Abstract

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 and orthogonal group, we moreover show that a broad class of circuits 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 unitary designs, 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.

Personal information

Lorenzo is a PhD student at QuSoft in Amsterdam, under the supervision of Jonas Helsen. He is interested in random systems, both quantum and classical. A random system in quantum information is usually described by Haar-random unitaries. While easy to work with in theory, Haar-random unitaries need exponentially many gates to be constructed. That is why researchers introduced unitary designs, ensembles over the unitary group that appear to be random to a restricted observer. Exciting results in [SHH24] show that unitary designs can be constructed in a depth logarithmic in the system size, using a technique called ``gluing". This talk will be about his recent paper called: Will it glue? On short-depth designs beyond the unitary group. We will discuss the possibilities to generalize the results in [SHH24] to other groups than the unitary group, as for example the Clifford group.

Reference

https://arxiv.org/abs/2506.23925