Controlled-Z gate

Definition

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

in the ordered basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$. The matrix is diagonal and, unlike $\mathrm{CX}$, symmetric under exchanging the two qubits: $\mathrm{CZ}$ has no distinguished "control" wire.

Action on basis states

$$\mathrm{CZ}|c,t\rangle = (-1)^{c\cdot t}|c,t\rangle$$

Only the $|11\rangle$ component picks up a sign; all other basis states are untouched. Applied to a product of superpositions it creates entanglement by correlating the relative phase with both qubits simultaneously, e.g. $\mathrm{CZ}\big(\tfrac{|0\rangle+|1\rangle}{\sqrt2}\otimes\tfrac{|0\rangle+|1\rangle}{\sqrt2}\big) = \tfrac12(|00\rangle+|01\rangle+|10\rangle-|11\rangle)$.

Key identities

• $\mathrm{CZ} = (I\otimes H)\,\mathrm{CX}\,(I\otimes H)$: conjugating the target of $\mathrm{CX}$ by Hadamard converts the bit-flip entangler into the phase-flip entangler and vice versa.
• $\mathrm{CZ}^2 = I\otimes I$: $\mathrm{CZ}$ is its own inverse (it is a Clifford involution).
• $\mathrm{CZ}$ is the $\lambda=\pi$ special case of the controlled-phase gate $\mathrm{CP}(\lambda)$.
• Applying $\mathrm{CZ}$ to every edge of a graph on qubits initialized in $|+\rangle$ prepares the corresponding cluster/graph state, the resource state for measurement-based quantum computation.