  The simulation of quantum dynamics on a digital quantum computer with
parameterized circuits has widespread applications in fundamental and applied
physics and chemistry. In this context, using the hybrid quantum-classical
algorithm, combining classical optimizers and quantum computers, is a
competitive strategy for solving specific problems. We put forward its use for
optimal quantum control. We simulate the wave-packet expansion of a trapped
quantum particle on a quantum device with a finite number of qubits. We then
use circuit learning based on gradient descent to work out the intrinsic
connection between the control phase transition and the quantum speed limit
imposed by unitary dynamics. We further discuss the robustness of our method
against errors and demonstrate the absence of barren plateaus in the circuit.
The combination of digital quantum simulation and hybrid circuit learning opens
up new prospects for quantum optimal control.
