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We analyze the expressivity of a universal deep neural network that can beorganized as a series of nested qubit rotations, accomplished by adjustabledata re-uploads. While the maximal expressive power increases with the depth ofthe network and the number of qubits, it is fundamentally bounded by the dataencoding mechanism. Focusing on regression problems, we systematicallyinvestigate the expressivity limits for different measurements andarchitectures. The presence of entanglement, either by entangling layers orglobal measurements, saturate towards this bound. In these cases, entanglementleads to an enhancement of the approximation capabilities of the networkcompared to local readouts of the individual qubits in non-entangling networks.We attribute this enhancement to a larger survival set of Fourier harmonicswhen decomposing the output signal.

Regressions on quantum neural networks at maximal expressivity