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 $k \in \Omega(n)$.

翻译：我们描述了一种基于路径积分蒙特卡洛方法估计$n$个元素上单纯复形第$k$个归一化贝蒂数的简洁算法。对于一般单纯复形，该算法的运行时间为$n^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$，其中$\gamma$衡量组合拉普拉斯算子的谱间隙，$\varepsilon \in (0,1)$为加法精度。在团复形情形下，算法运行时间改进为$\left(n/\lambda_{\max}\right)^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$，其中$\lambda_{\max} \geq k$为组合拉普拉斯算子的最大特征值。该算法为一系列估计贝蒂数的量子算法提供了经典基准。当$\gamma \in \Omega(1)$且$k \in \Omega(n)$时，在团复形上其运行时间可媲美量子算法。