  Operator controllability refers to the ability to implement an arbitrary
unitary in SU(N) and is a prerequisite for universal quantum computing.
Controllability tests can be used in the design of quantum devices to reduce
the number of external controls. Their practical use is hampered, however, by
the exponential scaling of their numerical effort with the number of qubits.
Here, we devise a hybrid quantum-classical algorithm based on a parametrized
quantum circuit. We show that controllability is linked to the number of
independent parameters, which can be obtained by dimensional expressivity
analysis. We exemplify the application of the algorithm to qubit arrays with
nearest-neighbour couplings and local controls. Our work provides a systematic
approach to the resource-efficient design of quantum chips.
