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DOI: 10.4230/LIPIcs.TQC.2026.2

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

Authors: Zhiyang He, Luke Robitaille, and Xinyu Tan

Published in: LIPIcs, Volume 389, 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)

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)

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@InProceedings{he_et_al:LIPIcs.TQC.2026.2,
  author =	{He, Zhiyang and Robitaille, Luke and Tan, Xinyu},
  title =	{{Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy}},
  booktitle =	{21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
  pages =	{2:1--2:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-439-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{389},
  editor =	{Arnon, Rotem and Harrow, Aram W.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.2},
  URN =		{urn:nbn:de:0030-drops-272997},
  doi =		{10.4230/LIPIcs.TQC.2026.2},
  annote =	{Keywords: Quantum fault-tolerance, Clifford hierarchy, permutation gates}
}

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DOI: 10.4230/LIPIcs.ESA.2021.52

Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs

Authors: Zhiyang He, Jason Li, and Magnus Wahlström

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract

Let G be a graph and S, T ⊆ V(G) be (possibly overlapping) sets of terminals, |S| = |T| = k. We are interested in computing a vertex sparsifier for terminal cuts in G, i.e., a graph H on a smallest possible number of vertices, where S ∪ T ⊆ V(H) and such that for every A ⊆ S and B ⊆ T the size of a minimum (A,B)-vertex cut is the same in G as in H. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlström (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier H with O(k³) vertices can be computed in randomized polynomial time, even for arbitrary digraphs G. However, since then, no improvements on the size O(k³) have been shown. In this paper, we draw inspiration from the renowned Bollobás’s Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlström’s methods. This new perspective allows us to construct a sparsifier H of Θ(k²) vertices for the case that G is a DAG. We also show how to compute H in time near-linear in the size of G, improving on the previous O(n^{ω+1}). Furthermore, H recovers the closest min-cut in G for every partition (A,B), which was not previously known. Finally, we show that a sparsifier of size Ω(k²) is required, both for DAGs and for undirected edge cuts.

Cite as

Zhiyang He, Jason Li, and Magnus Wahlström. Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{he_et_al:LIPIcs.ESA.2021.52,
  author =	{He, Zhiyang and Li, Jason and Wahlstr\"{o}m, Magnus},
  title =	{{Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.52},
  URN =		{urn:nbn:de:0030-drops-146331},
  doi =		{10.4230/LIPIcs.ESA.2021.52},
  annote =	{Keywords: graph theory, vertex sparsifier, representative family, matroid}
}

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Documents authored by He, Zhiyang

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

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Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs

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