  The SWAP gate has become an integral feature of quantum circuit architectures
and is designed to permute the states of two qubits through the use of the
well-known controlled-NOT gate. We consider the question of whether a two-qudit
quantum circuit composed entirely from instances of the generalised
controlled-NOT gate can be constructed to permute the states of two qudits.
Arguing via the signature of a permutation, we demonstrate the impossibility of
such circuits for dimensions $d \equiv 3$ (mod 4).
