Short-depth designs beyond the unitary group

Random quantum operations are a central tool in quantum information and beyond, having applications ranging from benchmarking and cryptography to many-body physics. While recent advances have shown that shallow-depth circuits can realize approximate unitary designs, which are ensembles that appear to be random to a restricted observer, it remained unclear whether similar constructions extend to other groups that play important roles in quantum computing. 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 these groups we prove that designs cannot be generated by any circuit ensemble with lightcones that are smaller than the system size. We moreover show that commonly considered circuit ensembles cannot generate designs in sub-linear depth on any circuit architecture. On the other hand, we prove that slightly weaker forms of randomness-including additive-error state designs and anti-concentration-nevertheless emerge at logarithmic depths in many cases. Our results reveal that the onset of randomness in shallow quantum circuits is a widespread yet subtle phenomenon, dependent on the interplay between the group itself and the context of its application.