Tsinghua University: Deng Dongling’s Research Group Pioneers Fibonacci Non-Abelian Topological State and Anyon Weaving

Recently, Deng Dongling’s research group at the Institute of Cross-Disciplinary Information Sciences of Tsinghua University, in collaboration with Wang Haohua and Song Chao’s research groups at the School of Physics of Zhejiang University, prepared the Fibonacci non-Abelian topological state in a superconducting system for the first time and realized the weaving operation of Fibonacci anyons.

Common basic particles in nature are divided into two types: bosons and fermions. Exchanging the positions of two basic particles will cause the system wave function to produce a phase of +1 (bosons, such as photons) or -1 (fermions, such as electrons). This is because in three-dimensional space, a loop of particle A around particle B (equivalent to exchanging positions twice) can be continuously deformed to disappear without passing through particle B. This restricts the system to return to the initial quantum state after exchanging particles twice, so each exchange of the system wave function can only produce a phase factor of +1 or -1. The corresponding particles are called bosons or fermions, satisfying the Bose-Einstein or Fermi-Dirac statistical laws. In two-dimensional space, a loop of particle A around particle B cannot be continuously deformed to disappear without passing through particle B, so there is no restriction that the particles must return to the initial quantum state after exchanging twice. In this case, the exchange of particles can produce arbitrary phases. Such particles are called Abelian anyons, and the process of exchanging positions is called braiding. More generally, if there is degeneracy in the system’s ground state, exchanging two particles can even change the amplitude of the system’s wave function, leading to a unitary evolution of the system as a whole instead of just obtaining a global phase. Such particles are called non-Abelian anyons.

The study of non-Abelian anyons has important basic theoretical significance and potential application value. Such particles satisfy non-Abelian statistical laws and are exotic particles that are fundamentally different from traditional bosons and fermions. Non-Abelian anyons are also the cornerstone of topological quantum computing. In topological quantum computing, quantum gates are realized by the weaving of non-Abelian anyons, and the measurement of calculation results is completed by the fusion of anyons. The topological properties of anyons make this quantum computer inherently immune to local errors, providing a fault-tolerant quantum computing solution at the hardware level.

Although there are many theoretical schemes, the experimental realization of non-Abelian anyons is very difficult. It was not until recent years that work on simulating non-Abelian anyons on quantum processors appeared. However, all previously simulated non-Abelian anyons have quantum gates corresponding to their weaving operations that do not have the ability for universal quantum computing. Fibonacci anyons have more complex statistical properties, and their experimental realization is even more difficult. The quantum dimension of Fibonacci anyons is the golden ratio of 1.618, which is closely related to the Fibonacci sequence in mathematics (Figure 1). Its weaving can realize arbitrary quantum gates and can be used to build a universal fault-tolerant quantum computer.

It is widely believed that it is extremely difficult to experimentally prepare Fibonacci non-Abelian topological states and realize the weaving operation of Fibonacci anyons. The experiment uses a string net condensation model, and through geometric transformation, the quantum bits on the square lattice of the superconducting quantum chip are matched with the honeycomb-shaped “strings” in the string net model (Figure 2). In this model, the system Hamiltonian consists of the sum of all vortex operators Qv and all block operators Bp. All strings in the ground state are closed, while Fibonacci anyons in the excited state are distributed at both ends of the open strings (Figure 2). The experiment used 27 superconducting quantum bits, with a single (double) bit gate accuracy of 99.96% (99.5%), and the system ground state was prepared through 115 layers of quantum circuits.

After preparing the ground state, the experiment measured the topological entanglement entropy by dividing the system into different regions, and the results were consistent with the theoretical predictions. On this basis, the experiment generated two pairs of Fibonacci anyons through string operator operations and demonstrated their weaving operations (Figure 3). The experiment designed a variety of different weaving orders to test the characteristics of Fibonacci anyons (Figure 3a), namely: (i) annihilation of Fibonacci anyons and their antiparticles; (ii) weaving changes the fusion results; (iii) and (iv) the same fusion results verify the Yang-Baxter equation; (v) measuring the quantum dimension of Fibonacci anyons. The experimental results are all in good agreement with the theoretical predictions (Figure 3b), among which the quantum dimension of Fibonacci anyons obtained according to the experimental results of weaving order (v) is 1.598, which is very close to the golden ratio of 1.618 predicted by theory.

As an important basic model in the field of topological quantum computing, the successful simulation and weaving of Fibonacci anyons is the basis for realizing universal topological quantum computing. This study prepared the Fibonacci non-Abelian topological state for the first time and realized the weaving operation of Fibonacci anyons, which is an important step towards the ultimate realization of universal topological quantum computing.

On July 1, the relevant research results were published online in the journal Nature Physics under the title ” Non-Abelian braiding of Fibonacci anyons with a superconducting processor . “

Assistant Professor Deng Dongling from the Institute of Cross-Disciplinary Information Sciences at Tsinghua University , Professor Wang Haohua from the School of Physics at Zhejiang University, and Researcher Song Chao are the corresponding authors of the paper . Xu Shibo and Wang Ke, PhD students of Zhejiang University in 2021, and Sun Zhengzhi, a postdoctoral fellow at the Institute of Cross-Disciplinary Information Sciences at Tsinghua University, are the co-first authors of the paper. Other authors include some other members of the superconducting quantum computing team at Zhejiang University, PhD students Li Weikang and Jiang Wenjie from the Institute of Cross-Disciplinary Information Sciences at Tsinghua University, and Yu Liwei, an associate researcher at the Chern Institute of Mathematics at Nankai University. The project is supported by the National Natural Science Foundation of China, the Ministry of Science and Technology, the Hefei National Laboratory, the Tsinghua University Dushi Project, and the Shanghai Institute of Intellectual Property.