Learning Unitaries by Gradient Descent
This addresses the challenge of optimizing non-convex landscapes in quantum computing, providing insights into parameter requirements for efficient learning, though it is incremental as it builds on existing gradient descent methods.
The paper tackles the problem of learning unitary transformations via gradient descent on time parameters, finding that convergence to the target unitary occurs with at least d^2 parameters, while fewer parameters lead to sub-optimal solutions, with rates indicating a computational phase transition.
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.