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.

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  • \([[4,2,2]]_{G}\) 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 two-qubit bit-flip with a two-qubit phase-flip group-based repetition code.

References

[1]
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
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Zoo Code ID: group_quantum_parity

Cite as:
“Group-based QPC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_quantum_parity
BibTeX:
@incollection{eczoo_group_quantum_parity, title={Group-based QPC}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_quantum_parity} }
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Cite as:

“Group-based QPC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_quantum_parity

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/group_gkp/group_quantum_parity.yml.