $R_X(0) = I$ (identity)
- $R_X(\pi/2) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix}$
- $R_X(\pi) = -i X$ (equivalent to Pauli X up to global phase)

## Properties

- **Periodicity**: $R_X(\theta + 2\pi) = R_X(\theta)$
- **Inverse**: $R_X(\theta)^\dagger = R_X(-\theta)$
- **Composition**: $R_X(\alpha) R_X(\beta) = R_X(\alpha + \beta)$

## Uses

- **Parameterized circuits**: commonly used in variational ansatze
- **Native pulses**: superconducting qubits naturally implement RX via resonant microwave pulses
- **Bloch sphere**: visualize quantum state evolution via x-axis rotation

## Decomposition

- **Exponential form**: $R_X(\theta) = e^{-i\theta X/2}$
- **Alternative**: $R_X(\theta) = H R_Z(\theta) H$ (rotation around x via z with basis changes)

## Implementation

- **Superconducting qubits**: pulse duration and amplitude control rotation angle; $R_X(\pi/2)$ ~20–40 ns
- **Trapped ions**: laser pulse at qubit transition frequency, tuned for desired angle
- **Photonic**: optical rotators or polarization controllers
- **Fidelity**: typically 99–99.9% depending on platform