Quantum Algorithm for Estimating Betti Numbers Using a Cohomology Approach

• URL: http://arxiv.org/abs/2309.10800v2
• Date: Fri, 20 Oct 2023 12:32:08 GMT
• Title: Quantum Algorithm for Estimating Betti Numbers Using a Cohomology Approach
• Authors: Nhat A. Nghiem, Xianfeng David Gu and Tzu-Chieh Wei
• Abstract summary: Calculating Betti numbers classically is a daunting task due to the massive volume of data. We consider the dual' approach, which is inspired by Hodge theory and de Rham cohomology. Our algorithm can calculate its $r$-th Betti number $beta_r$ up to some multiplicative error.
• Score: 2.2000560351723504
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Topological data analysis has emerged as a powerful tool for analyzing large-scale data. High-dimensional data form an abstract simplicial complex, and by using tools from homology, topological features could be identified. Given a simplex, an important feature is so-called Betti numbers. Calculating Betti numbers classically is a daunting task due to the massive volume of data and its possible high-dimension. While most known quantum algorithms to estimate Betti numbers rely on homology, here we consider the `dual' approach, which is inspired by Hodge theory and de Rham cohomology, combined with recent advanced techniques in quantum algorithms. Our cohomology method offers a relatively simpler, yet more natural framework that requires exponentially less qubits, in comparison with the known homology-based quantum algorithms. Furthermore, our algorithm can calculate its $r$-th Betti number $\beta_r$ up to some multiplicative error $\delta$ with running time $\mathcal{O}\big( \log(c_r) c_r^2 / (c_r - \beta_r)^2 \delta^2 \big)$, where $c_r$ is the number of $r$-simplex. It thus works best when the $r$-th Betti number is considerably smaller than the number of the $r$-simplex in the given triangulated manifold.

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