  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
$n^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$ 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
$\left(n/\lambda_{\max}\right)^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$
with $\lambda_{\max} \geq k$, 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 $k \in \Omega(n)$.

Data Structures and Algorithms;Quantum Physics