\(SU(3)\) Tverberg spin code[1][2; Ex. 6.4]
Description
\(SU(3)\) single-spin code family obtained from the two-step Tverberg construction [3] in the totally symmetric \(N\)-particle irrep \(\mathcal{H}=\mathrm{Sym}^N(\mathbb{C}^3)\) of \(\mathfrak{su}(3)\), which has dimension \(\binom{N+2}{2}\). Its weight basis is indexed by the discrete simplex \(\Delta_{3,N}\), whose centered form is the triangular \(A_2\) lattice.
A distance-two intermediate subspace can be chosen as \begin{align} \mathcal{B}=\mathrm{span}\{|a_1a_2a_3\rangle: a_1-a_2\equiv0\pmod 3\}~, \tag*{(1)}\end{align} giving \(\dim\mathcal{B}\approx\binom{N+2}{2}/3\) [1][2; Ex. 6.4]. A symmetric partition of this sublattice into pairs in the central hexagon and triples near the corners yields an error-detecting code for single Lie-algebra errors of dimension \(\frac{4}{27}\binom{N+2}{2}+O(N)\) [4; Sec. 6.2.1].
Protection
Detects errors in the \(\mathfrak{su}(3)\) Lie-algebra error set. More generally, for the Lie-type graph metric \(V_t=\mathrm{span}(\mathfrak{su}(3)\oplus \mathbb{C}I)^t\), a distance-\(d\) construction can be obtained by first choosing a graph-distance-\(d\) subset of the discrete simplex \(\Delta_{3,N}\) [4; Secs. 5.1,6.2].Rate
For general distance \(d\), the two-step construction uses an asymptotically optimal distance-\(d\) sublattice of the centered \(\Delta_{3,N}\) discrete simplex for the intermediate space. If \(d=2t\), then \(\dim\mathcal{B}=\dim\mathcal{H}/(3t^2)+O(N)\); if \(d=2t+1\), then \(\dim\mathcal{B}=\dim\mathcal{H}/(3t^2+3t+1)+O(N)\) [4; Sec. 6.2.2].Member of code lists
Primary Hierarchy
References
- [1]
- C. Bumgardner, “Codes in W\ast-metric Spaces: Theory and Examples”, (2012) arXiv:1205.4517
- [2]
- I. Shors, “Quantum Error Detection and Lie Theory”, undergraduate thesis, UC Davis Mathematics REU, 2022, URL
- [3]
- E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, (1999) arXiv:quant-ph/9908066
- [4]
- Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023
Page edit log
- Victor V. Albert (2026-05-26) — most recent
Cite as:
“\(SU(3)\) Tverberg spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/su3_tverberg_spin