  We consider a general model of unitary parameter estimation in presence of
Markovian noise, where the parameter to be estimated is associated with the
Hamiltonian part of the dynamics. In absence of noise, unitary parameter can be
estimated with precision scaling as $1/T$, where $T$ is the total probing time.
We provide a simple algebraic condition involving solely the operators
appearing in the quantum Master equation, implying at most $1/\sqrt{T}$ scaling
of precision under the most general adaptive quantum estimation strategies. We
also discuss the requirements a quantum error-correction like protocol must
satisfy in order to regain the $1/T$ precision scaling in case the above
mentioned algebraic condition is not satisfied. Furthermore, we apply the
developed methods to understand fundamental precision limits in atomic
interferometry with many-body effects taken into account, shedding new light on
the performance of non-linear metrological models.
