  In this paper we consider the problem of minimizing the ancillary systems
required to realize an arbitrary strategy of a quantum protocol, with the
assistance of classical memory. For this purpose we introduce the notion of
memory cost of a strategy, which measures the resources required in terms of
ancillary dimension. We provide a condition for the cost to be equal to a given
value, and we use this result to evaluate the cost in some special cases. As an
example we show that any covariant protocol for the cloning of a unitary
transformation requires at most one ancillary qubit. We also prove that the
memory cost has to be determined globally, and cannot be calculated by
optimizing the resources independently at each step of the strategy.

Quantum Physics