## Description

A code based on a continuous-variable (CV), or analog, cluster state. Such a state can be used to perform MBQC of logical modes, which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. The exact analog cluster state is non-normalizable, so approximate constructs have to be considered.

Analog cluster states are analog stabilizer states defined on a graph. There is one nullifier \(\hat{\eta}_j\) per graph vertex \(j\) of the form \begin{align} \hat{\eta}_j = \hat{p}_{j} - \sum_{k\in N(j)} V_{jk} \hat{x}_k~, \tag*{(1)}\end{align} where the neighborhood \(N(j)\) is the set of vertices which share an edge with \(j\), and where \(V_{jk}\) is a weighed (real-valued) adjacency matrix of a graph [4].

Analog cluster states, like cluster states, can be defined on various geometries. Analog cluster states defined on a 1D ladder are sometimes called dual-rail, not to be confused with the dual-rail code.

## Protection

## Encoding

## Gates

## Realizations

## Notes

## Parents

- Analog stabilizer code — Analog-cluster-state codes are particular analog stabilizer codes. Relaxing the real weighted adjacency matrix of an analog cluster state to be complex yields a description of a general analog (i.e., Gaussian) stabilizer code state [10].
- Group-based cluster-state code — Analog cluster states are group-based cluster states for \(G=\mathbb{R}\).

## Cousin

- GKP CV-cluster-state code — GKP CV-cluster-state codes reduce to analog-cluster-state codes when all physical modes are initialized in momentum states.

## References

- [1]
- J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states”, Physical Review A 73, (2006) DOI
- [2]
- N. C. Menicucci et al., “Universal Quantum Computation with Continuous-Variable Cluster States”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605198 DOI
- [3]
- M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
- [4]
- C. González-Arciniegas et al., “Cluster States from Gaussian States: Essential Diagnostic Tools for Continuous-Variable One-Way Quantum Computing”, PRX Quantum 2, (2021) arXiv:1912.06463 DOI
- [5]
- S. Yokoyama et al., “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain”, Nature Photonics 7, 982 (2013) arXiv:1306.3366 DOI
- [6]
- M. Chen, N. C. Menicucci, and O. Pfister, “Experimental Realization of Multipartite Entanglement of 60 Modes of a Quantum Optical Frequency Comb”, Physical Review Letters 112, (2014) arXiv:1311.2957 DOI
- [7]
- J. Yoshikawa et al., “Invited Article: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing”, APL Photonics 1, (2016) arXiv:1606.06688 DOI
- [8]
- Y. Miwa et al., “Demonstration of a universal one-way quantum quadratic phase gate”, Physical Review A 80, (2009) arXiv:0906.3141 DOI
- [9]
- A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement (Wiley, 2011) DOI
- [10]
- N. C. Menicucci, S. T. Flammia, and P. van Loock, “Graphical calculus for Gaussian pure states”, Physical Review A 83, (2011) arXiv:1007.0725 DOI

## Page edit log

- Victor V. Albert (2024-07-17) — most recent

## Cite as:

“Analog-cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cv_cluster_state