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

1D modular-qudit cluster states [1,2] are resources for universal MBQC.

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 cluster-state codes for \(G=\mathbb{Z}_q\).

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
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Zoo Code ID: qudit_cluster_state

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
BibTeX:
@incollection{eczoo_qudit_cluster_state, title={Modular-qudit cluster-state code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_cluster_state} }
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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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/qudit_cluster_state.yml.