Extracting useful information from large data sets can be a daunting task. Topological methods for analyzing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying such topological features -- connected components, holes, or voids -- and for determining how such features persist as the data is viewed at different scales. This paper provides quantum algorithms for calculating Betti numbers in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speedup over classical algorithms for topological data analysis.
Extracting useful information from large data sets can be a daunting task. Topological methods for analyzing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying such topological features -- connected components, holes, or voids -- and for determining how such features persist as the data is viewed at different scales. This paper provides quantum algorithms for calculating Betti numbers in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speedup over classical algorithms for topological data analysis.
Extracting useful information from large data sets can be a daunting task. Topological methods for analysing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying topological features and for determining how such features persist as the data is viewed at different scales. Here we present quantum machine learning algorithms for calculating Betti numbers--the numbers of connected components, holes and voids--in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speed-up over the best currently known classical algorithms for topological data analysis.
We introduce several new quantum algorithms for estimating homological invariants, specifically Betti numbers and persistent Betti numbers, of a simplicial complex given via a structured classical input. At the core of our algorithm lies the ability to efficiently construct the block-encoding of Laplacians (and persistent Laplacians) based on the classical description of the given complex. From such block-encodings, Betti numbers (and persistent Betti numbers) can be estimated. The complexity of our method is polylogarithmic in the number of simplices in both simplex-sparse and simplex-dense regimes, thus offering an advantage over existing works. Moreover, prior quantum algorithms based on spectral methods incur significant overhead due to their reliance on estimating the kernel of combinatorial Laplacians, particularly when the Betti number is small. We introduce a new approach for estimating Betti numbers based on homology tracking and homology property testing, which enables exponential quantum speedups over both classical and prior quantum approaches under sparsity and structure assumptions. We further initiate the study of homology triviality and equivalence testing as natural property testing problems in topological data analysis, and provide efficient quantum algorithms with time complexity nearly linear in the number of simplices when the rank of the boundary operator is large. In addition, we develop a cohomological approach based on block-encoded projections onto cocycle spaces, enabling rank-independent testing of homology equivalence. This yields the first quantum algorithms for constructing and manipulating r-cocycles in time polylogarithmic in the size of the complex. Together, these results establish a new direction in quantum topological data analysis and demonstrate that computing topological invariants can serve as a fertile ground for provable quantum advantage.
Persistence diagrams serve as a core tool in topological data analysis, playing a crucial role in pathological monitoring, drug discovery, and materials design. However, existing quantum topological algorithms, such as the LGZ algorithm, can only efficiently compute summary statistics like Betti numbers, failing to provide persistence diagram information that tracks the lifecycle of individual topological features, severely limiting their practical value. This paper proposes a novel quantum-classical hybrid algorithm that achieves, for the first time, the leap from "quantum computation of Betti numbers" to "quantum acquisition of practical persistence diagrams." The algorithm leverages the LGZ quantum algorithm as an efficient feature extractor, mining the harmonic form eigenvectors of the combinatorial Laplacian as well as Betti numbers, constructing specialized topological kernel functions to train a quantum support vector machine (QSVM), and learning the mapping from quantum topological features to persistence diagrams. The core contributions of this algorithm are: (1) elevating quantum topological computation from statistical summaries to pattern recognition, greatly expanding its application value; (2) obtaining more practical topological information in the form of persistence diagrams for real-world applications while maintaining the exponential speedup advantage of quantum computation; (3) proposing a novel hybrid paradigm of "classical precision guiding quantum efficiency." This method provides a feasible pathway for the practical implementation of quantum topological data analysis.
Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyze and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over previously known rigorous classical algorithms for computing the persistent Betti numbers to constant additive error – the salient task for applications. However, we also introduce a quantum-inspired classical power method with provable scaling only quadratically worse than the quantum algorithm. This gives a provable classical algorithm with scaling comparable to existing classical heuristics. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence for this being the case.
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 N-nilpotent boundary operators (∂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) 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.