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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.04993 |
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| _version_ | 1866914077852303360 |
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| author | He, Zhiyang Robitaille, Luke Tan, Xinyu |
| author_facet | He, Zhiyang Robitaille, Luke Tan, Xinyu |
| contents | The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates -- unitaries which permute the $2^n$ basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates $\{U_k\}_{k\geq 3}$ in staircase form such that each $U_k$ is in the third level but its inverse is not in the $k$-th level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04993 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of permutation gates in the third level of the Clifford hierarchy He, Zhiyang Robitaille, Luke Tan, Xinyu Quantum Physics The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates -- unitaries which permute the $2^n$ basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates $\{U_k\}_{k\geq 3}$ in staircase form such that each $U_k$ is in the third level but its inverse is not in the $k$-th level. |
| title | Characterization of permutation gates in the third level of the Clifford hierarchy |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2510.04993 |