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The Weyl-orbit embedding and a representation-theoretic characterization of the Clifford hierarchy

  • Asgar Jamneshan
  • Berke Ünal
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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↦(a↦ WaUWa*), 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.

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Cite

BibTeXRIS
BibTeX
@misc{jamneshan2026weylorbit,
  title = {{The Weyl-orbit embedding and a representation-theoretic characterization of the Clifford hierarchy}},
  author = {Asgar Jamneshan and Berke Ünal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.16862},
  eprint = {2606.16862},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.16862},
}
RIS
TY  - UNPB
TI  - The Weyl-orbit embedding and a representation-theoretic characterization of the Clifford hierarchy
AU  - Asgar Jamneshan
AU  - Berke Ünal
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2606.16862
UR  - https://arxiv.org/abs/2606.16862
AB  - 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↦(a↦ 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.
ER  -