Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.
Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.
This paper explores the application of quantum machine learning to the prediction of complex topological structures. Traditional machine learning methods often struggle with the inherent complexity and high dimensionality of topological data, requiring extensive feature engineering and potentially leading to suboptimal model performance. Quantum machine learning offers the potential to address these challenges by leveraging quantum phenomena to efficiently represent and process information, enabling the prediction of intricate topological landscapes. We propose a novel quantum algorithm designed to learn a predictive model based on the underlying topological structure, incorporating this information to enhance predictive accuracy. The core mechanism focuses on utilizing quantum state manipulation to capture the essential relationships within the topology, resulting in a more robust and accurate prediction. The research investigates the potential of this approach to significantly improve the prediction of topological structures, particularly for complex and high-dimensional datasets.
This paper explores the potential of quantum machine learning to enhance the generalization and robustness of machine learning models through the application of dynamic topological structure learning. We investigate how quantum algorithms can be utilized to implicitly learn and exploit the inherent topology of quantum systems, offering a novel approach to model representation and learning. The core claim is that this method, by leveraging quantum mechanics, can significantly improve model performance compared to traditional machine learning techniques. This work introduces a framework that dynamically generates and manipulates topological structures, facilitating a more robust and adaptable learning process. The research presents a conceptual framework for quantum-enhanced topological learning and highlights the potential benefits in areas requiring high-dimensional data and complex relationships. The paper details the mechanism, explores preliminary results, and concludes with future directions and implications of this approach.
A potential advantage of quantum machine learning stems from the ability of encoding classical data into high dimensional complex Hilbert space using quantum circuits. Recent studies exhibit that not all encoding methods are the same when representing classical data since certain parameterized circuit structures are more expressive than the others. In this study, we show the difference in encoding techniques can be visualized by investigating the topology of the data embedded in complex Hilbert space. The technique for visualization is a hybrid quantum based topological analysis which uses simple diagonalization of the boundary operators to compute the persistent Betti numbers and the persistent homology graph. To augment the computation of Betti numbers within a NISQ framework, we suggest a simple hybrid algorithm. Through an illuminating example of a synthetic data set and the methods of angle encoding, amplitude encoding, and IQP encoding, we reveal topological differences between the encoding methods, as well as differences with the original data. Consequently, our results suggest the encoding method needs to be considered carefully within different quantum machine learning models since it can strongly affect downstream analysis like clustering or classification.
This paper introduces the Topological Quantum Machine Learning Algorithm (TQMA), a novel approach to machine learning that leverages the inherent properties of topology and quantum mechanics to enhance model identification and prediction capabilities. Traditional machine learning methods often rely on extensive parameter tuning, hindering model adaptability and robustness. TQMA overcomes this limitation by incorporating a dynamically evolving, topologically structured representation within the model, enabling it to better capture complex relationships within data. We present a framework that integrates a quantum-inspired optimization process within a topological graph, allowing for exploration of intricate feature spaces and improved generalization. The algorithm's core mechanism centers around the construction and application of a quantum-enhanced topological graph, which facilitates the identification of subtle patterns and correlations often missed by conventional methods. We demonstrate the efficacy of TQMA through a series of benchmark datasets, showcasing significant improvements in classification accuracy and predictive performance compared to existing state-of-the-art techniques. This paper provides a comprehensive investigation into the potential of TQMA as a paradigm shift in machine learning, offering a pathway towards more intelligent and adaptable models.
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.
When it comes to realizing immediate quantum advantages in practical applications like machine learning and optimization techniques, numerous methods have emerged as a frontrunner technique. Such algorithms often involve quantum circuits for data encoding and the training of Quantum Neural Networks (QNNs) to minimize target functions when applied to tasks involving classical data. Quantum Machine Learning (QML) is a promising area of quantum computing because it may effectively handle difficult learning tasks by taking advantage of the high dimensional Hilbert space to train better representations from limited data. Despite QML's rising popularity, few academic works have addressed the language's security features. There has been a lot of buzz about the promising new discipline of quantum machine learning. Training a parameterized quantum circuit is a common practice in contemporary QML for performing data analysis on classical or quantum datasets. To make accurate predictions on an independent test dataset, state-of-the-art QML techniques first variationally optimize a parameterized quantum circuit using the training dataset. Quantum modeling in QML has also been shown to offer an exponential benefit in sample complexity. In the realms of pattern recognition, machine learning, and data mining, feature dimensionality reduction as a crucial link in the process has become one hot and tough spot. Most academics have gravitated towards this topic because of the difficulty of its studies. The goal of this research is to learn how to implement low loss in the procedure of feature dimension reduction, preserve the integrity of the original data, locate the most effective mapping, and obtain the best possible low-dimensional data. Preliminary findings suggest that QML models may offer advantages over classical models for classical data analysis. In this paper, a thorough investigation into generalization performance in QML is performed. This paper presents a brief survey on Qubits Encoding using Quantum Machine Learning models that help researchers to identify the limitations of the existing models and to design new efficient models for better performance levels.