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 ${\beta }_r\mathrm{/}\mathrm{\mid }{S}_r\mathrm{\mid }$ up to some additive error $\epsilon$ with running time $\mathcal{O}(\frac{\log (\mathrm{\mid }{S}_r^K\mathrm{\mid }\mathrm{\mid }{S}_{r+1}^K\mathrm{\mid })}{{\epsilon }^2}\log (\log \mathrm{\mid }{S}_r^K\mathrm{\mid })(r\log \mathrm{\mid }{S}_r^K\mathrm{\mid }\left)\right)$, where $\mathrm{\mid }{S}_r\mathrm{\mid }$ 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 ${\epsilon }^{\mathrm{\prime }}$, our algorithm has complexity $\mathcal{O}\left(\frac{\log (\mathrm{\mid }{S}_r^K\mathrm{\mid }\mathrm{\mid }{S}_{r+1}^K\mathrm{\mid })}{{\epsilon }^{\mathrm{\prime }2}}\right(\frac{\mathrm{\Gamma }}{{\beta }_r}{)}^2(\log \mathrm{\mid }{S}_r^K\mathrm{\mid })\log (r\log \mathrm{\mid }{S}_r^K\mathrm{\mid }))$, where $\mathrm{\Gamma }\le \mathrm{\mid }{S}_r^K\mathrm{\mid }$ 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 ${\beta }_r\ll \mathrm{\mid }{S}_r^K\mathrm{\mid }$, which can offer more flexibility and practicability than existing quantum algorithms that achieve the best performance in the regime ${\beta }_r\approx \mathrm{\mid }{S}_r^K\mathrm{\mid }$.