An Incremental Span-Program-Based Algorithm and the Fine Print of Quantum Topological Data Analysis
Mitchell Black, William Maxwell, Amir Nayyeri

TL;DR
This paper presents a new quantum algorithm for computing Betti numbers of sparse simplicial complexes, highlighting the limitations imposed by exponential bounds on effective resistance and capacitance, and analyzing the spectral gap's impact.
Contribution
Introduces a span-program-based quantum algorithm for Betti number computation that is optimized for sparse complexes and analyzes its limitations.
Findings
Quantum algorithm works best for sparse complexes.
Effective resistance and capacitance can be exponentially large.
All quantum algorithms for Betti numbers require exponential time for exact computation.
Abstract
We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. In contrast to previous quantum algorithms that work by estimating the eigenvalues of the combinatorial Laplacian, our algorithm is an instance of the generic Incremental Algorithm for computing Betti numbers that incrementally adds simplices to the simplicial complex and tests whether or not they create a cycle. In contrast to existing quantum algorithms for computing Betti numbers that work best when the complex has close to the maximal number of simplices, our algorithm works best for sparse complexes. To test whether a simplex creates a cycle, we introduce a quantum span-program algorithm. We show that the query complexity of our span program is parameterized by quantities called the effective resistance and effective capacitance of the boundary of the simplex. Unfortunately, we also prove…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Data Visualization and Analytics · Digital Image Processing Techniques
