The Weyl-orbit embedding and a representation-theoretic characterization of the Clifford hierarchy
Year: 2026
Type: preprint
Abstract: Bu, Gu, and Jaffe introduced the quantum uniformity measures to analytically characterize the Clifford hierarchy. We prove that the quantum uniformity measures are the pullbacks of the matrix-valued uniformity functionals of Gowers and Hatami under the map $U\mapsto(a\mapsto W_aUW_a^*)$, which we call the Weyl-orbit embedding. We study the structure of the Weyl-orbit embedding, and among other results, establish a canonical decomposition of the ambient matrix-valued uniformity space into uniformity preserving twisted copies of the quantum uniformity space. The Weyl-orbit embedding identifies the Clifford hierarchy with Leibman polynomial maps. This identification yields a representation theoretic characterization of the Clifford hierarchy in terms of unitary representations of the universal polynomial groups introduced by the first author and Thom.
Source: arXiv (Cornell University)
Authors: Jamneshan, Asgar, Ünal, Berke
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W7164892791