  Not all unitary operations upon a set of qubits can be implemented by
sequential interactions between each qubit and an ancillary system. We analyze
the specific case of sequential quantum cloning 1->M and prove that the minimal
dimension D of the ancilla grows linearly with the number of clones M. In
particular, we obtain D = 2M for symmetric universal quantum cloning and D =
M+1 for symmetric phase-covariant cloning. Furthermore, we provide a recipe for
the required ancilla-qubit interactions in each step of the sequential
procedure for both cases.
