  Gradient descent method, as one of the major methods in numerical
optimization, is the key ingredient in many machine learning algorithms. As one
of the most fundamental way to solve the optimization problems, it promises the
function value to move along the direction of steepest descent. For the vast
resource consumption when dealing with high-dimensional problems, a quantum
version of this iterative optimization algorithm has been proposed
recently[arXiv:1612.01789]. Here, we develop this protocol and implement it on
a quantum simulator with limited resource. Moreover, a prototypical experiment
was shown with a 4-qubit Nuclear Magnetic Resonance quantum processor,
demonstrating a optimization process of polynomial function iteratively. In
each iteration, we achieved an average fidelity of 94\% compared with
theoretical calculation via full-state tomography. In particular, the iterative
point gradually converged to the local minimum. We apply our method to
multidimensional scaling problem, further showing the potentially capability to
yields an exponentially improvement compared with classical counterparts. With
the onrushing tendency of quantum information, our work could provide a
subroutine for the application of future practical quantum computers.
