2024/10/15 by Zhiyang He, He, Zhiyang, Luke Robitaille +3 · 3 citations
Computer Science · #Coding theory and cryptography #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Quantum-Dot Cellular Automata
paper · pdf · doi:10.48550/arxiv.2410.11818
openalex publication_date 2024/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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 2n 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.