A (simple) classical algorithm for estimating Betti numbers

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 $n^{O(\frac{1}{\gamma}\log\frac{1}{\varepsilon})}$ with $\gamma$ measuring the spectral gap of the combinatorial Laplacian and $\varepsilon \in (0,1)$ the additive precision. In the case of a clique complex, the running time of our algorithm improves to $(n/\lambda_{\max})^{O(\frac{1}{\gamma}\log\frac{1}{\varepsilon})}$ with $\lambda_{\max} \geq k$ the maximum eigenvalue of the combinatorial Laplacian. Our algorithm provides a classical benchmark for a line of quantum algorithms for estimating Betti numbers, and it matches their running time on clique complexes when the spectral gap is constant and $k \in \Omega(n)$ or $\lambda_{\max} \in \Omega(n)$.