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PreprintCC BY 4.0arXiv

When is the Clifford hierarchy generalized semi-Clifford?

  • Rongbiao Thomas Wang
  • Bobby Zixuan Zhang
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Abstract

We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level k and qubit number n. We show that 𝒞k(n) is semi-Clifford if and only if k≤2, n≤2, or k=3 and n≤6. We also establish that 𝒞k(n) is generalized semi-Clifford when k≤4 or n≤3 and construct counterexamples when k,n ≥ 5. For n=4 and k≥5, we show that every gate defined over ℚ(i) up to a scalar is generalized semi-Clifford, reduce the general case to a single level, and propose a strategy to prove our conjecture that 𝒞k(n) is generalized semi-Clifford if and only if k≤4 or n≤4. Lastly, we go beyond this structure by introducing nc-generalized semi-Clifford gates, which we show are exactly the gates of the form ∑x|π(x)⟩⟨ x|⊗ Vx up to Clifford gates on both sides, where π permutes the computational basis states of nc control qubits. We conjecture that for k≥3 every gate in 𝒞k(n) has this form for some nc≥1 with every Vx∈𝒞k−3(n−nc), refining the conjecture of Zeng, Chen, and Chuang.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026when,
  title = {{When is the Clifford hierarchy generalized semi-Clifford?}},
  author = {Rongbiao Thomas Wang and Bobby Zixuan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06402},
  eprint = {2610.06402},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06402},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - When is the Clifford hierarchy generalized semi-Clifford?
AU  - Rongbiao Thomas Wang
AU  - Bobby Zixuan Zhang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06402
UR  - https://arxiv.org/abs/2610.06402
AB  - We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level k and qubit number n. We show that 𝒞_k(n) is semi-Clifford if and only if k≤2, n≤2, or k=3 and n≤6. We also establish that 𝒞_k(n) is generalized semi-Clifford when k≤4 or n≤3 and construct counterexamples when k,n ≥ 5. For n=4 and k≥5, we show that every gate defined over ℚ(i) up to a scalar is generalized semi-Clifford, reduce the general case to a single level, and propose a strategy to prove our conjecture that 𝒞_k(n) is generalized semi-Clifford if and only if k≤4 or n≤4. Lastly, we go beyond this structure by introducing n_c-generalized semi-Clifford gates, which we show are exactly the gates of the form ∑_x|π(x)⟩⟨ x|⊗ V_x up to Clifford gates on both sides, where π permutes the computational basis states of n_c control qubits. We conjecture that for k≥3 every gate in 𝒞_k(n) has this form for some n_c≥1 with every V_x∈𝒞_k−3(n−n_c), refining the conjecture of Zeng, Chen, and Chuang.
ER  -