One- and two-dimensional cluster states for topological phase simulation and measurement-based quantum computation

Nature Physics (2026)Cite this article 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.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutAll 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.The code used in this study is available from the corresponding authors upon reasonable request.Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information 10th Anniversary Edition (Cambridge Univ. Press, 2010).Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).Article ADS MathSciNet Google Scholar Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008).Article ADS MathSciNet Google Scholar McCutcheon, W. et al. Experimental verification of multipartite entanglement in quantum networks. Nat. Commun. 7, 13251 (2016).Article ADS Google Scholar Jozsa, R. Entanglement and quantum computation. in The Geometric Universe (eds Huggett, S. A. et al.) Ch. 27 (Oxford Univ. Press, 1998).Graham, T. M. et al. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature 602, 63–68 (2022).

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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.These authors contributed equally: Tao Jiang, Jianbin Cai, Junxiang Huang, Naibin Zhou.Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei, ChinaTao Jiang, Jianbin Cai, Naibin Zhou, Sirui Cao, Fusheng Chen, Kefu Chen, Zhiyuan Chen, Zihua Chen, Hui Deng, Daojin Fan, Yuanhao Fu, Dongxin Gao, Cheng Guo, Shaojun Guo, Lianchen Han, Tan He, Yisen Hu, Yong-Heng Huo, Dayu Li, Junyun Li, Na Li, Shaowei Li, Yuhuai Li, Yuan Li, Futian Liang, Jin Lin, Weiping Lin, Yancheng Liu, Yuwei Ma, Haoran Qian, Hao Rong, Tao Rong, Hong Su, Feifan Su, Chenyin Sun, Tianzuo Sun, Chen Wang, Chu Wang, Rui Wang, Zuolin Wei, Dachao Wu, Gang Wu, Yulin Wu, Yu Xu, Kai Yan, Xinpeng Yang, Yangsen Ye, Zhenping Ye, Chong Ying, Jiale Yu, Qinjing Yu, Chen Zha, Shaoyu Zhan, Yiming Zhang, Guming Zhao, Peng Zhao, Youwei Zhao, Qingling Zhu, Qiang Zhang, Chao-Yang Lu, Cheng-Zhi Peng, Ming Gong, Xiaobo Zhu & Jian-Wei PanShanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai, ChinaTao Jiang, Jianbin Cai, Naibin Zhou, Jiahao Bei, Guoqing Cai, Sirui Cao, Fusheng Chen, Jiang Chen, Kefu Chen, Xiawei Chen, Xiqing Chen, Zhiyuan Chen, Zihua Chen, Hui Deng, Zhibin Deng, Pei Ding, Zhuzhengqi Ding, Shuai Dong, Bo Fan, Daojin Fan, Yuanhao Fu, Dongxin Gao, Lei Ge, Cheng Guo, Shaojun Guo, Xiaoyang Guo, Lianchen Han, Tan He, Yisen Hu, Yong-Heng Huo, Zuokai Jiang, Honghong Jin, Yunxiang Leng, Dayu Li, Fangyu Li, Jiaqi Li, Junyan Li, Junyun Li, Na Li, Shaowei Li, Wei Li, Yuhuai Li, Yuan Li, Futian Liang, Nanxing Liao, Jin Lin, Weiping Lin, Hongxiu Liu, Yancheng Liu, Haoxin Lou, Yuwei Ma, Lingxin Meng, Hao Mou, Binghan Nie, Meijuan Nie, Le Niu, Haoran Qian, Hao Rong, Tao Rong, Qiong Shen, Hong Su, Feifan Su, Chenyin Sun, Tianzuo Sun, Yimeng Tan, Longyue Tang, Chen Wang, Chu Wang, Liangyuan Wang, Rui Wang, Zuolin Wei, Dachao Wu, Gang Wu, Yulin Wu, Yu Xu, Kai Yan, Xinpeng Yang, Yang Yang, Yangsen Ye, Zhenping Ye, Chong Ying, Jiale Yu, Qinjing Yu, Wenhu Yu, Xiangdong Zeng, Chen Zha, Shaoyu Zhan, Feifei