Three-rotor code[1]
Description
\([[3,1,2]]_{\mathbb Z}\) rotor code that is an extension of the \([[3,1,2]]_3\) qutrit CSS code to the integer alphabet, i.e., the angular momentum states of a planar rotor.
The code is \(U(1)\)-covariant and its ideal codewords, \begin{align} |\overline{x}\rangle = \sum_{y\in\mathbb{Z}} \left| -3y,y-x,2(y+x) \right\rangle~, \tag*{(1)}\end{align} where \(x\in\mathbb{Z}\), are not normalizable.
Protection
Normalized codewords approximately protect against erasure while maintaining covariance [2].Primary Hierarchy
Parents
Taking \(H_X=\begin{pmatrix}-3 & 1 & 2\end{pmatrix}\) and \(H_Z=\begin{pmatrix}4&6&3\end{pmatrix}\) yields the three-rotor code.
Three-rotor codewords are CV AME [3].
The three-rotor code is \(U(1)\)-covariant.
Three-rotor code
References
- [1]
- P. Hayden, S. Nezami, S. Popescu, and G. Salton, “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021) arXiv:1709.04471 DOI
- [2]
- P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [3]
- J. I. Kwon, A. J. Brady, and V. V. Albert, “Most continuous-variable cluster states are too entangled to be useless”, (2025) arXiv:2503.15698
Page edit log
- Victor V. Albert (2022-07-27) — most recent
Cite as:
“Three-rotor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rotor_3_1_2