A (simple) classical algorithm for estimating Betti numbers

TL;DR

This paper introduces a simple classical algorithm using path integral Monte Carlo to estimate Betti numbers of simplicial complexes, providing a benchmark for quantum algorithms with improved efficiency in certain cases.

Contribution

The paper presents a new classical algorithm for estimating Betti numbers that matches quantum algorithm performance in specific scenarios, with improved running time for clique complexes.

Findings

Algorithm's running time depends on spectral gap and eigenvalues.

For clique complexes, the running time improves with eigenvalue bounds.

The classical algorithm serves as a benchmark for quantum methods.

Abstract

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 $k \in \Omega(n)$.