Group-based QPC[1]
Description
An \([[m r,1,\min(m,r)]]_G\) generalization of the QPC.
Logical codewords for each group element \(g\) are \begin{align} |\overline{g}\rangle=\left({\textstyle \frac{1}{\sqrt{|G|^{m-1}}}}\sum_{h_{1},h_{2},\cdots,h_{m}\in G}\delta_{g,h_{1}h_{2}\cdots h_{m}}|h_{1},h_{2},\cdots,h_{m}\rangle\right)^{\otimes r}~. \tag*{(1)}\end{align} where \(\delta^{G}_{g,h}\) is the group Kronecker-delta function. For non-compact groups, the sum becomes an integral, and ideal codewords are no longer normalizable.
Cousin
- \([[4,2,2]]_{G}\) four group-qudit code— The \(|\overline{g_1=1,g_2}\rangle\) \([[4,1,2]]_{G}\) subcode is the smallest group-based QPC, i.e., a concatenation of a bit-flip with a phase-flip group-based repetition code for that group.
Member of code lists
Primary Hierarchy
Parents
A group-based QPC is a concatenation of a phase-flip group-based repetition code with a bit-flip group-based repetition code.
Group-based QPC
Children
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.
The \([[9,1,3]]_{G}\) group-based QPC reduces to the \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code for \(G=\mathbb{R}\).
A \([[m_1 m_2,1,\min(m_1,m_2)]]_G\) group-based QPC reduces to a QPC for \(G=\mathbb{Z}_2\).
The \([[9,1,3]]_{G}\) group-based QPC reduces to the \([[9,1,3]]_{\mathbb{Z}}\) modular-qudit code for \(G=\mathbb{Z}_q\).
References
- [1]
- 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
Page edit log
- Victor V. Albert (2024-04-05) — most recent
Cite as:
“Group-based QPC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_quantum_parity