Ball color code[1]
Description
A color code defined on a \(D\)-dimensional colex. This family includes hypercube color codes (color codes defined on balls constructed from hyperoctahedra) and 3D ball color codes (color 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.
Parents
Children
- \([[2^D,D,2]]\) hypercube quantum code — \([[2^D,D,2]]\) hypercube quantum codes can be thought of as small color codes defined on balls 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].
Cousin
- Polytope code — Polytopes dual to the hyperoctahedron, truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron are used to construct 3D ball color codes.
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.
Page edit log
- Victor V. Albert (2023-11-28) — most recent
Cite as:
“Ball color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ball_color