Quantum algorithms for topological data analysis provide significant advantages over the best known classical algorithms. Unlike previous work on simplicial complexes built from point clouds, path homology on digraphs is defined for directed graphs and provides a natural topological framework for analyzing data with intrinsic directional structures. Path homology has become an emerging area in Topological Data Analysis (TDA), attracting increasing attention in recent years. We propose a quantum algorithm for path homology on digraphs that offers a significant advantage over the best known classical algorithms. We design a universal encoding protocol for the paths and boundary operators of digraphs on quantum systems. We prove a property of path homology that provides the theoretical guarantee for the algorithm. The speedup of the quantum algorithm for path homology depends on input-access assumptions. The exponential speedup arises when the path space can be efficiently accessed, while for standard digraph input the algorithm provides polynomial speedup.
Quantum algorithms for topological data analysis provide significant advantages over the best known classical algorithms. Unlike previous work on simplicial complexes built from point clouds, path homology on digraphs is defined for directed graphs and provides a natural topological framework for analyzing data with intrinsic directional structures. Path homology has become an emerging area in Topological Data Analysis (TDA), attracting increasing attention in recent years. We propose a quantum algorithm for path homology on digraphs that offers a significant advantage over the best known classical algorithms. We design a universal encoding protocol for the paths and boundary operators of digraphs on quantum systems. We prove a property of path homology that provides the theoretical guarantee for the algorithm. The speedup of the quantum algorithm for path homology depends on input-access assumptions. The exponential speedup arises when the path space can be efficiently accessed, while for standard digraph input the algorithm provides polynomial speedup.
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.
Abstract Quantum algorithms for topological data analysis (TDA) provide significant advantages over the best known classical algorithms. Unlike previous work on simplicial complexes built from point clouds, path homology on digraphs is defined for directed graphs and provides a natural topological framework for analyzing data with intrinsic directional structures. Path homology has become an emerging area in TDA, attracting increasing attention in recent years. We propose a quantum algorithm for path homology on digraphs that offers a significant advantage over the best known classical algorithms. We design a universal encoding protocol for the paths and boundary operators of digraphs on quantum systems. We prove a property of path homology that provides the theoretical guarantee for the algorithm. The speedup of the quantum algorithm for path homology depends on input-access assumptions. The exponential speedup arises when the path space can be efficiently accessed, while for standard digraph input the algorithm provides polynomial speedup.
Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which the data are sampled. This inference is achieved by calculating topological invariants, some of which are difficult to compute classically. Meanwhile, quantum TDA utilizes quantum processes to extract the invariants used in making such inferences in an attempt to speed up the computations. Because applying transformations to the original classical dataset could alter the associated topological invariants, we investigate which quantum encodings would best preserve the invariants of the original dataset. This line of inquiry is distinct from standard approaches in quantum TDA, whose typical starting point is not from the classical dataset directly, but rather from the associated combinatorial objects, such as simplicial complexes, which typically demand a lot of resources to construct. We take the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible for applying quantum algorithms to extract topological features from classical datasets.
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.
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.
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.