2D subsystem color code

2D subsystem color code[1]

Alternative Names: 2D gauge color code.

Description

A subsystem version of the 2D color code. The original topological subsystem-code example is defined on the Union Jack lattice [1]; the square-octagon-lattice hypergraph construction of [2] reproduces the same code from a complementary viewpoint.

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, \(g\) is the genus of the surface embedding the two-colex, and the distance is bounded from below by the length of the smallest nontrivial homological cycle of the two-colex \(\Gamma\) [3; Construction B][4; Lemma 2].

Gates

Braiding twist defects [5].

Decoding

For the Union-Jack/square-octagon member, decoding can be reduced to correcting \(Z\) errors using stabilizers of the topological color code [2; Sec. 4.1].

Code Capacity Threshold

The threshold under ML decoding for depolarizing noise corresponds to the value of a critical point of a disordered spin model, calculated to be \(5.5(2)\%\) in Ref. [6].Erasure noise: \(50\%\) threshold error rate using the optimal erasure decoder [7], and \(9.7\%\) and \(44\%\) using gauge-fixing decoders [8,9].

Cousin

2D color code— Gauge fixing relates 2D subsystem color codes to 2D color codes on the same lattice [1,10]; the original Union-Jack member is reproduced by the square-octagon-lattice construction of [2].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Subsystem qubit codeSubsystem QECC Quantum

Subsystem qubit stabilizer codeSubsystem QECC Quantum

Subsystem qubit CSS codeSubsystem QECC Quantum

Subsystem color codeSubsystem QECC Quantum

Parents

Subsystem color code

Lattice subsystem codeSubsystem QECC Quantum

2D subsystem color code

Children

Doubled color code

References

[1]: H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
[2]: M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 DOI
[3]: P. Sarvepalli and K. R. Brown, “Topological subsystem codes from graphs and hypergraphs”, Physical Review A 86, (2012) arXiv:1207.0479 DOI
[4]: V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
[5]: H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
[6]: R. S. Andrist, H. Bombin, H. G. Katzgraber, and M. A. Martin-Delgado, “Optimal error correction in topological subsystem codes”, Physical Review A 85, (2012) arXiv:1204.1838 DOI
[7]: 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
[8]: H. M. Solanki and P. K. Sarvepalli, “Decoding Topological Subsystem Color Codes Over the Erasure Channel using Gauge Fixing”, (2022) arXiv:2111.14594
[9]: H. M. Solanki and P. Kiran Sarvepalli, “Correcting Erasures with Topological Subsystem Color Codes”, 2020 IEEE Information Theory Workshop (ITW) 1 (2021) DOI
[10]: H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-07-11)

Cite as:

“2D subsystem color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/2d_subsystem_color, arXiv:2606.11484

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