Also known as Gauge color code.

## Description

A subsystem version of the color code. One way to obtain it is by expanding the vertices of a two-colex embedded in a surface of genus \(g\). Vertex expansion consists of splitting every vertex into a triangle and splitting every edge into a pair of edges.

## 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 \(\Gamma\) [3; Construction B][4; Lemma 2].

## Gates

Braiding twist defects [5].

## Decoding

## 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. [9].Erasure noise: \(50\%\) noise threshold error rate under erasure noise using optimal erasure decoder [10], and \(9.7\%\) and \(44\%\) using gauge-fixing decoders [7,8].

## Threshold

Phenomenological noise: \(0.31\%\) under clustering decoder [6].

## Parents

- Subsystem CSS code
- Modular-qudit subsystem color code — Modular-qudit subsystem color codes reduce to subsystem color codes for \(q=2\).

## Children

## Cousins

- Color code
- Single-shot code — The 3D subsystem color code is a single-shot code [6,11].
- Symmetry-protected self-correcting quantum code — A particular gauge-fixed version of a subsystem code on a 3D lattice yields a self-correcting memory protected by one-form symmetries [12; Sec. IV D]. The symmetric energy barrier grows linearly with the length of a side of the lattice. When the system is coupled locally to a thermal bath respecting the symmetry and below a critical temperature, the memory time grows exponentially with the side length. The subsystem color code is not a self-correcting quantum memory if symmetry protection is removed [13].
- Honeycomb Floquet code — Both honeycomb and subsystem color codes are generated via periodic sequences of measurements. However, any measurement sequence can be performed on the color code without destroying the logical qubits, while honeycomb codes can be maintained only with specific sequences. Honeycomb codes require a shorter measurement cycle and use fewer qubits at the given code distance [14].
- Quantum pin code — Quantum pin codes have a subsystem version that can be viewed as a generalization of subsystem color codes [15].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH code for a certain boundary Hamiltonian is dual to the gauge color code [12; Sec. IV.C.1].

## References

- [1]
- H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
- [2]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [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]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [7]
- H. M. Solanki and P. Kiran Sarvepalli, “Correcting Erasures with Topological Subsystem Color Codes”, 2020 IEEE Information Theory Workshop (ITW) (2021) 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]
- R. S. Andrist et al., “Optimal error correction in topological subsystem codes”, Physical Review A 85, (2012) arXiv:1204.1838 DOI
- [10]
- 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
- [11]
- H. Bombín, “Single-Shot Fault-Tolerant Quantum Error Correction”, Physical Review X 5, (2015) arXiv:1404.5504 DOI
- [12]
- S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
- [13]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389
- [14]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [15]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI

## Page edit log

- Victor V. Albert (2022-05-18) — most recent
- Yi-Ting (Rick) Tu (2022-04-23)
- Victor V. Albert (2022-01-01)

## Cite as:

“Subsystem color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/subsystem_color