Will it glue? On short-depth designs beyond the unitary group

Understand

We study the formation of short-depth designs beyond the unitary group.

• We provide a range of results on several groups of broad interest in quantum information science: the Clifford group, the orthogonal group, the unitary symplectic groups, and the matchgate group.
• For all of these groups, we prove that analogues of unitary designs cannot be generated by any circuit ensemble with light-cones that are smaller than the system size.
• This implies linear lower bounds on the circuit depth in one-dimensional systems.