Description
Seven-qubit code designed to realize gates from the binary icosahedral group transversally. Can also be interpreted as a spin-\(7/2\) 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
Parents
- Qubit code
- Clifford code — The \(((7,2,3))\) permutation-invariant code can be interpreted as a spin-\(7/2\) Clifford code [2].
- Small-distance block quantum code
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 Nonadditive Quantum Code with Exotic Transversal Gate Set”, (2023) arXiv:2305.07023
Page edit log
- Victor V. Albert (2023-05-12) — most recent
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