3D subsystem color code[1]
Also known as 3D gauge color code.
Description
A subsystem version of the 3D color code.
Threshold
Phenomenological noise: \(0.31\%\) under clustering decoder [2].
Parents
Cousins
- 3D color code
- Single-shot code — The 3D subsystem color code defined on the cube-truncated rhombic dodecahedral honeycomb, i.e., a tesselation of cubes and chamfered cubes (a.k.a. tetratruncated rhombic dodecahedra) [2; Fig. 1], is a single-shot code [2,3].
- 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 [5][4; 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 [6].
- Two-gauge theory code — The 3D subsystem color code can be ungauged [7–9,9] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [5].
- Symmetry-protected topological (SPT) code — Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [5].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [5].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH code for a certain boundary Hamiltonian is dual to the 3D subsystem color code [4; Sec. IV.C.1].
References
- [1]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [2]
- 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
- [3]
- H. Bombín, “Single-Shot Fault-Tolerant Quantum Error Correction”, Physical Review X 5, (2015) arXiv:1404.5504 DOI
- [4]
- S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
- [5]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [6]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389
- [7]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [8]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [9]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
Page edit log
- Victor V. Albert (2024-07-11) — most recent
Cite as:
“3D subsystem color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/3d_subsystem_color