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Generation of three-dimensional cluster entangled state

Nature Photonics volume 19, pages 526–532 (2025)Cite this article

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Abstract

Measurement-based quantum computing is a promising paradigm of quantum computation, in which universal computing is achieved through a sequence of local measurements. The backbone of this approach is the preparation of multipartite entanglement, known as cluster states. Although a cluster state with two-dimensional connectivity is required for universality, a three-dimensional cluster state is necessary for additionally achieving fault tolerance. However, the challenge of making three-dimensional connectivity has limited cluster state generation capability up to two dimensions. Here we demonstrate the deterministic generation of a three-dimensional cluster state based on the photonic continuous-variable platform. To realize three-dimensional connectivity, we harness a crucial advantage of time–frequency modes of ultrafast quantum light: an arbitrary complex mode basis can be accessed directly, enabling connectivity as desired. We demonstrate the versatility of our method by generating cluster states with one-, two- and three-dimensional connectivities. For their complete characterization, we develop a quantum state tomography method for multimode Gaussian states. Moreover, we verify the cluster state generation by nullifier measurements as well as full inseparability tests. Our work paves the way towards fault-tolerant and universal-measurement-based quantum computing.

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Fig. 1: Experimental scheme.
Fig. 2: Full characterization of cluster states.
Fig. 3: Verification of cluster state generation.
Fig. 4: Ancilla-assisted cluster state generation.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The code used to produce the results are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank M. S. Kim and J. Park for fruitful discussions. This work was supported by the Ministry of Science and ICT (MSIT) of Korea (NRF-2020M3E4A1080028, NRF-2022R1A2C2006179 and RS-2023-NR119925) under the Information Technology Research Center (ITRC) support program (IITP-2025-2020-0-01606) and the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant (no. 2022-0-01029, Atomic ensemble based quantum memory) and by the Air Force Office of Scientific Research award (nos. FA2386-21-1-4020 and FA2386-22-1-4083).

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Authors and Affiliations

  1. Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea

    Chan Roh, Geunhee Gwak, Young-Do Yoon & Young-Sik Ra

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  1. Chan Roh
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Contributions

C.R. conducted the experiment with contributions from G.G. and Y.-D.Y. C.R. analysed the data and developed the theory. Y.-S.R. conceived and supervised the project. C.R. and Y.-S.R. wrote the manuscript with inputs from all authors.

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Correspondence to Young-Sik Ra.

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Nature Photonics thanks Jiangrui Gao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Ancilla-assisted generation of a cluster state with negative weights.

The same method as Fig. 4 is used to generate a 3D cluster state with negative weights. a, Four edges (6-7, 6-13, 9-10, and 10-11) in Gbasic have negative weights. b, Experimentally generated 3D cluster state after local phase correction in the range of (− 0.18, 0.18) rad. c, Nullifier measurements on the generated 3D cluster state. d, Conditional covariance matrix in the target modes. e, Inseparability test for the cluster state in (d), exhibiting negative minimum eigenvalues for all bipartitions. f, Nullifier variances of the target cluster state after removing the ancilla. In c and f, data are presented as scatter plots (from four repeated experiments), with mean values as bar charts and ± 1 s.d. as error bars.

Supplementary information

Supplementary Information

Supplementary Figs. 1–5, Sections 1–4 and References.

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Roh, C., Gwak, G., Yoon, YD. et al. Generation of three-dimensional cluster entangled state. Nat. Photon. 19, 526–532 (2025). https://doi.org/10.1038/s41566-025-01631-2

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