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Fortschritte der Physik

Exact Evaluation of Hexagonal Spin‐Networks for Topological Quantum Neural Networks

Matteo Lulli, Antonino Marcianò, Emanuele Zappala

Published Apr 20, 2025• 0 citations
Publisher

Abstract

Abstract The physical scalar product between spin‐networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNNs). These are a class of quantum neural networks supported on graphs and related to topological quantum field theory (TQFT), which have been previously introduced by the authors, recovering deep neural networks (DNNs) as their semiclassical limit. However, the effective evaluation of the scalar product remains an obstacle for the applicability of the theory. Inspired by decimation techniques for the computation of the partition function in statistical mechanics, an analytical technique is introduced for the exact evaluation of hexagonal spin‐networks of arbitrary size, and describe the corresponding algorithm for the evaluation of the physical scalar product defined by Noui and Perez. The transition amplitudes on certain classes of spin‐networks with both classical and quantum recoupling are investigated, obtaining a “continuous” spectrum of the transitions for the former and a discrete one for the latter. The theoretical and computational framework is expected to impact applications in string/tensor‐networks for solid state physics, lattice gauge theories, and quantum gravity approaches.

DOI
10.1002/prop.70005

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Abstract The physical scalar product between spin‐networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNNs). These are a class of quantum neural networks supported on graphs and related to topological quantum field theory (TQFT), which have been previously introduced by the authors, recovering deep neural networks (DNNs) as their semiclassical limit. However, the effective evaluation of the scalar product remains an obstacle for the applicability of the theory. Inspired by decimation techniques for the computation of the partition function in statistical mechanics, an analytical technique is introduced for the exact evaluation of hexagonal spin‐networks of arbitrary size, and describe the corresponding algorithm for the evaluation of the physical scalar product defined by Noui and Perez. The transition amplitudes on certain classes of spin‐networks with both classical and quantum recoupling are investigated, obtaining a “continuous” spectrum of the transitions for the former and a discrete one for the latter. The theoretical and computational framework is expected to impact applications in string/tensor‐networks for solid state physics, lattice gauge theories, and quantum gravity approaches.
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}}
Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories (TQFT); (2) Deep Neural Networks (DNN) are a subcase of QNN, in the sense that they emerge as the semiclassical limit of QNN; (3) A number of Machine Learning (ML) key-concepts can be rephrased by using the terminology of TQFT. Our framework provides as well a working hypothesis for understanding the generalization behavior of DNN, relating it to the topological features of the graphs structures involved.
Deep Neural Networks miss a principled model of their operation. A novel framework for supervised learning based on Topological Quantum Field Theory that looks particularly well suited for implementation on quantum processors has been recently explored. We propose using this framework to understand the problem of generalisation in Deep Neural Networks. More specifically, in this approach, Deep Neural Networks are viewed as the semi-classical limit of Topological Quantum Neural Networks. A framework of this kind explains the overfitting behavior of Deep Neural Networks during the training step and the corresponding generalisation capabilities. We explore the paradigmatic case of the perceptron, which we implement as the semiclassical limit of Topological Quantum Neural Networks. We apply a novel algorithm we developed, showing that it obtains similar results to standard neural networks, but without the need for training (optimisation).
Despite the huge theoretical potential of neural quantum states, their use in describing generic, highly-correlated quantum many-body systems still often poses practical difficulties. Customized network architectures are under active investigation to address these issues. For a guided search of suited network architectures a deepened understanding of the link between neural network properties and attributes of the physical system one is trying to describe, is imperative. Drawing inspiration from the field of machine learning, in this work we show how information propagation in deep neural networks impacts the physical entanglement properties of deep neural quantum states. In fact, we link a previously identified information propagation phase transition of a neural network to a similar transition of entanglement in neural quantum states. With this bridge we can identify optimal neural quantum state hyperparameter regimes for representing area as well as volume law entangled states. The former are easily accessed by alternative methods, such as tensor network representations, at least in low physical dimensions, while the latter are challenging to describe generally due to their extensive quantum entanglement. This advance of our understanding of network configurations for accurate quantum state representation helps to develop effective representations to deal with volume-law quantum states, and we apply these findings to describe the ground state (area law state) vs. the excited state (volume law state) properties of the prototypical next-nearest neighbor spin-1/2 Heisenberg model.