Quasi-hyperbolic color code[1]
Description
An extension of the color code construction to quasi-hyperbolic manifolds, e.g., a product of a 2D hyperbolic surface and a circle.
Protection
There exists a family with rate of order \(O(1/\log n)\) and minimum distance of order \(\Omega(\log n)\) which supports fault-tolerant non-Clifford gates [1]. A construction based on the Torelli mapping yields a code with constant rate with similar gates [1].
Rate
A construction based on the Torelli mapping yields a code with constant rate with similar gates [1].
Fault Tolerance
There exists a family with rate of order \(O(1/\log n)\) and minimum distance of order \(\Omega(\log n)\) which supports fault-tolerant non-Clifford gates [1]. A construction based on the Torelli mapping yields a code with constant rate with similar gates [1].
Parent
Cousins
- Homological code — Quasi-hyperbolic color codes are related to quasi-hyperbolic surface codes via a constant-depth Clifford circuit [1].
- Quantum rainbow code — Hypergraph products of color codes yield quantum rainbow codes with growing distance and transversal gates in the Clifford hierarchy. In particular, utilizing this construction for quasi-hyperbolic color codes yields an \([[n,O(n),O(\log n)]]\) triorthogonal code family with magic-state yield parameter \(\gamma\to 0\) [2].
References
- [1]
- G. Zhu et al., “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2023) arXiv:2310.16982
- [2]
- T. R. Scruby, A. Pesah, and M. Webster, “Quantum Rainbow Codes”, (2024) arXiv:2408.13130
Page edit log
- Guanyu Zhu (2024-08-27) — most recent
- Victor V. Albert (2024-08-27)
Cite as:
“Quasi-hyperbolic color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quasi_hyperbolic_color