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Ball code[1]

Description

A distance-two “morphed” color code defined on a \(D\)-dimensional colex [1; Appx. A]. This family includes hypercube codes (defined on balls constructed from hyperoctahedra) and 3D ball codes (defined on duals of certain Archimedean solids).

Protection

The hypercube code family has parameters \([[2^D,D,2]]\) [1; Exam. 3]. 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have parameters \([[24,11,2]]\), \([[48,23,2]]\), and \([[120,59,2]]\), respectively [1; Exam. 4].

Magic

The 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have \(\gamma\) close to one [1].

Transversal Gates

The 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have transveral \(CCZ\) gates.

Cousin

  • Polyhedron code— Polytopes dual to the hyperoctahedron, truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron are used to construct 3D ball codes.

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
Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Color code
Ball codes are color codes defined on a \(D\)-dimensional colex [1; Appx. A].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
Ball code
Children
\([[2^D,D,2]]\) hypercube quantum code
\([[2^D,D,2]]\) hypercube quantum codes can be thought of as small ball codes constructed from hyperoctahedra [1; Exam. 3], or on lattices with no bulk qubits and cubic boundaries [2,3].
\([[2m,2m-2,2]]\) error-detecting code
The \([[2m,2m-2,2]]\) error-detecting code is a ball color code [1; Sec. III.A].

References

[1]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[2]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[3]
E. Campbell, “The smallest interesting colour code,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09.
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Zoo Code ID: ball_color

Cite as:
“Ball code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ball_color
BibTeX:
@incollection{eczoo_ball_color, title={Ball code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ball_color} }
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Permanent link:
https://errorcorrectionzoo.org/c/ball_color

Cite as:

“Ball code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ball_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/ball_color.yml.