Lorenzo Grevink

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. For the Clifford, orthogonal, and unitary symplectic group, we moreover show that commonly considered circuit ensembles cannot generate designs in sub-linear depth on any circuit architecture. We show this by exploiting observables in the higher-order commutants of each group, which allow one to distinguish any short-depth circuit from truly random. While these no-go results rule out short-depth designs over these subgroups, we prove that slightly weaker forms of randomness---including additive-error state designs and anti-concentration in sampling distributions---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.