Exploring Channel Distinguishability in Local Neighborhoods of the Model Space in Quantum Neural Networks
Sabrina Herbst, Sandeep Suresh Cranganore, Vincenzo De Maio, Ivona Brandic
TL;DR
The paper tackles the trainability challenge of quantum neural networks by rethinking ansatz expressivity and focusing on local model-space behavior under small parameter updates. It introduces a channel-distinguishability bound, $c_U(\bm{\vartheta},\bm{\delta})=\|U(\bm{\vartheta})-U(\bm{\vartheta}+\bm{\delta})\|_\Diamond$, and shows $c_U(\bm{\vartheta},\bm{\delta}) \le \tfrac{1}{2}\sum_j |\delta_j|$, linking distinguishability to perturbation magnitude and the number of parameters. Numerical experiments with hardware-efficient architectures verify the bound and reveal significant variability across neighborhoods, with many early training updates yielding hardly distinguishable channels. The results suggest that architecture-driven trainability issues can arise even independent of data and loss, emphasizing the need for warm-starting and smarter initialization, and hinting at a broader paradigm shift in variational quantum computing for near-term devices. Overall, the work provides an ansatz-centered perspective on QNN training dynamics and highlights fundamental limitations imposed by local model-space properties.
Abstract
With the increasing interest in Quantum Machine Learning, Quantum Neural Networks (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 measure, fails at capturing this property. Hence, we look for alternative ways to characterize ansatzes 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 with few parameters 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.