Estimating the Jones polynomial for Ising anyons on noisy quantum computers

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 stabilizer 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 stabilizer circuits. To estimate the real and imaginary parts of the amplitudes up to additive error we use the Hadamard test, yielding non-Clifford circuits that nevertheless we can always efficiently compute the correct output of. Hence, we further argue that this setup defines a standard benchmark for near-term noisy quantum processors. Additionally, we study the benefit of performing quantum error mitigation with the method of zero-noise extrapolation.