  Heisenberg scaling characterizes the ultimate precision of parameter
estimation enabled by quantum mechanics, which represents an important quantum
advantage of both theoretical and technological interest. Here, we study the
attainability of strong, global notions of Heisenberg scaling in the
fundamental problem of phase estimation, from a practical standpoint. A main
message of this work is an asymptotic noise "threshold" for global Heisenberg
scaling. We first demonstrate that Heisenberg scaling is fragile to noises in
the sense that it cannot be achieved in the presence of phase damping noise
with strength above a stringent scaling in the system size. Nevertheless, we
show that when the noise does not exceed this threshold, the global Heisenberg
scaling in terms of limiting distribution (which we highlight as a practically
important figure of merit) as well as average error can indeed be achieved.
Furthermore, we provide a practical adaptive protocol using one qubit only,
which achieves global Heisenberg scaling in terms of limiting distribution
under such noise.
