Even with arbitrary ambient unitaries and ancillas, sublinear-depth nearest-neighbour circuits remain far from approximate 2-designs over the matchgate, orthogonal, and symplectic groups and from a Clifford 4-design.
Local random quantum circuits are approximate polynomial-designs,
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Ambient unitaries don't enable shallow group designs
Even with arbitrary ambient unitaries and ancillas, sublinear-depth nearest-neighbour circuits remain far from approximate 2-designs over the matchgate, orthogonal, and symplectic groups and from a Clifford 4-design.