Clifford subgroup-orbit QSC[1] 

Description

A \(((2^r,2,2-\sqrt{2},8))\) QSC for \(r \geq 2\) constructed using the real-Clifford subgroup-orbit code. Logical constellations are constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [2] to \(2\) different vectors on the complex sphere. The code is known as the Witting code for \(r=2\) because its two logical constellations form vertices of Witting polytopes.

Parent

Cousins

  • Real-Clifford subgroup-orbit code — Clifford group-orbit QSCs are quantum analogues of real Clifford subgroup-orbit codes.
  • Witting polytope code — Logical constellations of the Clifford subgroup-orbit code for \(r=2\) form vertices of Witting polytopes.
  • 24-cell code — Logical constellations of the Clifford subgroup-orbit code for \(r=1\) form vertices of 24-cells when mapped into the real sphere, while code constellations form vertices of a disphenoidal 288-cell.
  • Disphenoidal 288-cell code — Logical constellations of the Clifford subgroup-orbit code for \(r=1\) form vertices of 24-cells when mapped into the real sphere, while code constellations form vertices of a disphenoidal 288-cell.

References

[1]
S. P. Jain et al., “Quantum spherical codes”, (2023) arXiv:2302.11593
[2]
G. Nebe, E. M. Rains, and N. J. A. Sloane, “The invariants of the Clifford groups”, (2000) arXiv:math/0001038
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Zoo Code ID: clifford_group

Cite as:
“Clifford subgroup-orbit QSC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/clifford_group
BibTeX:
@incollection{eczoo_clifford_group, title={Clifford subgroup-orbit QSC}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/clifford_group} }
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Cite as:

“Clifford subgroup-orbit QSC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/clifford_group

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/coherent_state/clifford_group.yml.