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We study the Hamiltonian dynamics of a many-body quantum system subjected toperiodic projective measurements which leads to probabilistic cellular automatadynamics. Given a sequence of measured values, we characterize their dynamicsby performing a principal component analysis. The number of principalcomponents required for an almost complete description of the system, which isa measure of complexity we refer to as PCA complexity, is studied as a functionof the Hamiltonian parameters and measurement intervals. We consider differentHamiltonians that describe interacting, non-interacting, integrable, andnon-integrable systems, including random local Hamiltonians and translationalinvariant random local Hamiltonians. In all these scenarios, we find that thePCA complexity grows rapidly in time before approaching a plateau. The dynamicsof the PCA complexity can vary quantitatively and qualitatively as a functionof the Hamiltonian parameters and measurement protocol. Importantly, thedynamics of PCA complexity present behavior that is considerably less sensitiveto 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 afigure of merit that considers the local dynamics and the measurement directionto predict the sensitivity of the PCA complexity dynamics to the systemparameters.

Complexity of spin configurations dynamics due to unitary evolution and periodic projective measurements