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.
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).
Graph Neural Networks (GNNs) excel at learning from graph-structured data but are limited to modeling pairwise interactions, insufficient for capturing higher-order relationships present in many real-world systems. Topological Deep Learning (TDL) has allowed for systematic modeling of hierarchical higher-order interactions by relying on combinatorial topological spaces such as simplicial complexes. In parallel, Quantum Neural Networks (QNNs) have been introduced to leverage quantum mechanics for enhanced computational and learning power. In this work, we present the first Quantum Topological Deep Learning Model: Quantum Simplicial Networks (QSNs), being QNNs operating on simplicial complexes. QSNs are a stack of Quantum Simplicial Layers, which are inspired by the Ising model to encode higher-order structures into quantum states. Experiments on synthetic classification tasks show that QSNs can outperform classical simplicial TDL models in accuracy and efficiency, demonstrating the potential of combining quantum computing with TDL for processing data on combinatorial topological spaces.
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.
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 machine learning (QML) is promising for potential speedups and improvements in conventional machine learning (ML) tasks (e.g., classification/regression). The search for ideal QML models is an active research field. This includes identification of efficient classical-to-quantum data encoding scheme, construction of parametric quantum circuits (PQC) with optimal expressivity and entanglement capability, and efficient output decoding scheme to minimize the required number of measurements, to name a few. However, most of the empirical/numerical studies lack a clear path towards scalability. Any potential benefit observed in a simulated environment may diminish in practical applications due to the limitations of noisy quantum hardware (e.g., under decoherence, gate-errors, and crosstalk). We present a scalable quantum-classical hybrid deep neural network (DeepQMLP) architecture inspired by classical deep neural network architectures. In DeepQMLP, stacked shallow Quantum Neural Network (QNN) models mimic the hidden layers of a classical feed-forward multi-layer perceptron network. Each QNN layer produces a new and potentially rich representation of the input data for the next layer. This new representation can be tuned by the parameters of the circuit. Shallow QNN models experience less decoherence, gate errors, etc. which make them (and the network) more resilient to quantum noise. We present numerical studies on a variety of classification problems to show the trainability of DeepQMLP. We also show that DeepQMLP performs reasonably well on unseen data and exhibits greater resilience to noise over QNN models that use a deep quantum circuit. DeepQMLP provided up to 25.3% lower loss and 7.92% higher accuracy during inference under noise than QMLP.
Quantum machine learning models based on parametrized quantum circuits, also called quantum neural networks (QNNs), are considered to be among the most promising candidates for applications on near-term quantum devices. Here we explore the expressivity and inductive bias of QNNs by exploiting an exact mapping from QNNs with inputs x to classical perceptrons acting on x⊗x (generalised to complex inputs). The simplicity of the perceptron architecture allows us to provide clear examples of the shortcomings of current QNN models, and the many barriers they face to becoming useful general-purpose learning algorithms. For example, a QNN with amplitude encoding cannot express the Boolean parity function for n≥3, which is but one of an exponential number of data structures that such a QNN is unable to express. Mapping a QNN to a classical perceptron simplifies training, allowing us to systematically study the inductive biases of other, more expressive embeddings on Boolean data. Several popular embeddings primarily produce an inductive bias towards functions with low class balance, reducing their generalisation performance compared to deep neural network architectures which exhibit much richer inductive biases. We explore two alternate strategies that move beyond standard QNNs. In the first, we use a QNN to help generate a classical DNN-inspired kernel. In the second we draw an analogy to the hierarchical structure of deep neural networks and construct a layered non-linear QNN that is provably fully expressive on Boolean data, while also exhibiting a richer inductive bias than simple QNNs. Finally, we discuss characteristics of the QNN literature that may obscure how hard it is to achieve quantum advantage over deep learning algorithms on classical data.