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\([[16,4,3]]\) dodecahedral code[1]

Description

A \([[16,4,3]]\) non-CSS qubit stabilizer code whose encoder-respecting form is the graph of vertices of a dodecahedron [1].

A stabilizer tableau for the code, obtained from the graph-to-tableau map applied to the dodecahedral graph, is [1; Fig. 19 and Eq. (10)] \begin{align} \begin{smallmatrix} I & Z & X & Z & I & I & I & I & Z & I & I & I & I & I & I & I \\ I & I & Z & X & X & Z & I & I & I & Z & I & Z & I & I & I & I \\ I & I & I & I & Z & X & X & Z & I & I & I & I & Z & I & Z & I \\ Z & I & I & I & I & I & Z & X & I & I & I & I & I & I & I & Z \\ X & Z & Z & I & I & I & I & Z & X & I & Z & I & I & I & I & I \\ Z & X & Z & Z & I & I & I & I & I & X & I & Z & I & I & I & I \\ I & I & Z & X & I & I & I & I & Z & Z & X & I & Z & I & I & I \\ I & I & I & I & Z & I & I & I & I & Z & I & X & I & Z & I & I \\ I & I & I & I & I & Z & I & I & I & I & Z & I & X & I & Z & I \\ I & I & I & I & Z & X & I & I & I & I & I & Z & Z & X & I & Z \\ X & Z & I & I & I & I & Z & Z & I & I & I & I & Z & I & X & I \\ Z & X & Z & I & I & I & I & Z & I & I & I & I & I & Z & I & X \end{smallmatrix}~. \tag*{(1)}\end{align}

Protection

The code saturates the paper’s graph degree upper bound on distance, attaining distance 3 on a degree-3 graph [1].

Rate

The code has a rate of \(1/4\), higher than that of the five-qubit perfect code.

Encoding

Encoding circuits of depth 5 are possible and optimal; a direct graph-based construction gives depth 6 before further optimization [1].

Cousin

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Small-distance qubit stabilizer codeSmall-distance block quantum QECC Quantum
Parents
Small-distance qubit stabilizer code
\([[16,4,3]]\) dodecahedral code

References

[1]
A. B. Khesin, J. Z. Lu, and P. W. Shor, “Universal Graph Representation of Stabilizer Codes”, PRX Quantum 6, (2025) arXiv:2411.14448 DOI
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Zoo Code ID: quantum_dodecahedron

Cite as:
“\([[16,4,3]]\) dodecahedral code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_dodecahedron
BibTeX:
@incollection{eczoo_quantum_dodecahedron, title={\([[16,4,3]]\) dodecahedral code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_dodecahedron} }
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Cite as:

“\([[16,4,3]]\) dodecahedral code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_dodecahedron

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/16/quantum_dodecahedron.yml.