  We investigate the quantization of the complex-valued Berry phases in
non-Hermitian quantum systems with certain generalized symmetries. In Hermitian
quantum systems, the real-valued Berry phase is known to be quantized in the
presence of certain symmetries, and this quantized Berry phase can be regarded
as a topological order parameter for gapped quantum systems. In this paper, on
the other hand, we establish that the complex Berry phase is also quantized in
the systems described by a family of non-Hermitian Hamiltonians. Let
$H(\theta)$ be a non-Hermitian Hamiltonian parameterized by $\theta$. Suppose
that there exists a unitary and Hermitian operator $P$ such that $PH(\theta)P =
H(-\theta)$ or $PH(\theta)P = H^\dagger(-\theta)$. We prove that in the former
case, the complex Berry phase $\gamma$ is $\mathbb{Z}_2$-quantized, while in
the latter, only the real part of $\gamma$ is $\mathbb{Z}_2$-quantized. The
operator $P$ can be viewed as a generalized symmetry for $H(\theta)$, and in
practice, $P$ can be, for example, a spatial inversion. We also argue that this
quantized complex Berry phase is capable of classifying non-Hermitian
topological phases, and we demonstrate this in some one-dimensional strongly
correlated systems.
