In the accompanying paper [Y. Mitsuhashi, et al., Phys. Rev. Lett. 134, 180404 (2025)], we established the method of characterizing the maximal order of asymptotic unitary designs generated by symmetric local random circuits, and explicitly specified the order in the cases of ${\mathbb{Z}}_2, \mathrm{U}\left(1\right)$, and $\mathrm{SU}\left(2\right)$ symmetries. Here, we provide full details on the derivation of the main theorems for general symmetry and for concrete symmetries. Furthermore, we consider a general framework where we have access to a finite set of connected compact unitary subgroups, which includes symmetric local unitary gate sets.

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• Quantum circuits
• Quantum control
• Quantum information theory