Photonic Simulation of Majorana-Based Jones Polynomials

By braiding non-Abelian anyons it is possible to realize fault-tolerant quantum algorithms through the computation of Jones polynomials. So far, this has been an experimentally formidable task. In this Letter, a photonic quantum system employing two-photon correlations and nondissipative imaginary-time evolution is utilized to simulate two inequivalent braiding operations of Majorana zero modes. The resulting amplitudes are shown to be mathematically equivalent to Jones polynomials. The high fidelity of our optical platform allows us to distinguish between a wide range of links, such as Hopf links, Solomon links, Trefoil knots, Figure Eight knots and Borromean rings, through determining their corresponding Jones polynomials. Our photonic quantum simulator represents a significant step towards executing fault-tolerant quantum algorithms based on topological quantum encoding and manipulation.

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Photonic Simulation of Majorana-Based Jones Polynomials

Posted: 2025/1/6 | Category: Artificial Intelligence | 03/2021 - 12/2024 | by 郭光灿 教授, Prof Jiannis Pachos | with China University of Science and Technology, University of Leeds | Ref Link

Guo Guangcan, an academician of China Academy of Sciences and a professor at China University of Science and Technology, has made great progress in the field of topological quantum computing. The team members cooperated with researchers from the University of Leeds, UK, and used the self-built optical quantum simulator to calculate Jones polynomials based on majorana zero-mode topology. By simulating the weaving operation of majorana zero mode, the team calculated the Jones polynomials corresponding to kinks of different topological structures, and the Jones values obtained can distinguish different kink structures. On December 5, 2024, related research results were published in Physical Review Letters. Jones polynomial is an important topological invariant of kink, which can be used to distinguish different kink structures. At the same time, the Jones polynomial calculation of complex topological structure is difficult to solve by classical algorithm. Using majorana Zero Mode, a non-Abelian arbitrary subsystem, we can calculate the Jones polynomial of kink by constructing the corresponding knitting operation. Different from exchanging two identical bosons or fermions in three-dimensional space, the whole wave function of the system will only have one more whole phase; For the "non-Abelian anyon" with special properties in two-dimensional space, the whole wave function after exchange will undergo a unitary transformation. Therefore, quantum gates can be constructed by exchanging non-Abelian anyons, and topological quantum computation with natural fault tolerance can be realized. The physical properties of majorana zero mode have been studied in previous experiments. However, due to the high requirements of experimental materials and technology, it is quite challenging to realize a specific topological quantum algorithm by weaving majorana zero mode. Based on the quantum simulator of photon space mode, the team carried out a series of experimental studies to simulate the topological characteristics of non-Abelian anyons. On this basis, the team extended the previous encoding method based on single-photon spatial mode to two-photon spatial mode, and used the coincidence counting of two photons to encode, effectively increasing the number of encodable quantum states. At the same time, by introducing a quantum cooling device based on Sagnac interferometer, the dissipative evolution in the previous work is transformed into non-dissipative evolution, which improves the recycling ability of photon resources and helps to realize multi-step quantum evolution operation. These improved experimental techniques have improved the ability to independently develop the optical quantum simulator, and laid a technical foundation for the experimental simulation of the weaving operation of three Kitaiev chain models. In the experiment, the average fidelity of quantum state and weaving exchange process is above 97%. The team simulated five typical topological kinks by combining different weaving operations of three Kitaiev chain models, and obtained the numerical solution of Jones polynomial corresponding to the kink by projecting the quantum final state corresponding to the kink to the initial quantum state, and further distinguished the different kinks. This is instructive to the research fields where topological kinks frequently occur, such as statistical physics, chemical molecular synthesis and DNA replication.

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