On the Complexity of Random Quantum Computations and the Jones Polynomial

There is a natural relationship between Jones polynomials and quantum computation. We use this relationship to show that the complexity of evaluating relative-error approximations of Jones polynomials can be used to bound the classical complexity of approximately simulating random quantum computations. We prove that random quantum computations cannot be classically simulated up to a constant total variation distance, under the assumption that (1) the Polynomial Hierarchy does not collapse and (2) the average-case complexity of relative-error approximations of the Jones polynomial matches the worst-case complexity over a constant fraction of random links. Our results provide a straightforward relationship between the approximation of Jones polynomials and the complexity of random quantum computations.