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

Alternative names: \(((7,2,3))\) icosahedral code, \(((7,2,3))\) Kubischta-Teixeira 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 \(2I\). See Ref. [4] for other non-PI codes realizing \(2I\) gates transversally.

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]. See Ref. [4] for other non-PI codes realizing \(2I\) gates transversally.

Cousins

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
PI qubit codePI Cyclic quantum QECC Quantum
Combinatorial PI code
Parents
Combinatorial PI code
The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial PI code [6; Sec. 5.2].
Twisted \(1\)-group codePI Cyclic quantum Small-distance block quantum QECC Quantum
The \(((7,2,3))\) Pollatsek-Ruskai code admits a transversal representation of the twisted \(1\)-group \(2I\) [7].
Clifford spin codeQubit PI Cyclic quantum Monolithic quantum QECC Quantum
The \(((7,2,3))\) Pollatsek-Ruskai code can be interpreted as a spin-\(7/2\) Clifford code [3].
\(((7,2,3))\) Pollatsek-Ruskai 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, “Family of Quantum Codes with Exotic Transversal Gates”, Physical Review Letters 131, (2023) arXiv:2305.07023 DOI
[4]
C. Zhang, Z. Wu, S. Huang, and B. Zeng, “Transversal Gates in Nonadditive Quantum Codes”, (2025) arXiv:2504.20847
[5]
M. Du, C. Zhang, Y.-T. Poon, and B. Zeng, “Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions”, (2024) arXiv:2410.07983
[6]
A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI
[7]
E. Kubischta and I. Teixeira, “Quantum Codes from Twisted Unitary t -Groups”, Physical Review Letters 133, (2024) arXiv:2402.01638 DOI
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Zoo Code ID: icosahedral_permutation_invariant

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
BibTeX:
@incollection{eczoo_icosahedral_permutation_invariant, title={\(((7,2,3))\) Pollatsek-Ruskai 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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Permanent link:
https://errorcorrectionzoo.org/c/icosahedral_permutation_invariant

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/small_distance/small/7/icosahedral_permutation_invariant.yml.