Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 29 min ago
PreprintCC BY-NC-ND 4.0arXiv

The generalised semi-Clifford conjecture is false

  • Nadish de Silva
  • Oscar Lautsch
Open at sourcePDFDOICite

Abstract

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form C1 ΠD C2 for Clifford gates C1, C2, a permutation gate Π, and a diagonal gate D; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.

Related

  1. Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows
    G. R. Barrenechea, E. Castillo, R. Codina +2
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  2. Generalised Cone Conjecture, III: Non-generic Sakai surfaces
    Vladimir Lazić, Isabel Stenger, Zhixin Xie
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Semi-differentiability of generalised L∞ envelopes with applications to supremal functionals
    Nikos Katzourakis
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsOptimisation & controlarXiv
  4. The weighted union-set conjecture is false
    Yimeng Wang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. The List Total Colouring Conjecture is False
    Jonathan A. Noel
    arXivPreprintCC0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{silva2026generalised,
  title = {{The generalised semi-Clifford conjecture is false}},
  author = {Nadish de Silva and Oscar Lautsch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11903},
  eprint = {2609.11903},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11903},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The generalised semi-Clifford conjecture is false
AU  - Nadish de Silva
AU  - Oscar Lautsch
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11903
UR  - https://arxiv.org/abs/2609.11903
AB  - The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form C_1 ΠD C_2 for Clifford gates C_1, C_2, a permutation gate Π, and a diagonal gate D; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.
ER  -