Analog-cluster-state code

Analog-cluster-state code[1–3]

Also known as CV-cluster-state code, CV-graph-state code, Bosonic-cluster-state code.

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

Protection is related to the analog stabilizer code underlying the analog cluster state.

Encoding

Initialization of all modes in momentum eigenstates and action of gates of the form \(\exp(iV_{jk}\hat{x}_{j}\hat{x}_{k})\). The normalizable version substitutes momentum eigenstates with finitely squeezed states.

Gates

Combination of linear-optical gates and homodyne measurements on subsets of vertices [2,3].Gaussian operations can be realized as operations acting on graphs underlying a cluster state. They can be done in any order, demonstrating parallelism [2,3].Magic-state distillation is required for universal computation [2,3].

Realizations

Analog cluster states on a number of modes ranging from tens to millions [5–7] have been synthesized in photonic degrees of freedom.Required primitives for Gaussian gates have been realized [8].

Notes

See Ref. [9] for a review of analog cluster states and their applications.

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, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal Quantum Computation with Continuous-Variable Cluster States”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605198 DOI
[3]: M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
[4]: C. González-Arciniegas, P. Nussenzveig, M. Martinelli, and O. Pfister, “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, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “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, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “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, J. Yoshikawa, P. van Loock, and A. Furusawa, “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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/hyperplane/cv_cluster_state.yml.