  In this work, we address the central problem about how to effectively find
the available precision limit of unknown parameters. In the framework of the
quantum Ziv-Zakai bound (QZZB), we employ noiseless linear amplification
(NLA)techniques to an initial coherent state (CS) as the probe state, and focus
on whether the phase estimation performance is improved significantly in noisy
scenarios, involving the photon-loss and phase-diffusion cases. More
importantly, we also obtain two kinds of Heisenberg error limits of the QZZB
with the NLA-based CS in these noisy scenarios, making comparisons with both
the Margolus-Levitin (ML) type bound and the Mandelstam-Tamm (MT) type bound.
Our analytical results show that in cases of photon loss and phase diffusion,
the phase estimation performance of the QZZB can be improved remarkably by
increasing the NLA gain factor. Particularly, the improvement is more
pronounced with severe photon losses. Furthermore in minimal photon losses, our
Heisenberg error limit shows better compactness than the cases of the ML-type
and MT-type bounds. Our findings will provide an useful guidance for
accomplishing more complex quantum information processing tasks.
