Prime-qudit triorthogonal code[1]
Description
An \(m \times n\) matrix over \(\mathbb{F}_p=\mathbb{Z}_p\) 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 \(\mathbb{F}_p\). The triorthogonal prime-qudit CSS code associated with the matrix is constructed by mapping nonzero entries in self-orthogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [1,2].Transversal and Permutation-Based Gates
Admits a transversal gate from the third level of the qudit Clifford hierarchy [1].Cousin
- Prime-qudit RS code— Triorthogonal \(p\)-dimensional prime-qudit RS codes achieve a magic-state yield parameter \(\gamma = O(1/\log p)\) [1].
Primary Hierarchy
Parents
Prime-qudit triorthogonal code
Children
Prime-qudit triorthogonal codes reduce to triorthogonal codes when \(p=2\).
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”, Quantum 9, 1768 (2025) arXiv:2403.06228 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-06-07)
Cite as:
“Prime-qudit triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qudit_triorthogonal, arXiv:2606.11484