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}}
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}}
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.
Quantum computing is expected to fundamentally change computer systems in the future. Recently, a new research topic of quantum computing is the hybrid quantum–classical approach for machine learning, in which a parameterized quantum circuit, also called quantum neural network (QNN), is optimized by a classical computer. This hybrid approach can have the benefits of both quantum computing and classical machine learning methods. In this early stage, it is of crucial importance to understand the new characteristics of quantum neural networks for different machine learning tasks. In this paper, we will study quantum neural networks for the task of classifying images, which are high-dimensional spatial data. In contrast to previous evaluations of low-dimensional or scalar data, we will investigate the impacts of practical encoding types, circuit depth, bias term, and readout on classification performance on the popular MNIST image dataset. Various interesting findings on learning behaviors of different QNNs are obtained through experimental results. To the best of our knowledge, this is the first work that considers various QNN aspects for image data.
This paper presents the concept of quantum neural networks (QNN), a new framework that combines quantum computing with topological data analysis to process data defined on simplicial complexes. Based on the fundamentals of topological signal processing and variational quantum circuits, QNN aim to use the structure of simplicial complexes to perform efficient calculations and extract higher-order relational patterns from data. The proposed architecture uses quantum layers (QL), which map interactions between simplices into quantum circuits, providing both in-order and inter-order interactions. The paper describes two QL variants: basic and schematic, providing flexibility for applications with varying complexity. In addition, the paper explores the theoretical foundations of simplicial complexes, quantum neural networks, and the interaction between classical and quantum approaches in data encoding, state evolution and optimization. By leveraging quantum algorithms and eliminating current hardware limitations, QNN promise scalability and enhanced performance for applications in optimization, numerical simulation, and complex systems modeling. This research demonstrates the potential of QNN to advance the field of quantum machine learning, paving the way for future developments of quantum algorithms and their practical applications.
This paper introduces a novel type of Graph Neural Network (GNN) called Topological Quantum Graph Neural Networks (TQNN). TQNN leverages the inherent topological structure of graphs and the strengths of Graph Neural Networks (GNNs) to enhance model recognition and prediction capabilities. We propose a new architecture that utilizes a modified graph representation and a quantum-inspired transformation to improve the network's ability to capture complex relationships and patterns within data. The core mechanism involves incorporating a quantum-inspired transformation into the graph representation, which enables the network to represent and process information in a fundamentally different way, leading to significant improvements in performance across various tasks. This work presents a new approach to GNNs that addresses limitations of existing methods, offering a promising pathway for enhanced machine learning performance.
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.
Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge.The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network.When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of nonlocal properties.Here, we introduce quantum loop topography (QLT): a procedure of constructing a multidimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps.The loop configuration is guided by the characteristic response for defining the phase, which is Hall conductivity for the cases at hand.Feeding QLT to a fully connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish the Chern insulator and the fractional Chern insulator from trivial insulators with high fidelity.In addition to establishing the first case of obtaining a phase diagram with a topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.