The Controlled Z-gate

When we apply the $Z$-gate on separate qubits, we usually see the phase shift for those states where the respective qubit is in state $|1\rangle$. When both qubits are in-state $|1\rangle$, the phase shift adds up. Since the $Z$-gate accounts for a phase shift by half a wavelength, applying it twice results in the initial phase. Hence, the state $|11\rangle$ has the same phase as a state $|00\rangle$.

Sometimes we don’t want this effect. We might want to switch the phase of a single state. This is where, the controlled $Z$-gate comes in handy. Like its controlled peers, the controlled $Z$-gate applies the $Z$-gate on the target qubit only if the control qubit is in state $|1\rangle$.

The controlled $Z$-gate has the following transformation matrix.

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

As a result of the $CZ$-gate, we see state $|11\rangle$ has a different phase than the other three states. The qubit at position 0 must be in-state $|1\rangle$, for it is the control qubit. The qubit at position 1 must be on state $|1\rangle$ because the $Z$-gate applies the phase to this part of the qubit state.

The $CZ$-gate induces a phase shift for states where both qubits are in-state $|1\rangle$. It does not matter if we use qubit 0 or qubit 1 as the control qubit. Therefore, in Qiskit, we use a drawing that does not indicate the control qubit.