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# RZ Gate (Rotation around Z)

**RZ** rotates a qubit around the z-axis by angle $\theta$.

Matrix:

$$R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$

Action: $R_Z(\theta)|\psi\rangle$ rotates $|\psi\rangle$ by $\theta$ radians around z-axis on Bloch sphere, or equivalently, applies phase factors.

Special cases:

- $R_Z(0) = I$
- $R_Z(\pi/2) = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{-i\pi/4} & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$ (related to S gate)
- $R_Z(\pi) = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ (related to Z gate)

## Properties

- **Periodicity**: $R_Z(\theta + 2\pi) = R_Z(\theta)$
- **Inverse**: $R_Z(\theta)^\dagger = R_Z(-\theta)$
- **Diagonal**: only applies phases, doesn't mix basis states
- **Composition**: $R_Z(\alpha) R_Z(\beta) = R_Z(\alpha + \beta)$

## Uses

- **Phase control**: adjust global and relative phases in quantum circuits
- **Decomposition**: building block for $U(\theta, \phi, \lambda) = R_Z(\phi) R_Y(\theta) R_Z(\lambda)$
- **Parameter sweeps**: RZ with varying angles in parameterized circuits
- **Virtual gates**: "free" on many platforms via reference frame adjustment

## Relation to Phase Gates

- **S gate**: $S = R_Z(\pi/2)$
- **T gate**: $T = R_Z(\pi/4)$
- **Z gate**: $Z = R_Z(\pi)$

## Implementation

- **Superconducting qubits**: virtual gate (reference frame adjustment only, no pulse)
- **Trapped ions**: detuned laser (applies phase without excitation)
- **Photonic**: optical phase shifter
- **Cost**: negligible if virtual; otherwise same as single-qubit rotation