  Today's devices for quantum computing are still far from implementing useful
and powerful quantum algorithms. Decoherence and the wish to resist the effects
of errors in a system of quantum bits incurs a lot of overhead in the number of
gates and qubits. From a theoretical perspective, controlled quantum simulation
raises the hope to simulate the unitary quantum operationes generated by a
Hamiltonian with 3-body interaction with a suitably designed element that is
constructed of only 2-body interactions. That replacement would happen without
any additional gates, and its possibility would be due to the ambiguity of the
unit element of the Lie group connected with the algebra of traceless hermitian
matrices. We show that this hope is void, and give a general proof for this for
any order of interaction.
