The generalised semi-Clifford conjecture is false

• URL: http://arxiv.org/abs/2609.11903v1
• Date: Thu, 10 Sep 2026 17:55:10 GMT
• Title: The generalised semi-Clifford conjecture is false
• Abstract summary: The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation.<n>We show how our counterexample to the generalised semi-Clifford conjecture can be deduced.<n>Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form $C_1 ΠD C_2$ for Clifford gates $C_1, C_2$, a permutation gate $Π$, and a diagonal gate $D$; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.

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