Zhang, Kaili Zhang, Wen Zhang, Yiming Zhang, Guming Zhao, Peng Zhao, Xintao Zhao, Youwei Zhao, Luyuan Zheng, Na Zhou, Qingling Zhu, Haonan Zou, Qiang Zhang, Chao-Yang Lu, Cheng-Zhi Peng, Ming Gong, Xiaobo Zhu & Jian-Wei PanHefei National Laboratory, University of Science and Technology of China, Hefei, ChinaTao Jiang, Jianbin Cai, Naibin Zhou, Sirui Cao, Fusheng Chen, Kefu Chen, Zhiyuan Chen, Zihua Chen, Hui Deng, Xun Ding, Daojin Fan, Yuanhao Fu, Dongxin Gao, Jiacheng Gui, Cheng Guo, Shaojun Guo, Lianchen Han, Tan He, Yisen Hu, Yong-Heng Huo, Dayu Li, Jinjin Li, Junyun Li, Na Li, Shaowei Li, Yuhuai Li, Yuan Li, Futian Liang, Jin Lin, Weiping Lin, Dailin Liu, Xinyu Liu, Yancheng Liu, Yuwei Ma, Kailiang Nan, Wenyi Peng, Haoran Qian, Hao Rong, Tao Rong, Hong Su, Feifan Su, Chenyin Sun, Tianzuo Sun, Jun Tan, Chen Wang, Chu Wang, Jian Wang, Rui Wang, Shengtao Wang, Xinzhe Wang, Zuolin Wei, Dachao Wu, Gang Wu, Jin Wu, Yulin Wu, Shiyong Xie, Yu Xu, Kai Yan, Xinpeng Yang, Yangsen Ye, Zhenping Ye, Chong Ying, Jiale Yu, Qinjing Yu, Xiangdong Zeng, Chen Zha, Shaoyu Zhan, Haibin Zhang, Yiming Zhang, Yongzhuo Zhang, Guming Zhao, Peng Zhao, Youwei Zhao, Shifeng Zhou, Shuang Zhou, Zhengxiao Zhou, Chengjun Zhu, Qingling Zhu, Guihong Zou, Qiang Zhang, Chao-Yang Lu, Cheng-Zhi Peng, Ming Gong, Xiaobo Zhu & Jian-Wei PanCenter on Frontiers of Computing Studies, Peking University, Beijing, ChinaJunxiang Huang, Yukun Zhang & Xiao YuanSchool of Computer Science, Peking University, Beijing, ChinaJunxiang Huang, Yukun Zhang & Xiao YuanQuantumCTek Co., Ltd., Hefei, ChinaZhe Chen, Wenhao Chu, Linyin Hong, Dongdong Li, Xuemeng Liu, Huiyan Shen, Liangchao Sun, Yingxiu Sun, Wenbing Tu, Jiafei Wang, Biao Wang, Chang Wang, Jiazhou Wei, Chun Xue, Weifeng Yang, Lixiang Zhang, Zhong Zhao & Liang ZhouHenan Key Laboratory of Quantum Information and Cryptography, Zhengzhou, ChinaHe-Liang HuangNational Institute of Metrology, Beijing, ChinaJinjin LiJinan Institute of Quantum Technology and Hefei National Laboratory Jinan Branch, Jinan, ChinaXuelian Liang, Jie Ning, Xiaomin Wang, Xunxun Wang, Yeru Wang, Lianjie Xin, Fei Zhou, Qiang Zhang & Xiaobo ZhuSchool of Microelectronics, Xidian University, Xi’an, ChinaMaliang LiuUniversity of Science and Technology of China, Shanghai Research Institute, Shanghai, ChinaGang WuSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch 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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.Correspondence to Xiao Yuan, Ming Gong, Xiaobo Zhu or Jian-Wei Pan.The authors declare no competing interests.Nature Physics thanks Christian Andersen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary Sections I–VI, Figs. 1–25, Tables I–IX and Discussion.Statistical source data.Statistical source data.Statistical source data.Statistical source data.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsJiang, 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/s41567-026-03179-6Download citationReceived: 13 May 2025Accepted: 09 January 2026Published: 11 March 2026Version of record: 11 March 2026DOI: https://doi.org/10.1038/s41567-026-03179-6Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative