[Paper Review] Learning Unitaries by Gradient Descent

Bobak T. Kiani, Seth Lloyd+1 more|arXiv (Cornell University)|Jan 31, 2020

Quantum Computing Algorithms and Architecture25 references33 citations

TL;DR

The paper shows that gradient descent on alternating operator sequences can learn any Haar random unitary in U(d) when the sequence has at least d^2 parameters, revealing a computational phase transition at the critical parameter count.

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.

Motivation & Objective

• Assess the hardness of learning Haar random unitaries in U(d) using gradient descent on time-parameterized alternating operator sequences.
• Investigate how the number of parameters in the sequence affects convergence and identify a phase transition between under- and over-parameterized regimes.
• Examine learning both arbitrary and shallow-depth unitaries and characterize landscape behavior and convergence rates.

Proposed method

• Model unitaries as V(t,τ) = e^{-i A t_K} e^{-i B τ_K} ... e^{-i A t_1} e^{-i B τ_1} with parameters t_i, τ_i.
• Treat A, B as Gaussian Unitary Ensemble (GUE) random matrices to ensure controllability of the system.
• Formulate the learning problem as minimizing L(t,τ) = ||U − V(t,τ)||^2 using Frobenius norm.
• Perform gradient descent on all 2K parameters and analyze convergence for varying parameter counts (2K < d^2, 2K = d^2, 2K > d^2).
• Explore both arbitrary target unitaries and shallow-depth targets to assess learnability across depths.

Experimental results

Research questions

• RQ1 Does gradient descent on an alternating operator sequence reliably recover a Haar random target unitary in U(d) as a function of the number of parameters?
• RQ2 Is there a computational phase transition in convergence behavior at the critical parameter count 2K = d^2?
• RQ3 Can gradient descent learn shallow-depth unitaries, and if so, what parameter budget is required relative to d^2?
• RQ4 How do convergence rates differ between under-parameterized, critically parameterized, and over-parameterized regimes?
• RQ5 What landscape characteristics (local minima, saddle points) influence learnability for these quantum control tasks?

Key findings

• Gradient descent converges to the target unitary when the parameter count is at least d^2.
• Under-parameterized cases (2K < d^2) typically converge to a sub-optimal loss plateau.
• At the critical point 2K = d^2, learning exhibits a power-law convergence with slower rates.
• Over-parameterized cases (2K > d^2) show exponential convergence near the global minimum, consistent with a quadratic loss landscape.
• Learning shallow-depth unitaries generally requires 2K ≥ d^2 (or K ≥ d^2/2) parameters, indicating difficulty of low-depth learning with gradient descent.
• Loss landscapes are highly non-convex with traps when targeting shallow unitaries with insufficient parameters, yet become globally convergent in over-parameterized regimes.