Affine Equivalence in the Clifford Hierarchy
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911089803919360 |
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| author | Anderson, Jonas T. Connelly, Andrew |
| author_facet | Anderson, Jonas T. Connelly, Andrew |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14370 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine Equivalence in the Clifford Hierarchy Anderson, Jonas T. Connelly, Andrew Quantum Physics 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. |
| title | Affine Equivalence in the Clifford Hierarchy |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2507.14370 |