2D subsystem color code

2D subsystem color code[1]

Also known as 2D gauge color code.

Description

A subsystem version of the 2D color code.

Protection

One family of subsystem codes has parameters \([[3m,2g,2m+2g-2,d]]\), where \(m\) is the number of vertices of the original embedded two-colex, where \(g\) is the genus of the surface embedding the two-colex, and where the distance is bounded from below by the length of the smallest nontrivial homological cycle of the two-colex \(\Gamma\) [2; Construction B][3; Lemma 2].

Gates

Braiding twist defects [4].

Code Capacity Threshold

The threshold under ML decoding under depolarizing noise corresponds to the value of a critical point of a disordered spin model, calculated to be \(5.5(2)\%\) in Ref. [5].Erasure noise: \(50\%\) noise threshold error rate under erasure noise using optimal erasure decoder [6], and \(9.7\%\) and \(44\%\) using gauge-fixing decoders [7,8].

Parents

Child

Doubled color code

Cousin

2D color code

References

[1]: H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
[2]: P. Sarvepalli and K. R. Brown, “Topological subsystem codes from graphs and hypergraphs”, Physical Review A 86, (2012) arXiv:1207.0479 DOI
[3]: V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
[4]: H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
[5]: R. S. Andrist et al., “Optimal error correction in topological subsystem codes”, Physical Review A 85, (2012) arXiv:1204.1838 DOI
[6]: H. M. Solanki and P. K. Sarvepalli, “Decoding Topological Subsystem Color Codes Over the Erasure Channel Using Gauge Fixing”, IEEE Transactions on Communications 71, 4181 (2023) DOI
[7]: H. M. Solanki and P. K. Sarvepalli, “Decoding Topological Subsystem Color Codes Over the Erasure Channel using Gauge Fixing”, (2022) arXiv:2111.14594
[8]: H. M. Solanki and P. Kiran Sarvepalli, “Correcting Erasures with Topological Subsystem Color Codes”, 2020 IEEE Information Theory Workshop (ITW) 1 (2021) DOI

Page edit log

Victor V. Albert (2024-07-11) — most recent

Cite as:

“2D subsystem color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_subsystem_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/topological/color/2d_subsystem_color.yml.