CZ gate (controlled-Z) is a two-qubit gate that applies a 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 $\lvert 11\rangle$:

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

A Hadamard gate on both qubits followed by CZ and Hadamard on both again is equivalent to a 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.

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.