Modular-qudit cluster-state code[1]
Alternative names: Modular-qudit graph-state code.
Description
A code based on a modular-qudit cluster state.
Modular-qudit cluster states are modular-qudit stabilizer states defined on a graph. There is one modular-qudit stabilizer generator \(S_v\) per graph vertex \(v\) of the form [1; Eq. (1)] \begin{align} S_v = X^{\dagger}_{v} \prod_{w\in N(v)} Z_w~, \tag*{(1)}\end{align} where the neighborhood \(N(v)\) is the set of vertices which share an edge with \(v\).'
Encoding
Operators forming the information group can be used to track how logical information is encoded [2].Realizations
Quantum compututation with cluster states has been realized using photons in the time and frequency domains [4].Cousins
- Graph quantum code— A graph quantum code for \(G=\mathbb{Z}_q\) reduces to a modular-qudit cluster state when its logical dimension is one [5].
- Perfect-tensor code— MDS codes can be used to obtain cluster states that are AME with minimal support [6–11].
- Modular-qudit CWS code— A single modular-qudit cluster state is used to construct a modular-qudit CWS code.
Member of code lists
Primary Hierarchy
Parents
Group-based cluster-state codes reduce to modular-qudit cluster-state codes for \(G=\mathbb{Z}_q\).
Hopf-algebra cluster-state codes reduce to modular-qudit cluster-state codes when the Hopf algebra reduces the group \(\mathbb{Z}_q\).
Modular-qudit cluster-state code
Children
Modular-qudit cluster-state codes reduce to cluster-state codes for \(q=2\).
References
- [1]
- D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, “Quantum computation based ond-level cluster state”, Physical Review A 68, (2003) arXiv:quant-ph/0304054 DOI
- [2]
- V. Gheorghiu, S. Y. Looi, and R. B. Griffiths, “Location of quantum information in additive graph codes”, Physical Review A 81, (2010) arXiv:0912.2017 DOI
- [3]
- S. Clark, “Valence bond solid formalism ford-level one-way quantum computation”, Journal of Physics A: Mathematical and General 39, 2701 (2006) arXiv:quant-ph/0512155 DOI
- [4]
- C. Reimer et al., “High-dimensional one-way quantum processing implemented on d-level cluster states”, Nature Physics 15, 148 (2018) DOI
- [5]
- Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
- [6]
- A. V. Thapliyal, Multipartite maximally entangled states, minimal entanglement generating states and entropic inequalities unpublished presentation (2003).
- [7]
- W. Helwig and W. Cui, “Absolutely Maximally Entangled States: Existence and Applications”, (2013) arXiv:1306.2536
- [8]
- W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
- [9]
- D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, “Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices”, Physical Review A 92, (2015) arXiv:1506.08857 DOI
- [10]
- Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
- [11]
- D. Alsina, “PhD thesis: Multipartite entanglement and quantum algorithms”, (2017) arXiv:1706.08318
- [12]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [13]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [14]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, 45 arXiv:quant-ph/0703112 DOI
Page edit log
- Victor V. Albert (2024-06-13) — most recent
Cite as:
“Modular-qudit cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qudit_cluster_state