Recently, the Quantum Cloud group of BAQIS and the Institute of Physics, Chinese Academy of Sciences, in cooperation with Nankai University, South China University of Technology and RIKEN Center for Quantum Computing, realized the simulation of various Chern insulators with different Chern number and demonstrated the theoretical predicted bulk-edge correspondence. One September 5, 2023, the research results were published in Nature Communications entitied "Simulating Chern insulators on a superconducting quantum processor".

The Quantum Hall effect is a fundamental phenomenon in condensed matter physics, and researchers developed topological band theory to study this class of topological phase of matter. The existence of robust edge states is deeply related to the topology of gapped bulk band structures, which is the ubiquitous bulk-edge correspondence in topological systems. Different topological structures can be distinguished by the Chern number and the relevant topological state of matter can be characterized by Chern insulators. Chern insulators can be predicted by the first principle methods and synthesized by experiment. In the past few years, a series of innovative achievements have emerged and it is expected to develop devices with practical applications.

With the development of quantum control technology, the simulation of Chern insulators have been realized in many manually controllable physical platforms, including ultra-cold atoms in optical lattices, photonic systems, etc. Among these platforms, superconducting quantum computing system, which is stable and versatility, is expected to be an ideal platform for the simulation of Chern insulators.

At first, the cooperation team fabricated a high-quality quantum chip consisting of 30 transmon qubits and the research team controlled the coupling strengths and reduced the cross talks between qubits (Figure 1 and Figure 2) for realizing the 1D and ladder-typer model.  With a dimensional reduction procedure (Figure 3), a 2D integer quantum Hall system can be mapped to a 1D model with a periodic parameter as a synthetic dimension. Based on the same procedure, bilayer Chern insulators can be realized by a ladder-type quantum processor formed by two coupled parallel 1D chain. Different types of bilayer Chern insulators can be realized by modulating the phase in the synthetic dimension. The research team directly measured the band structures of the 2D Chern insulator along synthetic dimensions with various configurations of Aubry-André-Harper chains and observed dynamical localisation of edge excitations (Figure 4). With these two signatures of topology, the experiments implement the bulk-edge correspondence in the synthetic 2D Chern insulator (Figure 5). Moreover, the team simulated two different bilayer Chern insulators on the ladder-type superconducting processor. With the same and opposite periodically modulated on-site potentials for two coupled chains, a topologically nontrivial edge states with zero Hall conductivity and a Chern insulator with higher Chern numbers were simulated experimentally for the first time (Figure 6).

In this work, using a relatively large number of superconducting qubits with long coherence times and accurate control, the research team realized the reproduction and observation of the topological properties of quantum many-body systems, showed the future potential of using superconducting simulating platforms for investigating intriguing topological quantum phases and quantum many-body physics.

Zhong-Cheng Xiang, deputy chief engineer from the Institute of Physics, Chinese Academy of Sciences, Kaixuan Huang, postdoctor from BAQIS and Yu-Ran Zhang, professor from South China University of Technology are the co-first authors. Researcher Heng Fan, associate researcher Xu Kai, form Quantum Cloud group of BAQIS and the Institute of Physics, Chinese Academy of Sciences, and Franco Nori, professor from RIKEN Center for Quantum Computing are the Co-corresponding authors. Other collaborators are the following: Researcher Dongning Zheng, associate researcher Xiaohui Song, Ye Tian, from the Institute of Physics, Chinese Academy of Sciences. Researcher Haifeng Yu, associate researcher Guangming Xue, from BAQIS. Tao Liu, professor from South China University of Technology and Zhi-Bo Liu, professor Nankai University and so on. This work is supported by the National Natural Science Foundation of China, Ministry of Science and Technology, Beijing Natural Science Foundation, and the Strategic Priority Research Program of Chinese Academy of Sciences. The authors thank Hongming Weng for the detailed discussions.

URL：https://www.nature.com/articles/s41467-023-41230-9

Figure 1. Experimental measured coupling strengths between nearest-neighbour (NN), next-nearest-neighbour (NNN), and thirdnearest-neighbour (TNN) qubits, which are used for the numerical simulations of the 15-qubit experiment (a), and the 30-qubit experiment (b).