\(((7,2,3))\) permutation-invariant code[13] 

Also known as Pollatsek-Ruskai code.

Description

Seven-qubit code designed to realize gates from the binary icosahedral group transversally. Can also be interpreted as a spin-\(7/2\) single-spin code. The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\).

In terms of Dicke states, the unnormalized logical states of one version [3] of this code are \begin{align} \begin{split} |0_{L}\rangle&\propto15|D_{0}^{7}\rangle+3\sqrt{35}|D_{4}^{7}\rangle\\&\quad+\sqrt{105}\;|D_{2}^{7}\rangle-3\sqrt{35}|D_{6}^{7}\rangle\,,\\|1_{L}\rangle&\propto X^{\otimes7}|0_{L}\rangle\,. \end{split} \tag*{(1)}\end{align}

Transversal Gates

Binary icosahedral group \(2I\) gates can be realized transversally [3].

Parents

References

[1]
H. Pollatsek and M. B. Ruskai, “Permutationally Invariant Codes for Quantum Error Correction”, (2004) arXiv:quant-ph/0304153
[2]
J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021) arXiv:2005.10910 DOI
[3]
E. Kubischta and I. Teixeira, “A Family of Quantum Codes with Exotic Transversal Gates”, (2023) arXiv:2305.07023
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Zoo Code ID: icosahedral_permutation_invariant

Cite as:
\(((7,2,3))\) permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/icosahedral_permutation_invariant
BibTeX:
@incollection{eczoo_icosahedral_permutation_invariant, title={\(((7,2,3))\) permutation-invariant code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/icosahedral_permutation_invariant} }
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Cite as:

\(((7,2,3))\) permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/icosahedral_permutation_invariant

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/transversal/icosahedral_permutation_invariant.yml.