Sequential Quantum Cloning

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, $1\to M$, 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.