The Generalized Semi-Clifford Conjecture Holds at Level 4

Abstract:The Clifford hierarchy $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots$ 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 $\mathsf{C}_k$ 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 $\cup_k \mathsf{C}_k$ 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) $U \in \mathsf{C}_3$ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group $\Gamma_1(U)$ generated by $U \mathsf{P} U^\dagger$ and beyond, we prove the conjecture for $k \leq 4$ and any prime dimension $d$. Our proof centers on conjugation groups $\Gamma_1(U), \Gamma_2(U), \ldots$ of $U \in \mathsf{C}_k$, 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.

arXiv:2609.38751 [quant-ph]

(or arXiv:2609.38751v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.38751 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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