  We study the Hamiltonian dynamics of a many-body quantum system subjected to
periodic projective measurements which leads to probabilistic cellular automata
dynamics. Given a sequence of measured values, we characterize their dynamics
by performing a principal component analysis. The number of principal
components required for an almost complete description of the system, which is
a measure of complexity we refer to as PCA complexity, is studied as a function
of the Hamiltonian parameters and measurement intervals. We consider different
Hamiltonians that describe interacting, non-interacting, integrable, and
non-integrable systems, including random local Hamiltonians and translational
invariant random local Hamiltonians. In all these scenarios, we find that the
PCA complexity grows rapidly in time before approaching a plateau. The dynamics
of the PCA complexity can vary quantitatively and qualitatively as a function
of the Hamiltonian parameters and measurement protocol. Importantly, the
dynamics of PCA complexity present behavior that is considerably less sensitive
to the specific system parameters for models which lack simple local dynamics,
as is often the case in non-integrable models. In particular, we point out a
figure of merit that considers the local dynamics and the measurement direction
to predict the sensitivity of the PCA complexity dynamics to the system
parameters.
