RZ¶
RZ rotates a qubit by an angle \( \theta \) about the Z-axis of the Bloch sphere. It produces the same relative phase shift as P, but - unlike P - also applies an overall phase, making it a true rotation.
| Qubits | 1 |
| Parameters | \( \theta \) - rotation angle, in radians |
| Inverse | RZ(−θ) |
| Also called | Z-rotation |
What it does¶
RZ(θ) multiplies \( |0\rangle \) by \( e^{-i\theta/2} \) and \( |1\rangle \) by \( e^{i\theta/2} \). The difference between those two phases is θ - exactly what P(θ) produces - so RZ(θ) and P(θ) affect measurement probabilities identically: \( RZ(\theta) = e^{-i\theta/2}\,P(\theta) \).
Matrix¶
Action on basis states¶
- \( |0\rangle \rightarrow e^{-i\theta/2}\,|0\rangle \)
- \( |1\rangle \rightarrow e^{i\theta/2}\,|1\rangle \)
Phase and Bloch-sphere effect¶
Because global phase (a factor applied equally to every amplitude in a state) has no observable effect, RZ(θ) and P(θ) move a qubit's point on the Bloch sphere identically - both are rotations by θ about the Z-axis. The difference only shows up when the qubit is entangled with, or has its phase compared against, another qubit that wasn't rotated - since global-phase equivalence only holds per-qubit, not for a shared multi-qubit state.
Example¶
H then RZ with \( \theta = \pi/2 \) on q[0] gives the same measurement probabilities as H then S - the Probabilities panel is identical either way.
Other notations¶
Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.
| Language | Typical form |
|---|---|
| OpenQASM 2.0 | rz(theta) q[0]; |
| Qiskit | circuit.rz[theta, q[0]] |
| Cirq | cirq.rz(theta)(q[0]) |
| Q# | Rz(theta, q[0]); |