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Table of Contents
Abstract
1 Introduction
2 Quantum neural networks
3 Information geometry, effective dimension, and trainability of quantum neural networks
3.1 The Fisher information
3.2 The effective dimension
3.3 Generalisation error bounds
Summary
Fault-tolerant quantum computers offer the promise of dramatically improving machine learning through speed-ups in computation or improved model scalability. In the near-term, however, the benefits of quantum machine learning are not so clear. Understanding expressibility and trainability of quantum models—and quantum neural networks in particular—requires further investigation. In this work, we use tools from information geometry to define a notion of expressibility for quantum and classical models. The effective dimension, which depends on the Fisher information, is used to prove a novel generalization bound and establish a robust measure of expressibility. We show that quantum neural networks are able to achieve a significantly better effective dimension than comparable classical neural networks. To then assess the trainability of quantum models, we connect the Fisher information spectrum to barren plateaus, the problem of vanishing gradients...