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This research establishes a rigorous cohomological framework for the Clifford hierarchy, offering new structural insights into fault-tolerant quantum computation by linking quantum derivatives to non-abelian 1-cocycles. The work provides explicit characterizations for the third level of the hierarchy and delivers novel proofs regarding semi-Clifford properties for two- and three-qudit gates. These theoretical advances deepen our understanding of unitary operator structures essential for scalable quantum error correction.
A Cohomological Characterization of the Clifford Hierarchy
arXiv:2610.06699v1 Announce Type: cross Abstract: The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian \(1\)-cocycle, and conversely that every such \(1\)-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.
Abstract:The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian \(1\)-cocycle, and conversely that every such \(1\)-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.
50 pages, excluding references and appendices. Comments are welcome
Mathematical Physics (math-ph); Group Theory (math.GR); Quantum Physics (quant-ph)
arXiv:2610.06699 [math-ph]
(or arXiv:2610.06699v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2610.06699 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 262 words
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