🤖 AI Summary
This work addresses a central challenge in quantum topological data analysis (TDA): how to effectively preserve topological invariants—particularly persistent homology—when encoding classical datasets into quantum states. The authors propose a direct quantum encoding approach that bypasses the conventional construction of simplicial complexes, instead focusing on admissible encoding strategies derived directly from raw distance data. By integrating tools from algebraic topology, metric geometry, and quantum information theory, they systematically analyze how different encoding schemes affect persistent homology structures and identify several quantum encodings capable of efficiently preserving essential topological features. This framework substantially reduces computational resource requirements, offering a new paradigm for low-complexity quantum TDA.
📝 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.