A research team led by Prof.GUO Guangcan from the University of Science and Technology of China (USTC) of the Chinese Academy of Sciences, together with Prof.Jiannis K. Pachos from the University of Leeds, experimentally calculated the Jones polynomial based on the quantum simulation of braided Majorana zero modes, and determined the Jones polynomials of different links by simulating braiding operations of Majorana fermions. This study was published in Physical Review Letters.

Link or knot invariants, such as the Jones polynomials, serve as a powerful tool to determine whether or not two knots are topologically equivalent. Currently, there is much interest in determining Jones polynomials as they have applications in various disciplines such as DNA biology and condensed matter physics. 

Unfortunately, even approximating the value of Jones polynomials falls within the #P-hard complexity class, with the most efficient classical algorithms requiring an exponential amount of resources. Yet, quantum simulations offer an exciting way to experimentally investigate properties of non-Abelian anyons and Majorana zero modes (MZMs) are regarded as the most plausible candidate for experimentally realizing non-Abelian statistics.

In early work, researchers performed two distinct MZM braiding operations that generate anyonic worldlines of several links using a photonic quantum simulator that employed two-photon correlations and nondissipative imaginary-time evolution. They also using it conducted a series of experimental studies to simulate the topological properties of non-Abelian anyons.

Based on the findings, researchers expanded the single-photon encoding method to dual-photon spatial methods, and significantly increased the number of quantum states that can be encoded through coincidence counting of dual photons for encoding. Meanwhile, they transformed the dissipative evolution into a nondissipative evolution by introducing a Sagnac interferometer-based quantum cooling device, which enhanced the device’s capability to recycle photonic resources, contributing to achieving multi-step quantum evolution operations.

These techniques greatly improve the capability of the photonic quantum simulator and lay a solid foundation for the simulation of braiding Majorana zero modes in three Kitaev models. Researchers demonstrated that their experimental setup could faithfully realize the desired braiding evolutions of MZMs for the average fidelity of quantum states and braiding operation being above 97%.

Researchers simulated five typical topological knots by combining different braiding operations of Majorana zero modes in the three Kitaev chain models, which gave rise to Jones polynomials of five topologically distinct links, further distinguishing between topologically inequivalent links. This achievement provides insights into fields such as statistical physics, molecular synthesis technology and integrated DNA replication, where intricate topological links and knots emerge frequently.