Quantum encodings that preserve persistent homology
For researchers in quantum topological data analysis, this work provides a preliminary theoretical framework for selecting data encodings that preserve topological features, but it is an incremental step without empirical validation.
This paper investigates which quantum encodings of classical data preserve topological invariants (persistent homology) for quantum topological data analysis, proposing a direct encoding approach that avoids constructing simplicial complexes. No concrete numerical results are provided.
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.