  In the highly non-Gaussian regime, the quantum Ziv-Zakai bound (QZZB)
provides a lower bound on the available precision, demonstrating the better
performance compared with the quantum Cram\'er-Rao bound. However, evaluating
the impact of a noisy environment on the QZZB without applying certain
approximations proposed by Tsang [Phys. Rev. Lett. 108, 230401 (2012)] remains
a difficult challenge. In this paper, we not only derive the general form of
the QZZB with the photon loss and the phase diffusion by invoking the technique
of integration within an ordered product of operators, but also show its
estimation performance for several different Gaussian resources, such as a
coherent state (CS), a single-mode squeezed vacuum state (SMSVS) and a two-mode
squeezed vacuum state (TMSVS). Our results indicate that compared with the
SMSVS and the TMSVS, the QZZB for the CS always shows the better estimation
performance under the photon-loss environment. More interestingly, for the
phase-diffusion environment, the estimation performance of the QZZB for the
TMSVS can be better than that for the CS throughout a wide range of
phase-diffusion strength. Our findings will provide a useful guidance for
investigating the noisy quantum parameter estimation.
