Exploring Channel Distinguishability in Local Neighborhoods of the Model Space in Quantum Neural Networks

TL;DR

This paper analyzes the local distinguishability of quantum neural network ansatzes, revealing that models with few parameters are hard to differentiate after small updates, impacting training strategies.

Contribution

It introduces a new approach to characterize ansatz expressivity through local neighborhood analysis, challenging the adequacy of 2-design closeness as a measure.

Findings

Few-parameter QNNs are difficult to distinguish after small perturbations.

Numerical results support the derived upper bounds on model distinguishability.

Variability in model behavior suggests the importance of initialization strategies.

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.