Regressions on quantum neural networks at maximal expressivity

Considering a universal deep neural network organized as a series of nested qubit rotations, accomplished by adjustable data re-uploads we analyze its expressivity. This ability to approximate continuous functions in regression tasks is quantified making use of a partial Fourier decomposition of the generated output and systematically benchmarked with the aid of a teacher-student scheme. 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. However, we show that the measurement of the network generated output drastically modifies the attainability of this bound. Global-entangling measurements on the network can saturate the maximal expressive bound leading 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.