Description
BW lattice in dimension \(4\), which is the lattice corresponding to the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via Construction A.
Protection
The \(D_4\) lattice has a density of \(\pi^2/16\approx 0.6169\) and nominal coding gain of \(\sqrt{2}\). It exhibits the densest lattice packing in four dimensions [1].
Parents
Cousins
- Repetition code — The four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via Construction A.
- Construction-\(A\) code — The \(D_4\) lattice is constructed out of the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via the Construction A.
- \(D_4\) lattice-shell code
- \(D_4\) hyper-diamond GKP code
- \((1,3)\) 4D toric code — The \((1,3)\) 4D toric code on a hyper-diamond lattice admits a transversal logical \(CCCZ\) gate [2].
References
- [1]
- A. Korkine and G. Zolotareff, “Sur les formes quadratiques positives quaternaires”, Mathematische Annalen 5, 581 (1872) DOI
- [2]
- T. Jochym-O’Connor and T. J. Yoder, “Four-dimensional toric code with non-Clifford transversal gates”, Physical Review Research 3, (2021) arXiv:2010.02238 DOI
Page edit log
- Victor V. Albert (2022-11-08) — most recent
Cite as:
“\(D_4\) hyper-diamond lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dfour