No-go theorems for sublinear-depth group designs.

Sublinear-depth, constant-locality one-dimensional circuits cannot form an approximate k-design over groups G with an invariant state G ⊗ k Ψ = Ψ and specified entanglement structure.

• Why it matters: Efficiently approximating Haar-random group behavior is central in quantum information, but prior logarithmic-depth results for the unitary group leave open whether similar efficiency is possible more broadly.
• What they did: The work proves no-go theorems for sublinear-depth 1D circuits with constant-locality gates, and extends them to D-dimensional lattices, showing design depth must scale at least as n^(1/D).
• The result: It also establishes a constant separation between maximum depth and gate counts preventing approximation of even the second moment of random matchgate circuits, and shows exact matchgate Haar implementation needs deeper resources; additionally, Pauli-string circuits with sublinear gate counts cannot yield approximate designs over G.

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