  We introduce the first minimal and complete equational theory for quantum
circuits. Hence, we show that any true equation on quantum circuits can be
derived from simple rules, all of them being standard except a novel but
intuitive one which states that a multi-control $2\pi$ rotation is nothing but
the identity. Our work improves on the recent complete equational theories for
quantum circuits, by getting rid of several rules including a fairly
impractical one. One of our main contributions is to prove the minimality of
the equational theory, i.e. none of the rules can be derived from the other
ones. More generally, we demonstrate that any complete equational theory on
quantum circuits (when all gates are unitary) requires rules acting on an
unbounded number of qubits. Finally, we also simplify the complete equational
theories for quantum circuits with ancillary qubits and/or qubit discarding.
