We establish general limits on how precise a parameter, e.g. frequency or the
strength of a magnetic field, can be estimated with the aid of full and fast
quantum control. We consider uncorrelated noisy evolutions of N qubits and show
that fast control allows to fully restore the Heisenberg scaling (~1/N^2) for
all rank-one Pauli noise except dephasing. For all other types of noise the
asymptotic quantum enhancement is unavoidably limited to a constant-factor
improvement over the standard quantum limit (~1/N) even when allowing for the
full power of fast control. The latter holds both in the single-shot and
infinitely-many repetitions scenarios. However, even in this case allowing for
fast quantum control helps to increase the improvement factor. Furthermore, for
frequency estimation with finite resource we show how a parallel scheme
utilizing any fixed number of entangled qubits but no fast quantum control can
be outperformed by a simple, easily implementable, sequential scheme which only
requires entanglement between one sensing and one auxiliary qubit.Comment: 17 pages, 7 figures, 6 appendice
