  In this paper, we establish the quantitative mean ergodic theorems for two
subclasses of power bounded operators on a fixed noncommutative $L_p$-space
with $1<p<\infty$, which mainly concerns power bounded invertible operators and
Lamperti contractions. Our approach to the quantitative ergodic theorems is the
noncommutative square function inequalities. The establishment of the latter
involves several new ingredients such as the almost orthogonality and
Calder\'on-Zygmund arguments for non-smooth kernels from semi-commutative
harmonic analysis, the extension properties of the operators under
consideration from operator theory, and a noncommutative version of the
classical transference method due to Coifman and Weiss.
