Connecting channel expressiveness to gradient magnitudes and noise induced barren plateaus

The expressivity of parametrized quantum circuits has been directly linked to the presence of barren plateaus, so that more expressive circuits are likely to exhibit smaller gradients. Yet, expressiveness measures proposed in the literature are only applicable for noiseless circuits, as they compare moments of ensembles of unitaries. However, near-term hardware is noisy, meaning that current measures are not directly applicable to study the expressiveness of the more general set of completely-positive trace-preserving linear maps that arise in the presence of noise. In this work we propose several measures of expressiveness for quantum channels and study their properties, highlighting how average non-unitary channels differ from average unitary channels. Furthermore, we rigorously prove that highly-expressive noisy quantum circuits will suffer from exponential concentration, thus showing that the noise induced-barren plateau phenomenon is essentially an (channel) expressiveness induced barren plateau. We complement our analytical findings with numerical experiments that showcase that, in certain situations, noise can increase the expressiveness of parametrized quantum circuits.