\(((7,2,3))\) Pollatsek-Ruskai code

\(((7,2,3))\) Pollatsek-Ruskai code[1–3]

Also known as \(((7,2,3))\) icosahedral code.

Description

Seven-qubit PI code that realizes 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

Combinatorial PI code — The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial PI code [4; Sec. 5.2].
Twisted \(1\)-group code — The \(((7,2,3))\) Pollatsek-Ruskai code admits a transversal representation of the twisted \(1\)-group \(2I\) [5].
Clifford spin code — The \(((7,2,3))\) Pollatsek-Ruskai code can be interpreted as a spin-\(7/2\) Clifford code [2].

Cousin

\([[7,1,3]]\) Steane code — The Pollatsek-Ruskai code can be continuously deformed to the Steane code [6].

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, “Family of Quantum Codes with Exotic Transversal Gates”, Physical Review Letters 131, (2023) arXiv:2305.07023 DOI
[4]: A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI
[5]: E. Kubischta and I. Teixeira, “Quantum Codes from Twisted Unitary t -Groups”, Physical Review Letters 133, (2024) arXiv:2402.01638 DOI
[6]: M. Du et al., “Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions”, (2024) arXiv:2410.07983

Page edit log

Victor V. Albert (2023-05-12) — most recent

Cite as:

“\(((7,2,3))\) Pollatsek-Ruskai 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/qubits/permutation_invariant/icosahedral_permutation_invariant.yml.