Topological data analysis offers a robust way to extract useful information from noisy, unstructured data by identifying its underlying structure. Recently, an efficient quantum algorithm was proposed [Lloyd, Garnerone, Zanardi, Nat. Commun. 7, 10138 (2016)] for calculating Betti numbers of data points -- topological features that count the number of topological holes of various dimensions in a scatterplot. Here, we implement a proof-of-principle demonstration of this quantum algorithm by employing a six-photon quantum processor to successfully analyze the topological features of Betti numbers of a network including three data points, providing new insights into data analysis in the era of quantum computing.Topological data analysis offers a robust way to extract useful information from noisy, unstructured data by identifying its underlying structure. Recently, an efficient quantum algorithm was proposed [Lloyd, Garnerone, Zanardi, Nat. Commun. 7, 10138 (2016)] for calculating Betti numbers of data points -- topological features that count the number of topological holes of various dimensions in a scatterplot. Here, we implement a proof-of-principle demonstration of this quantum algorithm by employing a six-photon quantum processor to successfully analyze the topological features of Betti numbers of a network including three data points, providing new insights into data analysis in the era of quantum computing.

OpticaFeb 9, 201844 citations

Higher-Order Topological Kernels via Quantum Computation

Massimiliano Incudini, Francesco Martini, Alessandra Di Pierro

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) 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.

arXiv:2307.07383IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 17, 20234 citations

Quantum-Enhanced Topological Data Analysis: A Peep from an Implementation Perspective

Ankit Khandelwal, M Girish Chandra

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.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.

arXiv:2302.09553IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd ForumMay 1, 20234 citations

Hybrid quantum-classical framework for Betti number estimation with applications to topological data analysis

Nhat A. Nghiem, Tzu-Chieh Wei

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 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.

arXiv:2508.01516Aug 2, 20250 citations

Topological network analysis using a programmable photonic quantum processor

Yu Shang, Jinzhao Sun, Zhenghao Li +11 more

Understanding topological features in networks is crucial for unravelling complex phenomena across fields such as neuroscience, condensed matter, and high-energy physics. However, identifying higher-order topological structures -- such as $k$ -cliques, fundamental building blocks of complex networks -- remains a significant challenge. Here we develop a universal programmable photonic quantum processor that enables the encoding of arbitrary complex-weight networks, providing a direct pathway to uncovering their topological structures. We demonstrate how this quantum approach can identify weighted $k$ -cliques and estimate Betti numbers by leveraging the Gaussian boson sampling algorithm's ability to preferentially select high-weight, dense subgraphs. The unique capabilities of our programmable quantum processor allow us to observe topological phase transitions and identify clique percolation phenomena directly from the entropy of the sampling results. These findings showcase how photonic quantum computing can be applied to analyse the topological characteristics of real-world complex networks, opening new possibilities for quantum-enhanced data analysis.Understanding topological features in networks is crucial for unravelling complex phenomena across fields such as neuroscience, condensed matter, and high-energy physics. However, identifying higher-order topological structures -- such as $k$-cliques, fundamental building blocks of complex networks -- remains a significant challenge. Here we develop a universal programmable photonic quantum processor that enables the encoding of arbitrary complex-weight networks, providing a direct pathway to uncovering their topological structures. We demonstrate how this quantum approach can identify weighted $k$-cliques and estimate Betti numbers by leveraging the Gaussian boson sampling algorithm's ability to preferentially select high-weight, dense subgraphs. The unique capabilities of our programmable quantum processor allow us to observe topological phase transitions and identify clique percolation phenomena directly from the entropy of the sampling results. These findings showcase how photonic quantum computing can be applied to analyse the topological characteristics of real-world complex networks, opening new possibilities for quantum-enhanced data analysis.

arXiv:2507.08157Jul 10, 20250 citations

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

Jason Iaconis, Sayonee Ray, Samwel Sekwao +2 more

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 dataTopological 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

quant-pharXiv:2607.27206Jul 30, 20260 citations

k

-cliques, fundamental building blocks of complex networks -- remains a significant challenge. Here we develop a universal programmable photonic quantum processor that enables the encoding of arbitrary complex-weight networks, providing a direct pathway to uncovering their topological structures. We demonstrate how this quantum approach can identify weighted

k

-cliques and estimate Betti numbers by leveraging the Gaussian boson sampling algorithm's ability to preferentially select high-weight, dense subgraphs. The unique capabilities of our programmable quantum processor allow us to observe topological phase transitions and identify clique percolation phenomena directly from the entropy of the sampling results. These findings showcase how photonic quantum computing can be applied to analyse the topological characteristics of real-world complex networks, opening new possibilities for quantum-enhanced data analysis.