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\([[14,3,3]]\) Rhombic dodecahedron surface code[1]

Alternative names: Landahl jaunty code.

Description

A \([[14,3,3]]\) twist-defect surface code whose qubits lie on the vertices of a rhombic dodecahedron. Its non-CSS nature is due to twist defects [2] stemming from the geometry of the polytope. A local-Clifford-equivalent clean realization has only \(X\)- and \(Z\)-type operators on its four-valent vertices, and its symplectic double is a \([[28,6,3]]\) genus-three code [3].

A stabilizer tableau for the code is [1; Eq. (1)] \begin{align} \begin{smallmatrix} X & X & X & I & I & X & I & I & I & I & I & I & I & I \\ I & I & X & X & I & I & X & X & I & I & I & I & I & I \\ I & I & I & I & I & I & I & X & X & I & I & X & X & I \\ X & I & I & I & I & I & I & I & I & X & I & I & X & X \\ I & Y & Y & Y & Y & I & I & I & I & I & I & I & I & I \\ I & I & Y & I & I & Y & Y & I & I & I & Y & I & I & I \\ I & I & I & I & Y & I & I & I & Y & Y & I & I & Y & I \\ I & I & I & I & I & I & I & I & I & I & Y & Y & Y & Y \\ Z & Z & I & I & Z & I & I & I & I & Z & I & I & I & I \\ I & I & I & Z & Z & I & I & Z & Z & I & I & I & I & I \\ I & I & I & I & I & I & Z & Z & I & I & Z & Z & I & I \end{smallmatrix}~. \tag*{(1)}\end{align}

Cousin

  • Rhombic dodecahedron code— The qubits of the \([[14,3,3]]\) rhombic dodecahedron surface code lie on the vertices of the small rhombic dodecahedron.

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
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Twist-defect surface codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Parents
Twist-defect surface code
The rhombic dodecahedron surface code is a twist-defect surface code whose degree-three vertices can be interpreted as disclination twists [1].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[14,3,3]]\) Rhombic dodecahedron surface code

References

[1]
A. J. Landahl, “The surface code on the rhombic dodecahedron”, (2020) arXiv:2010.06628
[2]
H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
[3]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
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Zoo Code ID: rhombic_dodecahedron_surface

Cite as:
“\([[14,3,3]]\) Rhombic dodecahedron surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/rhombic_dodecahedron_surface
BibTeX:
@incollection{eczoo_rhombic_dodecahedron_surface, title={\([[14,3,3]]\) Rhombic dodecahedron surface code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rhombic_dodecahedron_surface} }
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Permanent link:
https://errorcorrectionzoo.org/c/rhombic_dodecahedron_surface

Cite as:

“\([[14,3,3]]\) Rhombic dodecahedron surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/rhombic_dodecahedron_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/14/rhombic_dodecahedron_surface.yml.