Short-depth designs beyond the unitary group.

No circuit ensemble with lightcones smaller than the system size can generate approximate designs for the Clifford, orthogonal, unitary symplectic, or matchgate groups.

• Why it matters: Shallow random-circuit constructions are known for approximate unitary designs, but it was unclear whether comparable randomness emerges for other symmetry groups central to quantum computing.
• What they did: The authors prove lower bounds showing these groups’ designs cannot be produced in sub-linear depth by commonly used circuit ensembles on any architecture, while logarithmic-depth circuits yield weaker randomness notions.
• The result: They show additive-error state designs and anti-concentration can appear at logarithmic depth in many cases, clarifying when and how randomness emerges beyond the unitary group.

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