Quantum Algorithm for Estimating Betti Numbers Using Cohomology Approach

Quantum 9, 1955 (2025).

https://doi.org/10.22331/q-2025-12-23-1955

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 $mathcal{O}Big(frac{log(|S_r^K| |S_{r+1}^K|)}{epsilon^2} log (log |S_r^K|) big( rlog |S_r^K| big) Big)$, 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( rlog |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 $beta_r approx |S_r^K|$.