  The evaluation of the Jones polynomial at roots of unity is a paradigmatic
problem for quantum computers. In this work we present experimental results
obtained from existing noisy quantum computers for special cases of this
problem, where it is classically tractable. Our approach relies on the
reduction of the problem of evaluating the Jones polynomial of a knot at
lattice roots of unity to the problem of computing quantum amplitudes of qudit
stabiliser circuits, which are classically efficiently simulatable. More
specifically, we focus on evaluation at the fourth root of unity, which is a
lattice root of unity, where the problem reduces to evaluating amplitudes of
qubit stabiliser circuits. To estimate the real and imaginary parts of the
amplitudes up to additive error we use the Hadamard test. We further argue that
this setup defines a standard benchmark for near-term noisy quantum processors.
Furthermore, we study the benefit of performing quantum error mitigation with
the method of zero noise extrapolation.
