Description
An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [2]. Certain double covers of hyperbolic tilings also yield admissible tilings [1]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [4]; see also a construction based on the more general quantum pin codes [3].
Protection
The use of hyperbolic surfaces allows one to circumvent bounds on code parameters (such as the BPT bound) that are valid for lattice geometries. Hyperbolic color codes can have high rate but tend to have small distance. For example, a \(\{4g,4g\}\) tiling with periodic boundary conditions (i.e., a \(g\)-torus) yields a \([[4g+8,4g,4]]\) code family [2]. More examples, such as the \([[160,20,8]]\) code on the 4.10.10 tiling, are provided in [3; Sec. V.A].
Rate
In the double-cover construction [1], an \(\{\ell,m\}\) input tiling yields a code family with an asymptotic rate of \(1 - 1/\ell - 1/m\).
Parent
Child
- \([[8,2,2]]\) hyperbolic color code — The \([[8,2,2]]\) hyperbolic color code is defined on the projective plane.
Cousins
- Hyperbolic surface code
- Small-distance block quantum code — Many hyperbolic color codes have distance \(\leq 6\).
References
- [1]
- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1301.6588 DOI
- [2]
- E. B. da Silva and W. S. Soares Jr, “Hyperbolic quantum color codes”, (2018) arXiv:1804.06382
- [3]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [4]
- H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
Page edit log
- Victor V. Albert (2024-04-02) — most recent
Cite as:
“Hyperbolic color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hyperbolic_color