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 $|11\rangle$).

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.

• 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

• 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

• 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

• 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