# 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 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

## 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