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 weighted (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 array of modes are called quantum wires [5,6], not to be confused with the Kitaev quantum wire, a fermion code. 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.Squeezers and beam-splitters [7].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 [8–10] have been synthesized in photonic degrees of freedom. An \(12\times N\) mode cluster state, where \(N\) is the number of clock cycles of the experiment, has been realized in a photonic device by Xanadu [11].Required primitives for Gaussian gates have been realized [12].Notes
See Ref. [13] for a review of analog cluster states and their applications.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.
Primary Hierarchy
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]
- N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-Way Quantum Computing in the Optical Frequency Comb”, Physical Review Letters 101, (2008) arXiv:0804.4468 DOI
- [6]
- S. T. Flammia, N. C. Menicucci, and O. Pfister, “The optical frequency comb as a one-way quantum computer”, Journal of Physics B: Atomic, Molecular and Optical Physics 42, 114009 (2009) arXiv:0811.2799 DOI
- [7]
- N. C. Menicucci, “Temporal-mode continuous-variable cluster states using linear optics”, Physical Review A 83, (2011) arXiv:1007.3434 DOI
- [8]
- 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
- [9]
- 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
- [10]
- 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
- [11]
- H. Aghaee Rad et al., “Scaling and networking a modular photonic quantum computer”, Nature (2025) DOI
- [12]
- 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
- [13]
- A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement (Wiley, 2011) DOI
- [14]
- 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