Valence-bond-solid (VBS) code

Valence-bond-solid (VBS) code[1–3]

Description

An \(n\)-qubit approximate \(q\)-dimensional spin code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) [4] matrix product states with various boundary conditions. The codes become exact when either \(n\) or \(q\) go to infinity. The original work on these codes studied the \(q=2\) case [1].

Transversal Gates

Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates [3; Tab. III].

Cousins

Covariant block quantum code— Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates, i.e., are \(SU(q)\)-covariant [3; Tab. III].
Symmetry-protected topological (SPT) code— VBS codewords [2] are associated with 1D SPT orders [5–8].

Member of code lists

Primary Hierarchy

Quantum Domain

Spin Kingdom

Spin codeQECC Quantum

Parents

Spin code

VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [2,3].

Frustration-free Hamiltonian codeHamiltonian-based QECC Quantum

VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [2,3].

Approximate quantum error-correcting code (AQECC)Quantum

VBS codes approximately protect against erasures in the thermodynamic limit.

Valence-bond-solid (VBS) code

References

[1]: N. E. Bonesteel, “Chiral spin liquids and quantum error-correcting codes”, Physical Review A 62, (2000) arXiv:quant-ph/0006092 DOI
[2]: D.-S. Wang, G. Zhu, C. Okay, and R. Laflamme, “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
[3]: 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
[4]: 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
[5]: 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
[6]: 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
[7]: 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
[8]: 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.