Description
An \(n\)-qubit approximate \(q\)-dimensional spin code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) [3] matrix product states with various boundary conditions. The codes become exact when either \(n\) or \(q\) go to infinity.
Transversal Gates
Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates [2; Tab. III].
Parents
- Spin code — VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [1,2].
- Frustration-free Hamiltonian code — VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [1,2].
- Approximate quantum error-correcting code (AQECC) — VBS codes approximately protect against erasures in the thermodynamic limit.
Cousins
- Covariant block quantum code — Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates, i.e., are \(SU(q)\)-covariant [2; Tab. III].
- Symmetry-protected topological (SPT) code — VBS codewords [1] are associated with 1D SPT orders [4–7].
References
- [1]
- D.-S. Wang, G. Zhu, C. Okay, and R. Laflamme, “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
- [2]
- D.-S. Wang, Y.-J. Wang, N. Cao, B. Zeng, and R. Laflamme, “Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes”, New Journal of Physics 24, 023019 (2022) arXiv:2105.14777 DOI
- [3]
- I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Rigorous Results on Valence-Bond Ground States in Antiferromagnets”, Condensed Matter Physics and Exactly Soluble Models 249 (2004) DOI
- [4]
- X. Chen, Z.-C. Gu, and X.-G. Wen, “Classification of gapped symmetric phases in one-dimensional spin systems”, Physical Review B 83, (2011) arXiv:1008.3745 DOI
- [5]
- N. Schuch, D. Pérez-García, and I. Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states”, Physical Review B 84, (2011) arXiv:1010.3732 DOI
- [6]
- X. Chen, Z.-C. Gu, and X.-G. Wen, “Complete classification of one-dimensional gapped quantum phases in interacting spin systems”, Physical Review B 84, (2011) arXiv:1103.3323 DOI
- [7]
- X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group”, Physical Review B 87, (2013) arXiv:1106.4772 DOI
Page edit log
- Victor V. Albert (2024-05-27) — most recent
Cite as:
“Valence-bond-solid (VBS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/vbs
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/many_spin/vbs.yml.