Related papers: Characterization of permutation gates in the third level of the Clifford hierarchy

Characterization of permutation gates in the third level of the Clifford hierarchy

URL: http://arxiv.org/abs/2510.04993v1
Date: Mon, 06 Oct 2025 16:28:35 GMT
Title: Characterization of permutation gates in the third level of the Clifford hierarchy
Authors: Zhiyang He, Luke Robitaille, Xinyu Tan,
Abstract summary: We characterize permutation gates in the third level of the Clifford hierarchy.<n>As a corollary, we construct a family of non-semi-Clifford permutation gates $U_k_kgeq 3$ in staircase form.
Score: 1.5639019650876584
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: 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.

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