Affine Equivalence in the Clifford Hierarchy

• URL: http://arxiv.org/abs/2507.14370v2
• Date: Sun, 03 Aug 2025 21:10:47 GMT
• Title: Affine Equivalence in the Clifford Hierarchy
• Authors: Jonas T. Anderson, Andrew Connelly,
• Abstract summary: We use cryptography literature on affine equivalence classes of 4-bit permutations to find all 4-qubit permutations in the Clifford Hierarchy.<n>We then use the classification of 4-qubit permutations and previous results on the structure of diagonal gates to prove that all 4-qubit gates in the third level of the Clifford Hierarchy are semi-Clifford.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper we prove a collection of results on the structure of permutations in the Clifford Hierarchy. First, we leverage results from the cryptography literature on affine equivalence classes of 4-bit permutations which we use to find all 4-qubit permutations in the Clifford Hierarchy. We then use the classification of 4-qubit permutations and previous results on the structure of diagonal gates in the Clifford Hierarchy to prove that all 4-qubit gates in the third level of the Clifford Hierarchy are semi-Clifford. Finally, we introduce the formalism of cycle structures to permutations in the Clifford Hierarchy and prove a general structure theorem about them. We also classify many small cycle structures up to affine equivalence. Interestingly, this classification is independent of the number of qubits.

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