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(1/root gamma log 1/epsilon)) with gamma measuring the spectral gap of the combinatorial Laplacian and epsilon is an element of (0,1) the additive precision. In the case of a clique complex, the running time of our algorithm improves to (n/ lambda max) (O(1/root gamma log 1/epsilon)) with lambda(max )>= 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 is an element of Omega(1) and k is an element of Omega(n).