Absence of censoring inequalities in random quantum circuits

Ref. 1 asked whether deleting gates from a random quantum circuit architecture can ever make the architecture a better approximate $t$-design. We show that it can. In particular, we construct a family of architectures such that the approximate $2$-design depth decreases when certain gates are deleted. We also give some intuition for this construction and discuss the relevance of this result to the approximate $t$-design depth of the 1D brickwork. Deleting gates always decreases scrambledness in the short run, but can sometimes cause it to increase in the long run. Finally, we give analogous results for spectral gaps and when deleting edges of interaction graphs.