The Jones polynomial, discovered in 1984, is an important knot invariant
in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient (namely, polynomial) quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, __\( e^{2\pi i/5} \)__, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et al. are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.

We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an __\( n \)__ strands braid with __\( m \)__ crossings at any primitive root of unity __\( e^{2\pi i/k} \)__, where the running time of the algorithm is polynomial in __\( m \)__, __\( n \)__ and __\( k \)__. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperley–Lieb algebra). By the results of Freedman et al., our algorithm solves a BQP complete problem.

Our algorithm works by encoding the local structure of the problem into the local
unitary gates which are applied by the circuit. This structure is significantly different
from previous quantum algorithms, which are mostly based on the Quantum Fourier transform. Since the results of the current paper were presented in their preliminary form, these ideas have been extended and generalized in several interesting directions. Most notably, Aharonov, Arad, Eban and Landau give a simplification and extension of these results that provides additive approximations for all points of the Tutte polynomial, including the Jones polynomial at any point, and the Potts model partition function at any temperature and any set of coupling strengths. We hope and believe that the ideas presented in this work will have other extensions and generalizations.