Pauli Z Gate

Pauli Z applies a phase flip: $|0\rangle$ unchanged, $|1\rangle$ gets a $-1$ phase.

Matrix:

$$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Action: $Z|0\rangle = |0\rangle$, $Z|1\rangle = -|1\rangle$.

• Self-inverse: $Z^2 = I$
• Eigenvalues: $+1, -1$ with eigenvectors $|0\rangle$ and $|1\rangle$ (computational basis)
• Diagonal: doesn't mix basis states, only applies phases
• Commutation: anticommutes with X and Y, commutes with Z

• Parity measurement: Z eigenvalues relate to parity of measurement outcomes
• Phase correction: apply controlled phases in quantum algorithms
• Basis preparation: Z measurements are the default computational basis
• Bloch sphere: rotation by $\pi$ around z-axis

• Building block: many parametric gates use RZ as foundation: $U(\theta, \phi, \lambda) = R_Z(\phi) R_Y(\theta) R_Z(\lambda)$

• Superconducting qubits: RZ($\pi$) often virtual (no pulse needed, adjust reference frame only); saves gate time
• Trapped ions: phase rotation via detuned laser (phase without excitation)
• Photonic: optical phase shifter
• Cost: typically fastest or “free” (virtual) on most platforms