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

## Properties

- **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

## Uses

- **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

## Composition

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

## Implementation

- **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