One- and two-dimensional cluster states for topological phase simulation and measurement-based quantum computation
Tao Jiang, J. Cai, Junxiang Huang, Naibin Zhou, Yukun Zhang, Jiahao Bei, Guoqing Cai, S. Cao, Fusheng Chen, Jiang Chen, Kefu Chen, Xiawei Chen, Xiqing Chen, Zhe Chen, Zhiyuan Chen, Zihua Chen, Wenhao Chu, Hui Deng, Zhibin Deng, Pei Ding, Xun Ding, Zhuzhengqi Ding, Shuai Dong, Bo Fan, D. Fan, Yu Fu, Dongxin Gao, Lei Ge, J. Gui, Cheng Guo, Shaojun Guo, Xiaoyan Guo, L. Han, Tan He, Linyin Hong, Yisen Hu, Hengjie Huang, Yongting Huo, Zuokai Jiang, Hong-wei Jin, Yunxiang Leng, Dayu Li, Dongdong Li, Fang-Ke Li, Jiaqi Li, Jinjin Li, Junyan Li, Junyun Li, Na Li, Shaowei Li, Wei Li, Yuhuai Li, Yuan Li, Futian Liang, Xue-yan Liang, Na Liao, Jin Lin, Weiping Lin, Dailin Liu, Hongxiu Liu, Maliang Liu, Xinyu Liu, Xuemeng Liu, Yanchen Liu, Hao Lou, Yuwei Ma, Ling-Mei Meng, Hao Mou, Kailiang Nan, Binghan Nie, M. Nie, Jie Ning, Le Niu, Wenyi Peng, H. Qian, H. Rong, Tao Rong, Hui Shen, Qiong Shen, Hong-Bo Su, Feifan Su, Chenyin Sun, Liang-liang Sun, Tianzuo Sun, Yingxiu Sun, Yimeng Tan, Jun Tan, Longyue Tang, Wenbing Tu, Jiafei Wang, Biao Wang, Chang Wang, Chen Wang, Chu Wang, Jian Wang, Lian-Qiang Wang, Rui Wang, Sheng-xing Wang, Xiaomin Wang, Xinzhe Wang, Xunxun Wang, Yeru Wang, Zuolin Wei, Jia-Ning Wei, Dachao Wu, Gang Wu, Jin Yu Wu, Yulin Wu, Shiyong Xie, Lianjie Xin, Yu Xu, Chun Xue, Kai Yan, Weifeng Yang, Xinpeng Yang, Yang Yang, Y. Ye, Zhen Ye, C. Ying, Jiale Yu, Qi-Ming Yu, Wenhu Yu, Xiangdong Zeng, C. Zha, Shaoyu Zhan, Feifei Zhang, Haibin Zhang, Kaili Zhang, W. Zhang, Yiming Zhang, Yongzhuo Zhang, Lixiang Zhang, Guming Zhao, Peng Zhao, Xintao Zhao, You-Wei Zhao, Zhong Zhao, Luyuan Zheng, Fei Zhou, Liang Zhou, Naibin Zhou, Shifeng Zhou, Shuang Zhou, Zheng-Ye Zhou, Chen-Xi Zhu, Qi-xing Zhu, Gui-Zhou Zou, Haonan Zou, Qiang Zhang, Chaochun Lu, Cheng-Zhi Peng, Xiao Yuan, M. Gong, Xiaobo Zhu, Jian-Wei Pan·May 4, 2025·DOI: 10.1038/s41567-026-03179-6
Physics
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Abstract
Symmetry-protected topological (SPT) phases are fundamental features of cluster states, serving as key resources for measurement-based quantum computation (MBQC). Generating large-scale cluster states and verifying their SPT phases are essential steps toward practical MBQC, which however still presents significant experimental challenges. In this work, we address these challenges by utilizing advanced superconducting hardware with optimized gate operations, enhanced readout fidelity, and error mitigation techniques. We successfully generate and verify 95-qubit one-dimensional and 72-qubit two-dimensional genuine entangled cluster states, achieving fidelities of $0.5603 \pm 0.0084$ and $0.5519 \pm 0.0054$, respectively. Leveraging these high-fidelity cluster states, we investigate SPT phases through quantum teleportation across all 95 qubits and demonstrate input-state-dependent robustness against symmetry-breaking perturbations, highlighting the practicality and intrinsic robustness of MBQC enabled by the SPT order. Our results represent a significant advancement in large-scale entanglement generation and topological phase simulation, laying the foundation for scalable and practical MBQC using superconducting quantum systems.