Permutation gates in the third level of the Clifford hierarchy

Abstract: The Clifford hierarchy is a fundamental structure in quantum computation, classifying unitary operators based on their commutation relations with the Pauli group. Despite its significance, the mathematical structure of the hierarchy is not well understood at the third level and higher. In this work, we study permutations in the hierarchy: gates which permute the $2^n$ basis states. We fully characterize all the semi-Clifford permutation gates. Moreover, we prove that any permutation gate in the third level, not necessarily semi-Clifford, must be a product of Toffoli gates in what we define as staircase form, up to left and right multiplications of Clifford permutations. Finally, we show that the smallest number of qubits for which there exists a non-semi-Clifford permutation in the third level is $7$.