RZ Gate (Z-Rotation)

RZ shifts the relative phase between $|0⟩$ and $|1⟩$ by a continuously tunable angle $\theta$. Relative phase is the difference in the complex-number angles of the two amplitudes. It determines how the $|0⟩$ and $|1⟩$ components interfere when combined by a later gate. Like Z, RZ does not change measurement probabilities in the computational basis. The effect is purely in the phase. It becomes visible only after a gate like H converts the phase difference into a probability change. On many real quantum processors, RZ is implemented as a "virtual gate" by adjusting the classical control frame, making it essentially free of errors.

Tuning phase for interference

1. 1Start with $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$.
2. 2Apply RZ$(\pi /2)$. Each basis state picks up a different phase factor. The relative phase between $|0⟩$ and $|1⟩$ is now $\pi /2$.
3. 3Measure in the computational basis. The probabilities are still 50/50 because RZ only changes phase, not amplitudes.
4. 4Apply H. Hadamard converts the phase difference into a probability bias, shifting the measurement outcome away from 50/50.

RZ is arguably the most practical gate. On many superconducting quantum processors, Z-rotations are implemented by adjusting the classical reference frame rather than pulsing the qubit physically. This makes them virtually error-free and instantaneous. Understanding RZ is key to efficient circuit compilation.