Description
A subsystem version of the color code.
Subsystem color codes form a \((d,e)\) family on punctured \(D\)-colexes, or equivalently on suitably colored simplicial \(D\)-balls, encoding one logical qubit and interpolating between conventional and subsystem color codes via gauge fixing [2]. Examples include 2D subsystem color codes obtained by expanding the vertices of a two-colex embedded in a surface of genus \(g\), where each vertex is split into a triangle and each edge into a pair of edges.
The stabilizer group may contain generators of unbounded weight, distinguishing these codes from stabilizer codes with bounded-weight generators for which some logical qubits were re-assigned to be gauge qubits.
Gauge fixing between subsystem color codes defined on the same lattice can be implemented using local measurements and classical processing analogous to error correction [2].
Transversal Gates
For a \(D\)-dimensional \((d,e)\) gauge color code, \(CNOT\) is transversal, Hadamard is transversal when \(d=e\), and \(R_n=\operatorname{diag}(1,e^{2\pi i/2^n})\) is transversal whenever \(D \geq n(D-e)\) [2].Cousins
- Color code— Gauge fixing relates subsystem color codes to conventional color codes defined on the same lattice [2].
- EAOA qubit stabilizer code— The 15-qubit subsystem color code yields several EAOA qubit stabilizer constructions, including \([[13,1,3;6,2,3]]\), \([[15,1,3;5,1,2]]\), and \([[15,1,3;4,1,4]]\) examples obtained via clean-qubits and entanglement-assisted gauge-fixing constructions [6].
- 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 [7].
- Majorana subsystem stabilizer code— A particular self-dual stabilizer Hamiltonian within the 3D subsystem color code admits a Majorana variant whose boundaries support 2D Majorana color codes [8].
- Quantum pin code— Quantum pin codes have a subsystem version that can be viewed as a generalization of subsystem color codes [9].
Primary Hierarchy
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]
- 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
- [4]
- H. M. Solanki and P. Kiran Sarvepalli, “Correcting Erasures with Topological Subsystem Color Codes”, 2020 IEEE Information Theory Workshop (ITW) 1 (2021) DOI
- [5]
- H. M. Solanki and P. K. Sarvepalli, “Decoding Topological Subsystem Color Codes Over the Erasure Channel using Gauge Fixing”, (2022) arXiv:2111.14594
- [6]
- P. J. Nadkarni, S. Adonsou, G. Dauphinais, D. W. Kribs, and M. Vasmer, “Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction”, (2024) arXiv:2411.14389
- [7]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [8]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [9]
- 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