Characterization of randomness in quantum circuits of continuous gate sets
Yosuke Mitsuhashi, Ryotaro Suzuki, Tomohiro Soejima, Nobuyuki Yoshioka · Physical Review A · 2025
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 ${\mathbb{Z}}_{2}, \mathrm{U}(1)$, and $\mathrm{SU}(2)$ 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.