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 \(d^2\) or more parameters. Rates of convergence indicate a "computational phase transition." With less than \(d^2\) parameters, gradient descent converges to a sub-optimal solution, whereas with more than \(d^2\) parameters, gradient descent converges exponentially to an optimal solution.