  We analyze the expressivity of a universal deep neural network that can be
organized as a series of nested qubit rotations, accomplished by adjustable
data re-uploads. While the maximal expressive power increases with the depth of
the network and the number of qubits, it is fundamentally bounded by the data
encoding mechanism. Focusing on regression problems, we systematically
investigate the expressivity limits for different measurements and
architectures. The presence of entanglement, either by entangling layers or
global measurements, saturate towards this bound. In these cases, entanglement
leads to an enhancement of the approximation capabilities of the network
compared to local readouts of the individual qubits in non-entangling networks.
We attribute this enhancement to a larger survival set of Fourier harmonics
when decomposing the output signal.
