The generalised semi-Clifford conjecture is false

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 $C_1 \Pi D C_2$ for Clifford gates $C_1, C_2$, a permutation gate $\Pi$, 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.

Preliminary draft. Improved exposition forthcoming

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

arXiv:2609.11903 [quant-ph]

(or arXiv:2609.11903v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.11903 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.