Group-based quantum repetition code[1]
Description
An \([[n,1]]_G\) generalization of the quantum repetition code.
The code encodes one group-valued qudit into \(n\). There are two variants, a bit- and a phase-flip code, whose codewords for any \(g\in G\) and for \(n=3\) are \begin{align} |\overline{g}_{\text{bit}}\rangle&\rightarrow|g,g,g\rangle\tag*{(1)}\\ |\overline{g}_{\text{phase}}\rangle&\rightarrow\frac{1}{|G|}\sum_{h_{1},h_{2},h_{3}\in G}\delta^{G}_{g,h_{1}h_{2}h_{3}}|h_{1},h_{2},h_{3}\rangle~, \tag*{(2)}\end{align} where \(\delta^{G}_{g,h}\) is the group Kronecker-delta function.
Parent
- Group-based QPC — A \([[m_1 m_2,1,\min(m_1,m_2)]]_G\) group-based QPC reduces to a group-based quantum repetition code when \(m_1\) or \(m_2\) is one.
Child
- Quantum repetition code — Group-based quantum repetition codes reduce to quantum repetition codes for \(G = \mathbb{Z}_2\).
Cousin
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code is a concatenation of a bit-flip with a phase-flip group repetition code for \(G=\mathbb{Z}_q\).
References
- [1]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
Page edit log
- Victor V. Albert (2024-04-04) — most recent
Cite as:
“Group-based quantum repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_quantum_repetition