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Chin. Phys. B, 2023, Vol. 32(1): 010307    DOI: 10.1088/1674-1056/aca7f3

Variational quantum simulation of thermal statistical states on a superconducting quantum processer

Xue-Yi Guo(郭学仪)1, Shang-Shu Li(李尚书)1,2, Xiao Xiao(效骁)3, Zhong-Cheng Xiang(相忠诚)1, Zi-Yong Ge(葛自勇)1,2, He-Kang Li(李贺康)1,2, Peng-Tao Song(宋鹏涛)1,2, Yi Peng(彭益)1,2, Zhan Wang(王战)1,2, Kai Xu(许凯)1,4, Pan Zhang(张潘)5,6,7,†, Lei Wang(王磊)1,8,‡, Dong-Ning Zheng(郑东宁)1,2,4,8,§, and Heng Fan(范桁)1,2,4,9,¶
1 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
2 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China;
3 The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China;
4 CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China;
5 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China;
6 School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China;
7 International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China;
8 Songshan Lake Materials Laboratory, Dongguan 523808, China;
9 Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Abstract  Quantum computers promise to solve finite-temperature properties of quantum many-body systems, which is generally challenging for classical computers due to high computational complexities. Here, we report experimental preparations of Gibbs states and excited states of Heisenberg $XX$ and $XXZ$ models by using a 5-qubit programmable superconducting processor. In the experiments, we apply a hybrid quantum-classical algorithm to generate finite temperature states with classical probability models and variational quantum circuits. We reveal that the Hamiltonians can be fully diagonalized with optimized quantum circuits, which enable us to prepare excited states at arbitrary energy density. We demonstrate that the approach has a self-verifying feature and can estimate fundamental thermal observables with a small statistical error. Based on numerical results, we further show that the time complexity of our approach scales polynomially in the number of qubits, revealing its potential in solving large-scale problems.
Keywords:  superconducting qubit      quantum simulation      variational quantum algorithm      quantum statistical mechanics      machine learning  
Received:  27 November 2022      Revised:  01 December 2022      Accepted manuscript online:  02 December 2022
PACS:  03.67.Lx (Quantum computation architectures and implementations)  
  03.67.-a (Quantum information)  
  05.30.-d (Quantum statistical mechanics)  
Fund: We thank Zhengan Wang, Ruizhen Huang and Tao Xiang for useful discussions. Project supported by the State Key Development Program for Basic Research of China (Grant No. 2017YFA0304300), the National Natural Science Foundation of China (Grant Nos. 11934018, 11747601, and 11975294), Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), Scientific Instrument Developing Project of Chinese Academy of Sciences (Grant No. YJKYYQ20200041), Beijing Natural Science Foundation (Grant No. Z200009), the Key-Area Research and Development Program of Guangdong Province, China (Grant No. 2020B0303030001), and Chinese Academy of Sciences (Grant No. QYZDB-SSW-SYS032).
Corresponding Authors:  Pan Zhang, Lei Wang, Dong-Ning Zheng, Heng Fan     E-mail:;;;

Cite this article: 

Xue-Yi Guo(郭学仪), Shang-Shu Li(李尚书), Xiao Xiao(效骁), Zhong-Cheng Xiang(相忠诚), Zi-Yong Ge(葛自勇), He-Kang Li(李贺康), Peng-Tao Song(宋鹏涛), Yi Peng(彭益), Zhan Wang(王战), Kai Xu(许凯), Pan Zhang(张潘), Lei Wang(王磊), Dong-Ning Zheng(郑东宁), and Heng Fan(范桁) Variational quantum simulation of thermal statistical states on a superconducting quantum processer 2023 Chin. Phys. B 32 010307

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