Absence of censoring inequalities in random quantum circuits

TL;DR

This paper demonstrates that removing gates from certain random quantum circuit architectures can unexpectedly improve their approximate t-design properties, challenging previous assumptions about the effects of gate deletion.

Contribution

It constructs specific architectures where deleting gates reduces the approximate 2-design depth, revealing non-monotonic effects of gate removal on circuit scrambling.

Findings

Deleting gates can decrease approximate 2-design depth.

Gate deletion may increase scrambling in the long run.

Analogous results are shown for spectral gaps and interaction graphs.

Abstract

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.