Modular-qudit cluster-state code[1]
Also known as 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 \(v\).'
Gates
Realizations
Quantum compututation with cluster states has been realized using photons in the time and frequency domains [3].
Parents
- Modular-qudit stabilizer code — Modular-qudit cluster-state codes are particular modular-qudit stabilizer codes. Any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [4] (see also [5,6]).
- Group-based cluster-state code — Group-based cluster-state codes reduce to modular-qudit cluster-state codes for \(G=\mathbb{Z}_q\).
- Hopf-algebra cluster-state code — Hopf-algebra cluster-state codes reduce to modular-qudit cluster-state codes when the Hopf algebra reduces the group \(\mathbb{Z}_q\).
Child
- Cluster-state code — Modular-qudit cluster-state codes reduce to cluster-state codes for \(q=2\).
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 [7].
- Perfect-tensor code — Since any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [4] (see also [5,6]), stabilizer AME states can be understood as modular-qudit cluster states [8].
- Modular-qudit CWS code — A single modular-qudit cluster state is used to construct a modular-qudit CWS code.
References
- [1]
- D. L. Zhou et al., “Quantum computation based ond-level cluster state”, Physical Review A 68, (2003) arXiv:quant-ph/0304054 DOI
- [2]
- 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
- [3]
- C. Reimer et al., “High-dimensional one-way quantum processing implemented on d-level cluster states”, Nature Physics 15, 148 (2018) DOI
- [4]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [5]
- 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
- [6]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
- [7]
- Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
- [8]
- W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
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