  Jones polynomials were introduced as a tool to distinguish between
topologically different links. Recently, they emerged as the central building
block of topological quantum computation: by braiding non-Abelian anyons it is
possible to realise quantum algorithms through the computation of Jones
polynomials. So far, it has been a formidable task to evaluate Jones
polynomials through the control and manipulation of non-Abelian anyons. In this
study, a photonic quantum system employing two-photon correlations and
non-dissipative 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 at a
particular value of their parameter. 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.
