Note
This is a Note for the project about quantum critical state collaborated with my undergraduate classmates in SUSTC / SUSTech (present abbrev.). Wenhui Huang, Xinchi Zhou, and Ziyu Tao all belong to the Class of 2015 in the Department of Physics at Southern University of Science and Technology, and graduated in 2019.
The critical state project is inspired by the theory proposed by Xiong-jun Liu group when the theory paper (Exact New Mobility Edges between Critical and Localized States, Phys. Rev. Lett. 131, 176401 (2023)) was initially submitting to Physical Review Letters.
In March 2023, initiated by Prof. Liu and Prof. Zhong’s group meeting, Xinchi Zhou, Ziyu Tao and Wenhui Huang started numerical research and run severval testing experiments in the 1D chain 36-qubit superconducting processor following Phys. Rev. Lett. 131, 176401 that was under review, which also provides experimental evidence as supporting material for the theoretical manuscript during peer review before acceptance.
Following the online publication of Phys. Rev. Lett. 131, 176401, Prof. Youpeng Zhong suggested in a group meeting that the advantages of the 2D processor could be leveraged to re-run the experiment, potentially uncovering new phenomena beyond the scope of the original PRL theory. For this follow-up study, Wenhui Huang designed and characterized the 66-qubit 2D processor under the supervision of Prof. Zhong Subsequently, Ziyu Tao and Xinchi Zhou engaged in prolonged, in-depth discussions (2023-2025) encompassing both the details of Phys. Rev. Lett. 131, 176401 and extensions beyond it. Guided by these theoretical considerations, Ziyu Tao conducted the several rounds of experiments on the 2D 66-qubit processor with fruitful inputs from Xinchi Zhou, Prof. Xiong-jun Liu and Prof. Youpeng Zhong on each detail of measured results and figure, which stimulated both a unified theory paper (The fundamental localization phases in quasiperiodic systems: a unified framework and exact results) and the final experimental work (Observation of exact quantum critical states). A large number of revisions are made in the exp. manuscript, where some of the nearly final versions can be viewed in the arxiv v1, v2, v3. During the manuscript preparation and the peer review, Bing-Chen Yao (undergrad student of Prof. X.-J. Liu group) and Xin-Chi Zhou made a large progress on theoretical analysis, e.g., the renormalization group analysis (see the Methods, and Supplementary Material), which largely increase the novelty of this work and made him deserved to be one of the co-first author in v3. The following Acknowledgements are adapted from the website of spring nature: https://www.nature.com/articles/s41567-026-03333-0.
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Acknowledgements
We thank Qi Zhou and Yucheng Wang for helpful discussions. This work was supported by National Key Research and Development Program of China no. 2021YFA1400900 (X.-J.L.), Quantum Science and Technology-National Science and Technology Major Project nos. 2021ZD0302000 (X.-J.L.) and 2021ZD0301703 (Y. Zhong and S.L.), the Science, Technology and Innovation Commission of Shenzhen Municipality no. KQTD20210811090049034 (Y. Zhong), the National Natural Science Foundation of China nos. 12425401 (X.-J.L.), 12174178 (Y. Zhong) and 12261160368 (X.-J.L.), respectively, and Shanghai Municipal Science and Technology Major Project no. 2019SHZDZX01 (X.-J.L.). X.-J.L. was also supported by the New Cornerstone Science Foundation through the XPLORER PRIZE.
These authors contributed equally: Wenhui Huang, Xin-Chi Zhou, Libo Zhang, Jiawei Zhang, Yuxuan Zhou, Bing-Chen Yao.
X.-J.L. conceived of the project. W. Huang designed and tested the device under the supervision of Y. Zhong. Z. Tao collected the data, and analysed the data with the help of X.-C.Zhou. X.-C.Zhou and B.-C.Yao provided theoretical and numerical studies under the supervision of X.-J.L. L.Zhang and Y. Zhou fabricated the device. Jiawei Zhang built the microwave electronics under the supervision of Y. Zhong. Y. Zhong, X.-J.L., and D.Yu supervised the project. All authors contributed to the discussion and preparation of the paper.
Corresponding authors
Correspondence to Ziyu Tao, Song Liu, Youpeng Zhong, Xiong-Jun Liu or Dapeng Yu.