Learning Unitaries by Gradient Descent

TL;DR

This paper investigates the effectiveness of gradient descent in learning unitary transformations, revealing a phase transition at $d^2$ parameters where convergence behavior changes significantly.

Contribution

It provides numerical evidence that gradient descent reliably learns unitaries when the sequence has at least $d^2$ parameters, highlighting a computational phase transition.

Findings

Gradient descent converges to the target unitary with $d^2$ or more parameters.

Below $d^2$ parameters, convergence is sub-optimal.

Above $d^2$ parameters, convergence is exponential and optimal.

Abstract

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.