English

On Clifford hierarchy testing and near-extremizers of noncommutative uniformity norms

Quantum Physics 2026-05-27 v1

Abstract

We consider the problem of testing whether an unknown unitary is close to a specified level of the Clifford hierarchy. Bu, Gu, and Jaffe proposed a candidate tester for this task based on a connection with noncommutative analogues of the Gowers uniformity norms. The complexity of this tester -- whose analysis depends on a robust characterization of the near-extremizers of these norms -- was left open. We establish such a characterization for the fourth noncommutative uniformity norm and, as a consequence, obtain an efficient tester for the third level of the Clifford hierarchy. We further discuss possible routes toward resolving the problem of testing for all higher levels, highlighting the main barriers that remain.

Keywords

Cite

@article{arxiv.2605.26983,
  title  = {On Clifford hierarchy testing and near-extremizers of noncommutative uniformity norms},
  author = {Zongbo and Bao and Jop Briët and Davi Castro-Silva and Philippe van Dordrecht and Jonas Helsen},
  journal= {arXiv preprint arXiv:2605.26983},
  year   = {2026}
}

Comments

17 pages, 99% inverse theorem for Gowers norm for unitaries. Comments welcome!