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)

• 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: $R_Z(\alpha) R_Z(\beta) = R_Z(\alpha + \beta)$

• 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

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

• 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