The exact evaluation of hexagonal spin-networks and topological quantum neural networks

TL;DR

This paper presents an efficient algorithm for evaluating the physical scalar product in topological quantum neural networks with hexagonal spin-networks, facilitating their practical application in quantum machine learning.

Contribution

The authors develop a novel, efficient algorithm for computing the scalar product between hexagonal spin-networks, advancing the computational tools for topological quantum neural networks.

Findings

The algorithm effectively computes the scalar product using recoupling theory.

The evaluation method is validated on classical and quantum recoupling cases.

Results are reproducible via the provided 'idea.deploy' framework.

Abstract

The physical scalar product between spin-networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNN), which are quantum neural networks previously introduced by the authors in the context of quantum machine learning. However, the effective evaluation of the scalar product remains a bottleneck for the applicability of the theory. We introduce an algorithm for the evaluation of the physical scalar product defined by Noui and Perez between spin-network with hexagonal shape. By means of recoupling theory and the properties of the Haar integration we obtain an efficient algorithm, and provide several proofs regarding the main steps. We investigate the behavior of the TQNN evaluations on certain classes of spin-networks with the classical and quantum recoupling. All results can be independently reproduced through the "idea.deploy" framework~\href{https://github.com/lullimat/idea.deploy}{\nolinkurl{https://github.com/lullimat/idea.deploy}}