  We introduce the first complete equational theory for quantum circuits. More
precisely, we introduce a set of circuit equations that we prove to be sound
and complete: two circuits represent the same unitary map if and only if they
can be transformed one into the other using the equations. The proof is based
on the properties of multi-controlled gates -- that are defined using
elementary gates -- together with an encoding of quantum circuits into linear
optical circuits, which have been proved to have a complete axiomatisation.
