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

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