Learning Unitaries by Gradient Descent

We study the hardness of learning unitary transformations in U (d) via gradient descent on time parameters of alternating operator sequences. We provide numerical evidence that, despite the non-convex nature of the loss landscape, gradient descent always converges to the target unitary when the sequence contains d² or more parameters. Rates of convergence indicate a "computational phase transition. " With less than d² parameters, gradient descent converges to a sub-optimal solution, whereas with more than d² parameters, gradient descent converges exponentially to an optimal solution.