Global Heisenberg scaling in noisy and practical phase estimation

TL;DR

This paper investigates the conditions under which global Heisenberg scaling in phase estimation can be achieved in noisy environments, identifying a noise threshold and proposing a practical adaptive protocol for quantum phase estimation.

Contribution

It establishes a noise threshold for attaining global Heisenberg scaling and introduces a one-qubit adaptive protocol effective below this threshold.

Findings

Heisenberg scaling is fragile to phase damping noise above a certain threshold.

Global Heisenberg scaling can be achieved below the noise threshold in terms of limiting distribution.

A practical one-qubit adaptive protocol attains Heisenberg scaling under manageable noise levels.

Abstract

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.