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, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
[2]
T. R. Scruby, A. Pesah, and M. Webster, “Quantum Rainbow Codes”, (2024) arXiv:2408.13130
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Zoo Code ID: quasi_hyperbolic_color

Cite as:
“Quasi-hyperbolic color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quasi_hyperbolic_color
BibTeX:
@incollection{eczoo_quasi_hyperbolic_color, title={Quasi-hyperbolic color code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quasi_hyperbolic_color} }
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Cite as:

“Quasi-hyperbolic color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quasi_hyperbolic_color

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