  We study the entanglement dynamics of quantum automaton (QA) circuits in the
presence of U(1) symmetry. We find that the second R\'enyi entropy grows
diffusively with a logarithmic correction as $\sqrt{t\ln{t}}$, saturating the
bound established by Huang [IOP SciNotes 1, 035205 (2020)]. Thanks to the
special feature of QA circuits, we understand the entanglement dynamics in
terms of a classical bit string model. Specifically, we argue that the
diffusive dynamics stems from the rare slow modes containing extensively long
domains of spin 0s or 1s. Additionally, we investigate the entanglement
dynamics of monitored QA circuits by introducing a composite measurement that
preserves both the U(1) symmetry and properties of QA circuits. We find that as
the measurement rate increases, there is a transition from a volume-law phase
where the second R\'enyi entropy persists the diffusive growth (up to a
logarithmic correction) to a critical phase where it grows logarithmically in
time. This interesting phenomenon distinguishes QA circuits from non-automaton
circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an
area-law phase transition exists, and any non-zero rate of projective
measurements in the volume-law phase leads to a ballistic growth of the R\'enyi
entropy.
