论文标题
一种用于估计贝蒂数字的（简单）经典算法
A (simple) classical algorithm for estimating Betti numbers
论文作者
论文摘要
我们描述了一种简单的算法，用于估计使用路径积分蒙特卡洛方法在$ n $元素上估算$ k $ thth的betti数字。 For a general simplicial complex, the running time of our algorithm is $n^{O\left(\frac{1}{\sqrtγ}\log\frac{1}{\varepsilon}\right)}$ with $γ$ measuring the spectral gap of the combinatorial Laplacian and $\varepsilon \in （0,1）$添加精度。在一个集团复合体的情况下，我们算法的运行时间提高到$ \左（n/λ_ {\ max} \ right） \ geq k $，其中$λ_{\ max} $是组合laplacian的最大特征值。我们的算法为估算贝蒂数字的量子算法提供了经典的基准。在集团复合体上，它与他们的运行时间相匹配，例如，在ω（1）$中$γ\ inω（n）$中的$γ\。
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 $n^{O\left(\frac{1}{\sqrtγ}\log\frac{1}{\varepsilon}\right)}$ with $γ$ 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 $\left(n/λ_{\max}\right)^{O\left(\frac{1}{\sqrtγ}\log\frac{1}{\varepsilon}\right)}$ with $λ_{\max} \geq k$, where $λ_{\max}$ 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, $γ\in Ω(1)$ and $k \in Ω(n)$.
