Global Heisenberg scaling in noisy and practical phase estimation

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 present a comprehensive and rigorous study of the attainability of strong, global notions of Heisenberg scaling (in contrast to the commonly studied local estimation based on e.g. quantum Fisher information) in the fundamental problem of quantum metrology, in noisy environments. As our first contribution, we formally define two useful notions of Heisenberg scaling in global estimation respectively based on the average estimation error and the limiting distribution of estimation error (which we highlight as a practically important figure of merit). A main result of this work is that for the standard phase damping noise, an O(n (-1)) noise rate is a necessary and sufficient condition for attaining global Heisenberg scaling. We first prove that O(n (-1)) is an upper bound on the noise rate for Heisenberg scaling to be possible, and then show by constructing a 'robust' estimation procedure that global Heisenberg scaling in both senses can indeed be achieved under Theta(n (-1)) noise. In addition, we provide a practically more friendly adaptive protocol using only an one-qubit memory, which achieves global Heisenberg scaling in terms of limiting distribution as well under O(n (-1)) noise.