Tightness of the CBNE sample complexity upper bound

Determine whether the published upper bound on the sample complexity of the classical Betti number estimation (CBNE) algorithm of Apers et al. (2023) for estimating normalized Betti numbers is tight. Specifically, ascertain if the Hoeffding-inequality-based bound provided for CBNE cannot be improved in general, or if sharper bounds exist that reduce the required number of Monte Carlo samples across simplicial complexes and clique complexes.

Background

The paper analyzes a classical algorithm (CBNE) from Apers et al. (2023) for estimating normalized Betti numbers and compares it to a recently proposed quantum algorithm with a similar Monte Carlo structure. The quantum algorithm achieves exponentially smaller sample complexity than the upper bound reported for CBNE.

Apers et al. derive CBNE’s sample complexity using Hoeffding’s inequality, yielding asymptotic bounds such as O((log(2/η)/ε^2) n^{2ℓ}) for general complexes and O((log(2/η)/ε^2) 4^{ℓ}) for clique complexes. The present paper provides refined variance-based bounds and an improved algorithm (CBNE-Var), motivating the unresolved question of whether Apers et al.’s original upper bound is optimal or can be tightened.