# CZ gate
**CZ gate** (controlled-Z) is a two-qubit gate that applies a [[z-gate|Pauli Z]] to the target qubit if and only if the control qubit is $\lvert 1\rangle$, and does nothing otherwise. Because Z only adds a phase of $-1$ to $\lvert 1\rangle$, the gate is symmetric: either qubit can be treated as the control.

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

The rows and columns are ordered $\lvert 00\rangle, \lvert 01\rangle, \lvert 10\rangle, \lvert 11\rangle$. The gate leaves the first three basis states unchanged and flips the sign of $\lvert 11\rangle$:

$$\text{CZ}\lvert 00\rangle = \lvert 00\rangle \qquad \text{CZ}\lvert 01\rangle = \lvert 01\rangle \qquad \text{CZ}\lvert 10\rangle = \lvert 10\rangle \qquad \text{CZ}\lvert 11\rangle = -\lvert 11\rangle$$

## Creating entanglement

A Hadamard gate on both qubits followed by CZ and Hadamard on both again is equivalent to a [[cx-gate|CX gate]]: $\text{CX} = (I \otimes H) \cdot \text{CZ} \cdot (I \otimes H)$. The CZ gate is the native two-qubit gate on several hardware platforms (superconducting qubits, neutral atoms) because the symmetric interaction is easier to implement physically than the directional [[cx-gate|CX gate]].

## Properties

The CZ gate is its own inverse: $\text{CZ}^2 = I$. Its symmetry under qubit exchange means the circuit identity $\text{CZ}_{01} = \text{CZ}_{10}$ holds exactly, with no additional SWAP needed. Together with all single-qubit gates it forms a universal gate set for quantum computation.