In particular, we find (see table): no mixed-unitary one-designs, orthogonal/symplectic/matchgate 2-designs, and Clifford 8-designs can be achieved in sublinear depth with local gates. These findings imply many known (and some new) constructions are depth-optimal.

In this work, we derive general no-go theorems that rule out the existence of group designs with certain restrictions, e.g. depth or gate-count. Our results apply to a wide class of groups including the symplectic unitaries, matchgates, mixed-unitaries, Cliffords and other.