Topological Data Analysis (TDA) is a powerful but computationally expensive tool for data analysis. Quantum Topological Data Analysis (QTDA) promises to reduce this cost while giving comparable results. This paper explores using QTDA for quantum machine learning (QML) and provides a pipeline for QML where QTDA's quantum state output is fed to the QML module. We also show how reduced Hamiltonians can be used to recover most of the performance for machine learning while using much fewer resources in terms of the number of qubits, depth, and gate counts.
Topological Data Analysis (TDA) is a powerful but computationally expensive tool for data analysis. Quantum Topological Data Analysis (QTDA) promises to reduce this cost while giving comparable results. This paper explores using QTDA for quantum machine learning (QML) and provides a pipeline for QML where QTDA's quantum state output is fed to the QML module. We also show how reduced Hamiltonians can be used to recover most of the performance for machine learning while using much fewer resources in terms of the number of qubits, depth, and gate counts.
Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical algorithms for computing TDA are exorbitant, and quickly become impractical for high-order characteristics. Quantum computers offer the potential of achieving significant speedup for certain computational problems. Indeed, TDA has been purported to be one such problem, yet, quantum computing algorithms proposed for the problem, such as the original Quantum TDA (QTDA) formulation by Lloyd, Garnerone and Zanardi, require fault-tolerance qualifications that are currently unavailable. In this study, we present NISQ-TDA, a fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise.
This paper investigates the integration of Topological Data Analysis (TDA) with quantum algorithms to enhance feature extraction processes. Traditional feature extraction methods often struggle with noisy data and limited representation, while TDA offers a powerful framework for identifying inherent patterns and structures within complex datasets. We propose a novel approach that leverages the topological properties of data to generate robust and informative features, ultimately improving the performance of machine learning models. The paper explores the potential of quantum algorithms, specifically Quantum Feature Extraction (QFE), to accelerate and refine this process. We present a preliminary implementation demonstrating the efficacy of this combined technique, showcasing improved accuracy in a simulated dataset. The research explores the benefits of using quantum algorithms to discover hidden patterns that classical methods miss, offering a pathway toward more accurate and resilient machine learning.
There is heightened interest in quantum algorithms for Topological Data Analysis (TDA) as it is a powerful tool for data analysis, but it can get highly computationally expensive. Even though there are different propositions and observations for Quantum Topological Data Analysis (QTDA), the necessary details to implement them on software platforms are lacking. Towards closing this gap, the present paper presents an implementation of one such algorithm for calculating Betti numbers.The step-by-step instructions for the chosen quantum algorithm and the aspects of how it can be used for machine learning tasks are provided. We provide encouraging results on using Betti numbers for classification and give a preliminary analysis of the effect of the number of shots and precision qubits on the outcome of the quantum algorithm.
Quantum-driven Topological Data Analysis (Q-TDA) offers a fundamentally different approach to analyzing sparse data compared to traditional methods. This paper investigates the potential of leveraging quantum mechanics to identify patterns and relationships within these datasets, with the goal of significantly reducing the computational burden associated with data preprocessing and enhancing the detection of hidden structure. We introduce a novel quantum algorithm designed to efficiently explore topological spaces, which are inherently structured and provide a powerful framework for understanding data. This work explores the theoretical foundations and preliminary implementation of this quantum approach, highlighting its potential to revolutionize the field of topological data analysis. The core claim is that Q-TDA can achieve superior performance and efficiency for sparse data, offering a paradigm shift in the analysis of complex datasets.
Quantum computing offers the potential of exponential speedups for certain classical computations. Over the last decade, many quantum machine learning (QML) algorithms have been proposed as candidates for such exponential improvements. However, two issues unravel the hope of exponential speedup for some of these QML algorithms: the data-loading problem and, more recently, the stunning dequantization results of Tang et al. A third issue, namely the fault-tolerance requirements of most QML algorithms, has further hindered their practical realization. The quantum topological data analysis (QTDA) algorithm of Lloyd, Garnerone and Zanardi was one of the first QML algorithms that convincingly offered an expected exponential speedup. From the outset, it did not suffer from the data-loading problem. A recent result has also shown that the generalized problem solved by this algorithm is likely classically intractable, and would therefore be immune to any dequantization efforts. However, the QTDA algorithm of Lloyd et~al. has a time complexity of O(n4/(ε2δ)) (where n is the number of data points, ε is the error tolerance, and δ is the smallest nonzero eigenvalue of the restricted Laplacian) and requires fault-tolerant quantum computing, which has not yet been achieved. In this paper, we completely overhaul the QTDA algorithm to achieve an improved exponential speedup and depth complexity of O(nlog(1/(δε))). Our approach includes three key innovations: (a) an efficient realization of the combinatorial Laplacian as a sum of Pauli operators; (b) a quantum rejection sampling approach to restrict the superposition to the simplices in the complex; and (c) a stochastic rank estimation method to estimate the Betti numbers. We present a theoretical error analysis, and the circuit and computational time and depth complexities for Betti number estimation.
Topological data analysis (TDA) provides a powerful framework for extracting information about the shape of complex, unstructured data, but the classical cost of computing high dimensional topological features limits its application. Quantum algorithms for TDA offer a route around this bottleneck, yet existing approaches typically focus on exact or high precision Betti number estimation, making the regime for practical quantum advantage appear narrow. Here, we instead frame quantum TDA as a feature-extraction method for downstream data analysis by extracting low-order spectral information from the combinatorial Laplacian as a proxy for high-dimensional topology.
We support this perspective from both the application and algorithmic sides. First, we show that higher-order TDA features improve predictive performance in two time-series applications: functional MRI analysis for neurodegenerative disease classification and financial time-series analysis for identifying market instability. Second, we develop a moment-based quantum algorithm and show that low-order moments, including the relative trace, are strongly correlated with high-dimensional Betti information, even when the relative Betti number is small. Finally, we present circuit constructions, resource estimates, quantum-classical crossover projections, and experimental results from a Barium development system similar to the forthcoming IonQ Tempo line, extracting Laplacian-derived observables from graph instances and quantitatively comparing them with exact Betti information.
Together, these results establish quantum TDA as a practical approach for extracting topological features from classically challenging data