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arXiv:2607.27206

Practical Quantum Topological Data Analysis with Applications to High-Dimensional Feature Extraction and Time Series Analysis

Jason Iaconis, Sayonee Ray, Samwel Sekwao, Claudio Girotto, Martin Roetteler

quant-phSubmitted Jul 29, 2026• Announced Jul 30, 2026• 0 citations
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Abstract

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

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38 pages, 24 figures

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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.
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Together, these results establish quantum TDA as a practical approach for extracting topological features from classically challenging data
Prior work has explored quantum algorithms for topological data analysis (TDA), revealing the possibility of exponential quantum speedups in estimating the ratios of Betti numbers to the dimension of the combinatorial Laplacian. However, this quantity is only non-vanishing and efficient-to-quantumly-estimate when Betti numbers are exponentially large, a case for which concrete examples are rarely known. Furthermore, certain randomized classical algorithms are sometimes efficient in this regime. Thus, the prospect of achieving quantum advantage in conventional TDA appears fairly narrow. Here, we address these challenges to the quantum advantage in TDA by developing quantum algorithms for Mayer homology, which generalize simplicial homology to NNN-nilpotent boundary operators (∂N=0\partial^N =0∂N=0) and have recently been successfully applied to real-world TDA contexts. We introduce an efficient quantum algorithm for estimating Mayer Betti numbers and their persistent counterparts. We then prove that for high-order simplices, Mayer Betti numbers are often exponentially large in the dense regime, which ameliorates the normalization bottleneck of conventional quantum TDA. In the same regime, we argue that existing dequantization algorithms developed for conventional TDA, when applied to Mayer homology, generally lose theoretical guaranties, facing certain structural barriers that prevent their practical utilities. We also provide logical resource estimates revealing that a quantum computer with roughly a few hundred qubits and sixty million Toffoli gates could solve Mayer homology problems beyond the capabilities of known classical approaches. Finally, we discuss real-world applications of Mayer homology in genomics, supersymmetry, drug discovery, and neuroscience, revealing the potential of our quantum algorithm to deliver real-world impacts via Mayer homology.
Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. TDA enhances the analysis of objects by embedding them into a simplicial complex and extracting useful global properties such as the Betti numbers, i.e. the number of multidimensional holes, which can be used to define kernel methods that are easily integrated with existing machine-learning algorithms. These kernel methods have found broad applications, as they rely on powerful mathematical frameworks which provide theoretical guarantees on their performance. However, the computation of higher-dimensional Betti numbers can be prohibitively expensive on classical hardware, while quantum algorithms can approximate them in polynomial time in the instance size. In this work, we propose a quantum approach to defining topological kernels, which is based on constructing Betti curves, i.e. topological fingerprint of filtrations with increasing order. We exhibit a working prototype of our approach implemented on a noiseless simulator and show its robustness by means of some empirical results suggesting that topological approaches may offer an advantage in quantum machine learning.
Topological data analysis (TDA) is a rapidly growing area that applies techniques from algebraic topology to extract robust features from large-scale data. A key task in TDA is the estimation of (normalized) Betti numbers, which capture essential topological invariants. While recent work has led to quantum algorithms for this problem, we explore an alternative direction: combining classical and quantum resources to estimate the Betti numbers of a simplicial complex more efficiently. Assuming the classical description of a simplicial complex, that is, its set of vertices and edges, we propose a hybrid quantum-classical algorithm. The classical component enumerates all simplices, and this combinatorial structure is subsequently processed by a quantum algorithm to estimate the Betti numbers. We analyze the performance of our approach and identify regimes where it potentially achieves polynomial to exponential speedups over existing quantum methods, at the trade-off of using more ancilla qubits. We further demonstrate the utility of normalized Betti numbers in concrete applications, highlighting the broader potential of hybrid quantum algorithms in topological data analysis.
Topological data analysis (TDA) is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA are limited if we want to learn from high-dimensional topological features because the number of high-dimensional simplices grows exponentially in the size of the data. In the context of quantum computation, it has been previously shown that there exists an efficient quantum algorithm for estimating the Betti numbers even in high dimensions. However, the Betti numbers are less general than the persistent Betti numbers, and there have been no quantum algorithms that can estimate the persistent Betti numbers of arbitrary dimensions. This paper shows the first quantum algorithm that can estimate the (normalized) persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for simplicial complexes such as the Vietoris-Rips complex and demonstrates exponential speedup over the known classical algorithms.
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.
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.