Pauli-Z¶
The Pauli-Z gate leaves \( |0\rangle \) alone and flips the sign of \( |1\rangle \). It's the simplest possible "phase flip."
| Qubits | 1 |
| Parameters | None |
| Inverse | Itself (Z² = I) |
| Also called | Z gate, phase-flip gate; equal to P(π) |
What it does¶
Z does nothing visible to a qubit that's definitely \( |0\rangle \) or \( |1\rangle \) - measuring gives the same result either way. Its effect only becomes visible on a superposition, where it flips the relative phase between the \( |0\rangle \) and \( |1\rangle \) components by \( \pi \).
Matrix¶
Action on basis states¶
- \( |0\rangle \rightarrow |0\rangle \)
- \( |1\rangle \rightarrow -|1\rangle \)
Phase and Bloch-sphere effect¶
On the Bloch sphere, Z is a 180° rotation about the Z-axis, which passes straight through the poles - so the poles (\( |0\rangle \), \( |1\rangle \)) don't move, but any point on the equator (an equal superposition) rotates halfway around. See Phase disks for how this shows up visually.
Example¶
H then Z on q[0] turns \( |{+}\rangle = (|0\rangle+|1\rangle)/\sqrt{2} \) into \( |{-}\rangle = (|0\rangle-|1\rangle)/\sqrt{2} \) - same measurement probabilities as plain H, but a different relative phase.
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 | z q[0]; |
| Qiskit | circuit.z[q[0]] |
| Cirq | cirq.Z(q[0]) |
| Q# | Z(q[0]); |