Quantum Algorithm for Estimating Betti Numbers Using Cohomology Approach

Topological data analysis has emerged as a powerful tool for analyzing large-scale data. An abstract simplicial complex, in principle, can be built from data points, and by using tools from homology, topological features could be identified. Given a simplex, an important feature is called the Betti numbers, which roughly count the number of `holes' in different dimensions. Calculating Betti numbers exactly can be $\#$P-hard, and approximating them can be NP-hard, which rules out the possibility of any generic efficient algorithms and unconditional exponential quantum speedup. Here, we explore the specific setting of a triangulated manifold. In contrast to most known methods to estimate Betti numbers, which rely on homology, we exploit the `dual' approach, namely, cohomology, combining the insight of the Hodge theory and de Rham cohomology. Our proposed algorithm can calculate its $r$-th normalized Betti number $\beta_r/|S_r|$ up to some additive error $\epsilon$ with running time $\mathcal{O}\Big(\frac{\log(|S_r^K| |S_{r+1}^K|)}{\epsilon^2} \log (\log |S_r^K|) \big( r\log |S_r^K| \big) \Big)$, where $|S_r|$ is the number of $r$-simplexes in the given complex. For the estimation of $r$-th Betti number $\beta_r$ to a chosen multiplicative accuracy $\epsilon'$, our algorithm has complexity $\mathcal{O}\Big(\frac{\log(|S_r^K| |S_{r+1}^K|)}{\epsilon'^2} \big( \frac{ \Gamma}{\beta_r}\big)^2 (\log |S_r^K|) \log \big( r\log |S_r^K| \big) \Big)$, where $\Gamma \leq |S_r^K|$ can be chosen. A detailed analysis is provided, showing that our cohomology framework can even perform exponentially faster than previous homology methods in several regimes. In particular, our method is most effective when $\beta_r \ll |S_r^K|$, which can offer more flexibility and practicability than existing quantum algorithms that achieve the best performance in the regime $\beta_r \approx |S_r^K|$.