Prime-qudit triorthogonal code[1]
Description
An \(m \times n\) matrix over \(GF(p)=\mathbb{Z}_p\) is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\) modulo \(p\), where addition and multiplication are done on \(GF(p)\). The triorthogonal prime-qudit CSS code associated with the matrix is constructed by mapping non-zero entries in self-orhogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [1,2].
Transversal Gates
Admits a transversal gate from the third level of the qudit Clifford hierarchy [1].
Parent
Children
- Triorthogonal code — Prime-qudit triorthogonal codes reduce to triorthogonal codes when \(p=2\).
- \([[9m-k,k,2]]_3\) triorthogonal code
Cousin
- Prime-qudit RS code — Triorthogonal \(p\)-dimensional prime-qudit RS codes achieve a magic-state yield parameter \(\gamma = O(1/\log p)\) [1].
References
- [1]
- A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
- [2]
- S. Prakash and T. Saha, “Low Overhead Qutrit Magic State Distillation”, (2024) arXiv:2403.06228
Page edit log
- Victor V. Albert (2024-06-07) — most recent
Cite as:
“Prime-qudit triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qudit_triorthogonal