Will it glue? On short-depth designs beyond the unitary group

TL;DR

This paper investigates the limitations of generating short-depth quantum designs beyond the unitary group, establishing lower bounds on circuit depth for various groups and showing that weaker forms of randomness can still emerge at logarithmic depths.

Contribution

It proves that certain quantum groups cannot be generated by shallow circuits, and demonstrates that weaker randomness properties appear at logarithmic depths.

Findings

Linear lower bounds on circuit depth for design generation.

No sub-linear depth design generation for Clifford and orthogonal groups.

Weaker randomness forms emerge at logarithmic depths.

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.