A (simple) classical algorithm for estimating Betti numbers
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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/γlog1/ε) with γ measuring the spectral gap of the combinatorial Laplacian and ε∈ (0,1) the additive precision. In the case of a clique complex, the running time of our algorithm improves to (n/λ_max)^O(1/γlog1/ε) with λ_max≥ 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 ∈Ω(n) or λ_max∈Ω(n).
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