Researchers achieve calculation of Jones polynomial based on Majorana zero modes

The research team experimentally calculated the Jones polynomial based on quantum simulations of a braided Majorana zero mode. The research team determined the Jones polynomials for the various links by simulating the braiding operation of Majorana fermions. The study was published in Physical Review Letters.

Link or knot invariants, such as Jones polynomials, serve as powerful tools to determine whether two knots are topologically equivalent. Currently, there is great interest in determining the Jones polynomial, as it has applications in a variety of fields, such as DNA biology and condensed matter physics.

Unfortunately, even approximating the value of the Jones polynomial falls into the #P-hard complexity class, and the most efficient classical algorithms require an exponential amount of resources. However, quantum simulation provides an exciting way to experimentally investigate the properties of non-abelian anions, and the Majorana zero mode (MZM) is the most plausible candidate for experimentally realizing non-abelian statistics. It is considered.

The research team used a photonic quantum simulator employing two-photon correlation and non-dissipative imaginary time evolution to perform two different MZM braiding operations to generate arbitrary worldlines of several links. Based on this simulator, the team conducted a series of experimental studies simulating the topological properties of non-Abelian anyons.

They successfully simulated the exchange behavior of a single Kitaev chain MZM, detected the non-Abelian geometric phase of the MZM in a two-Kitaev chain model, and further extended it to higher-dimensional semion zero-order modes to locally We have researched a braiding process that is not affected by We removed noise and maintained preservation of quantum context resources.

Building on this work, the team extended their previous single-photon encoding method to a two-photon space method, which takes advantage of two-photon coincidences for encoding. This has significantly increased the number of quantum states that can be encoded.

Meanwhile, the introduction of Sagnac interferometer-based quantum cooling devices transformed dissipative evolution into non-dissipative evolution, enhanced the device’s ability to recycle photonic resources, and contributed to the realization of multi-step quantum evolution operations. These techniques have significantly improved the capabilities of photonic quantum simulators and laid a solid technical foundation for the simulation of braiding Majorana zero modes in three Kitaev models.

The research team demonstrated that the experimental setup can faithfully realize the desired braided evolution of MZMs, as the average fidelity of quantum states and braiding behavior was over 97%.

The research team simulated five typical topological knots by combining different braiding operations of the Majorana zero mode in three Kitaev chain models. This resulted in a Jones polynomial of five topologically distinct links, further distinguishing topologically unequal links.

Such advances could greatly contribute to the fields of statistical physics, molecular synthesis techniques, and integrated DNA replication, where complex topological links and knots frequently appear.