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)

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

• 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

• 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)

• 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