Any group with an invariant state cannot have approximate k-designs built from sublinear-depth local circuits; linear depth is necessary for matchgate, orthogonal, symplectic, Clifford (k=8), and mixed-unitary group designs.
This is our first example of a group for which there is no invariant state atk= 2, ruling out the application of Theorem 1
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No-go theorems for sublinear-depth group designs
Any group with an invariant state cannot have approximate k-designs built from sublinear-depth local circuits; linear depth is necessary for matchgate, orthogonal, symplectic, Clifford (k=8), and mixed-unitary group designs.