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arXiv:2605.28927

Quantum encodings that preserve persistent homology

Arthur J. Parzygnat, Andrew Vlasic

quant-phcs.CGmath.ATSubmitted May 27, 2026• Announced May 29, 2026• 0 citations
v1 · May 27, 2026v2 · Aug 18, 2026
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Abstract

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.

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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.
Abstract: Quantum Topological Data Analysis (QTDA) is introduced as a hybrid classical–quantum methodology for accelerating persistent homology on complex manifolds while preserving topological interpretability. The approach encodes simplicial complexes into quantum states, applies boundary operators via unitary transforms, and estimates invariants with variational and linear‑algebraic quantum routines. An end‑to‑end pipeline is formalized that couples deep representation learning with quantum‑assisted homological primitives, yielding stable persistence diagrams aligned with structural robustness. Experiments on synthetic and semi‑synthetic filtrations quantify latency, scaling, and stability, showing empirical reductions in runtime growth under modest qubit budgets. The results indicate that QTDA enables time‑critical analytics such as anomaly detection and manifold clustering, while meeting explainability and governance requirements through auditable topological summaries. Keywords Quantum Topological Data Analysis, persistent homology, Betti numbers, simplicial complexes, variational quantum algorithms, manifold learning, interpretability, auditability
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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.
Quantum state tomography (QST) is a crucial technique for characterizing the state of a quantum system. However, the data generated by QST—typically a large, dense measurement matrix—presents significant challenges for analysis and interpretation. This work proposes a novel approach leveraging Topological Data Analysis (TDA), specifically persistent homology, to extract meaningful insights from these complex quantum state data. We demonstrate that the underlying topological structure of the measurement matrices can reveal fundamental characteristics of the quantum state, including entanglement properties and the degree of coherence. The core idea involves treating the measurement matrix as a high-dimensional point cloud and applying persistent homology to uncover persistent loops and connected components, which directly correspond to topological features of the quantum state. This method offers a new perspective on QST data analysis, potentially leading to improved state characterization and a deeper understanding of quantum entanglement. The application of TDA provides a robust and geometric interpretation of the measurement data, moving beyond traditional statistical methods.
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.