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Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy

Authors Zhiyang He , Luke Robitaille , Xinyu Tan

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ACM Subject Classification
  • Theory of computation → Quantum information theory
Keywords
  • Quantum fault-tolerance
  • Clifford hierarchy
  • permutation gates

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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 2ⁿ 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 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 ≥ 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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Zhiyang He, Luke Robitaille, and Xinyu Tan. Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.TQC.2026.2

Author Details

Zhiyang He
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Luke Robitaille
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Xinyu Tan
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • He, Zhiyang: National Science Foundation Graduate Research Fellowship under Grant No. 2141064.
  • Tan, Xinyu: U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704.

Acknowledgements

The work was conducted as a part of the 2024 Summer Program for Undergraduate Research (SPUR) and 2024-2025 Undergraduate Research Opportunities Program (UROP) at MIT. We thank Jonathan Bloom, Isaac Chuang, David Jerison, and Peter Shor for their mentorship. X. Tan would like to thank Robert Calderbank for introducing the problem of characterizing the third-level Clifford hierarchy to her in 2022 and many insightful discussions afterwards. We thank Jonas Anderson, Jeongwan Haah, Aram Harrow, Greg Kahanamoku-Meyer, Andrey Khesin and Anirudh Krishna for helpful discussions.

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