  Random quantum circuits are paradigmatic models of minimally structured and
analytically tractable chaotic dynamics. We study a family of Floquet unitary
circuits with Haar random $U(1)$ charge conserving dynamics; the minimal such
model has nearest-neighbor gates acting on spin 1/2 qubits, and a single layer
of even/odd gates repeated periodically in time. We find that this minimal
model is not robustly thermalizing at numerically accessible system sizes, and
displays slow subdiffusive dynamics for long times. We map out the
thermalization dynamics in a broader parameter space of charge conserving
circuits, and understand the origin of the slow dynamics in terms of proximate
localized and integrable regimes in parameter space. In contrast, we find that
small extensions to the minimal model are sufficient to achieve robust
thermalization; these include (i) increasing the interaction range to
three-site gates (ii) increasing the local Hilbert space dimension by appending
an additional unconstrained qubit to the conserved charge on each site, or
(iii) using a larger Floquet period comprised of two independent layers of
gates. Our results should inform future numerical studies of charge conserving
circuits which are relevant for a wide range of topical theoretical questions.
