On Groups in the Qubit Clifford Hierarchy

Jonas T. Anderson

doi:10.22331/q-2024-06-13-1370

On Groups in the Qubit Clifford Hierarchy

Northrop Grumman Corporation

Published:	2024-06-13, volume 8, page 1370
Eprint:	arXiv:2212.05398v2
Doi:	https://doi.org/10.22331/q-2024-06-13-1370
Citation:	Quantum 8, 1370 (2024).

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Abstract

Here we study the unitary groups that can be constructed using elements from the qubit Clifford Hierarchy. We first provide a necessary and sufficient canonical form that semi-Clifford and generalized semi-Clifford elements must satisfy to be in the Clifford Hierarchy. Then we classify the groups that can be formed from such elements. Up to Clifford conjugation, we classify all such groups that can be constructed using generalized semi-Clifford elements in the Clifford Hierarchy. We discuss a possible minor exception to this classification in the appendix. This may not be a full classification of all groups in the qubit Clifford Hierarchy as it is not currently known if all elements in the Clifford Hierarchy must be generalized semi-Clifford. In addition to the diagonal gate groups found by Cui et al., we show that many non-isomorphic (to the diagonal gate groups) generalized symmetric groups are also contained in the Clifford Hierarchy. Finally, as an application of this classification, we examine restrictions on transversal gates given by the structure of the groups enumerated herein which may be of independent interest.

► BibTeX data

@article{Anderson2024groupsinqubit,
  doi = {10.22331/q-2024-06-13-1370},
  url = {https://doi.org/10.22331/q-2024-06-13-1370},
  title = {On {G}roups in the {Q}ubit {C}lifford {H}ierarchy},
  author = {Anderson, Jonas T.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1370},
  month = jun,
  year = {2024}
}

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Cited by

[1] Yusei Mori, Hideaki Hakoshima, and Keisuke Fujii, "Nontrivial Multiproduct Commutation Relation Toward Reducing T -Count in Sequential Pauli-Based Computation", PRX Quantum 7 2, 020345 (2026).

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[3] Bhanu Pratap Yadav, Mahdi Bayanifar, and Olav Tirkkonen, 2025 IEEE Information Theory Workshop (ITW) 1 (2025) ISBN:979-8-3315-3142-3.

[4] Oliver Weißl and Evgenii Egorov, 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) 672 (2025) ISBN:979-8-3315-3159-1.

[5] Yijia Xu, Yixu Wang, Christophe Vuillot, and Victor V. Albert, "Letting the Tiger out of Its Cage: Bosonic Coding without Concatenation", Physical Review X 15 4, 041025 (2025).

[6] Dominik Hangleiter, Marcin Kalinowski, Dolev Bluvstein, Madelyn Cain, Nishad Maskara, Xun Gao, Aleksander Kubica, Mikhail D. Lukin, and Michael J. Gullans, "Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes", PRX Quantum 6 2, 020338 (2025).

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 22:43:16) and SAO/NASA ADS (last updated successfully 2026-08-10 22:43:19). The list may be incomplete as not all publishers provide suitable and complete citation data.

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