# CZ Gate (Controlled-Z)

**CZ** applies a phase to the $|11\rangle$ state. Symmetric two-qubit gate (control and target are interchangeable).

Matrix:

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

Action: $|ab\rangle \to (-1)^{ab}|ab\rangle$ (applies $-1$ phase to $|11\rangle$).

## Relation to CNOT

CZ and CNOT are related via Hadamards on the target:

$$\text{CZ}_{01} = (I \otimes H) \text{CX}_{01} (I \otimes H)$$

$$\text{CNOT} = (I \otimes H) \text{CZ} (I \otimes H)$$

Either can be converted to the other using two Hadamard gates, so CNOT and CZ have equivalent power.

## Properties

- **Self-inverse**: $\text{CZ}^2 = I$
- **Symmetric**: control and target are equivalent; $\text{CZ}_{01} = \text{CZ}_{10}$
- **Diagonal**: only applies phases, like Z gates
- **Commutes with Z**: CZ and Z operators commute

## Uses

- **Alternative entangling gate**: some platforms find CZ easier to implement than CNOT
- **Measurement**: CZ can measure parity (CNOT more common)
- **Algorithm equivalence**: CZ and CNOT interconvertible; can substitute in any algorithm

## Parametric Version

- **CZ($\theta$)**: applies phase $e^{i\theta}$ to $|11\rangle$; $\text{CZ}(\theta) = \text{diag}(1, 1, 1, e^{i\theta})$
- **Use**: tunable coupling systems with parametric interaction strength

## Implementation

- **Superconducting qubits**: capacitive or tunable coupling; gate time ~20–100 ns; fidelity 98–99.5%
- **Trapped ions**: entangling laser pulses naturally produce CZ-like interactions
- **Photonic**: beam splitters or nonlinear effects