Comparing quantum and classical Monte Carlo algorithms for estimating Betti numbers of clique complexes

• Published in: arXiv.org
• Main Authors: Ismail Yunus Akhalwaya, Bhayat, Ahmed, Connolly, Adam, Herbert, Steven, Horesh, Lior, Sorci, Julien, