Abstract
Quantum entanglement is a fundamental resource for quantum information processing and serves as a critical benchmark for quantum hardware performance. Cluster states are a special class of entangled states that serve as universal resources for measurement-based quantum computation and possess an intrinsic symmetry-protected topological order, which confers robustness against symmetry-respecting noise. Here we report the scalable preparation and verification of genuine multipartite cluster states on the 105-qubit Zuchongzhi 3.1 superconducting processor. We achieve one-dimensional cluster states of up to 95 qubits and two-dimensional cluster states of up to 72 qubits. The symmetry-protected topological cluster states exhibit input-state-dependent robustness under symmetry-breaking perturbations due to an operational parity structure that enhances the performance of measurement-based quantum computation. Furthermore, we use our two-dimensional cluster states to implement the Deutsch–Jozsa algorithm within the measurement-based quantum computation framework, achieving higher output-state fidelity compared with traditional circuit-based models and a query efficiency advantage over classical approaches. Our work establishes a scalable platform that combines large-scale entanglement generation, symmetry-protected topological order and practical quantum algorithms to enable robust, fault-tolerant measurement-based quantum computation.
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Data availability
All source data supporting the findings of this study are available via figshare at https://doi.org/10.6084/m9.figshare.30774326. Source data are provided with this paper.
Code availability
The code used in this study is available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank R. Jozsa and D. Azses for valuable discussions, the University of Science and Technology of China (USTC) Center for Micro- and Nanoscale Research and Fabrication for supporting the sample fabrication and QuantumCTek Co., Ltd. for the manufacture and maintenance of room-temperature electronics equipment. This research was supported by the Quantum Science and Technology-National Science and Technology Major Project (grant numbers 2021ZD0300200 and 2023ZD0300200), Anhui Initiative in Quantum Information Technologies, special funds from Jinan Science and Technology Bureau and Jinan High Tech Zone Management Committee, Shanghai Municipal Science and Technology Major Project (grant number 2019SHZDZX01), National Natural Science Foundation of China (grant numbers 92476203, 92476001, 12175003 and 12361161602), NSAF (grant number U2330201), Shandong Provincial Natural Science Foundation (grant number ZR2022LLZ008), Cultivation Project of Shanghai Research Center for Quantum Sciences (grant number LZPY2024), Key-Area Research and Development Program of Guangdong Province (2020B0303060001). M.G. was sponsored by National Natural Science Foundation of China (grant numbers T2322024 and 12474495), Shanghai Rising-Star Program (grant number 23QA1410000) and the Youth Innovation Promotion Association of CAS (grant number 2022460). X. Zhu acknowledges support from the New Cornerstone Science Foundation through the Xplorer Prize and the Taishan Scholars Program.
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X. Yuan, M.G., X. Zhu and J.-W.P. conceived the research and designed the experiment. Development of the experimental system, device fabrication, experimentation, data analysis, and the writing and revision of the manuscript were carried out collaboratively by all authors.
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Jiang, T., Cai, J., Huang, J. et al. One- and two-dimensional cluster states for topological phase simulation and measurement-based quantum computation. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03179-6
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DOI: https://doi.org/10.1038/s41567-026-03179-6
