Chaos and Complexity from Quantum Neural Network: A study with Diffusion Metric in Machine Learning

TL;DR

This paper investigates quantum chaos and complexity in Quantum Neural Networks using diffusion metrics, establishing bounds on generalization and linking system dynamics to learning stability.

Contribution

It introduces a geometric framework for analyzing QNN dynamics, defining quantum chaos and complexity in terms of physically relevant quantities, and relates these to generalization bounds.

Findings

Limit cycles enhance QNN generalization capability.

Quantum chaos correlates with stability and learning performance.

Derived bounds on parameter variance in steady state.

Abstract

In this work, our prime objective is to study the phenomena of quantum chaos and complexity in the machine learning dynamics of Quantum Neural Network (QNN). A Parameterized Quantum Circuits (PQCs) in the hybrid quantum-classical framework is introduced as a universal function approximator to perform optimization with Stochastic Gradient Descent (SGD). We employ a statistical and differential geometric approach to study the learning theory of QNN. The evolution of parametrized unitary operators is correlated with the trajectory of parameters in the Diffusion metric. We establish the parametrized version of Quantum Complexity and Quantum Chaos in terms of physically relevant quantities, which are not only essential in determining the stability, but also essential in providing a very significant lower bound to the generalization capability of QNN. We explicitly prove that when the system executes limit cycles or oscillations in the phase space, the generalization capability of QNN is maximized. Finally, we have determined the generalization capability bound on the variance of parameters of the QNN in a steady state condition using Cauchy Schwartz Inequality.