Exploring channel distinguishability in local neighborhoods of the model space in quantum neural networks

With the increasing interest in Quantum Machine Learning, Quantum Neural Net-
works (QNNs) have emerged and gained significant attention. These models have,
however, been shown to be notoriously difficult to train, which we hypothesize is
partially due to the architectures, called ansatzes, that are hardly studied at this
point. Therefore, in this paper, we take a step back and analyze ansatzes. We
initially consider their expressivity, i.e., the space of operations they are able to
express, and show that the closeness to being a 2-design, the primarily used mea-
sure, fails at capturing this property. Hence, we look for alternative ways to char-
acterize ansatzes, unrelated to expressivity, by considering the local neighborhood
of the model space, in particular, analyzing model distinguishability upon small
perturbation of parameters. We derive an upper bound on their distinguishability,
showcasing that QNNs using the Hardware Efficient Ansatz with few parame-
ters are hardly discriminable upon update. Our numerical experiments support
our bounds and further indicate that there is a significant degree of variability,
which stresses the need for warm-starting or clever initialization. Altogether, our
work provides an ansatz-centric perspective on training dynamics and difficulties
in QNNs, ultimately suggesting that iterative training of small quantum models
may not be effective, which contrasts their initial motivation.