A (simple) classical algorithm for estimating Betti numbers

Abstract

We describe a simple algorithm for estimating the $k$-th normalized Betti number of a simplicial complex over $n$ elements using the path integral Monte Carlo method. For a general simplicial complex, the running time of our algorithm is $n^{O\left(\frac{1}{\sqrt{\gamma }}\log \frac{1}{\epsilon }\right)}$ with $\gamma$ measuring the spectral gap of the combinatorial Laplacian and $\epsilon \in (0,1)$ the additive precision. In the case of a clique complex, the running time of our algorithm improves to ${\left(n/{\lambda }_{\max }\right)}^{O\left(\frac{1}{\sqrt{\gamma }}\log \frac{1}{\epsilon }\right)}$ with ${\lambda }_{\max }\ge k$, where ${\lambda }_{\max }$ is the maximum eigenvalue of the combinatorial Laplacian. Our algorithm provides a classical benchmark for a line of quantum algorithms for estimating Betti numbers. On clique complexes it matches their running time when, for example, $\gamma \in \Omega (1)$ and $k\in \Omega (n)$.

arXiv commentv3: final version, accepted to Quantum

References in corpus (6)

• Persistent Laplacians: properties, algorithms and implications
• Towards quantum advantage via topological data analysis
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• A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits
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