We describe quantum circuits generating four-qubit maximally entangled states, the amount of entanglement being quantified by using the absolute value of the Cayley hyperdeterminant as an entanglement monotone. More precisely we show that this type of four-qubit entangled states can be obtained by the action of a family of $\mathtt{CNOT}$ circuits on some special states of the LU orbit of the state $|0000\rangle$ .