The Generalized Semi-Clifford Conjecture Holds at Level 4
Abstract
The Clifford hierarchy C1 ⊂ C2 ⊂ ⋯ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about Ck for k > 3. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension d or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in ∪k Ck is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case d=2, k=3 (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) U ∈ C3 on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group Γ1(U) generated by U P U† and beyond, we prove the conjecture for k ≤ 4 and any prime dimension d. Our proof centers on conjugation groups Γ1(U), Γ2(U), … of U ∈ Ck, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.
Related
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BibTeX
@misc{marcus2026generalized,
title = {{The Generalized Semi-Clifford Conjecture Holds at Level 4}},
author = {Maxwell Marcus and Sathyawageeswar Subramanian and Marcel Dall'Agnol},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38751},
eprint = {2609.38751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38751},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Generalized Semi-Clifford Conjecture Holds at Level 4
AU - Maxwell Marcus
AU - Sathyawageeswar Subramanian
AU - Marcel Dall'Agnol
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38751
UR - https://arxiv.org/abs/2609.38751
AB - The Clifford hierarchy C_1 ⊂ C_2 ⊂ ⋯ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about C_k for k > 3. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension d or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in ∪_k C_k is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case d=2, k=3 (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) U ∈ C_3 on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group Γ_1(U) generated by U P U^† and beyond, we prove the conjecture for k ≤ 4 and any prime dimension d. Our proof centers on conjugation groups Γ_1(U), Γ_2(U), … of U ∈ C_k, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.
ER -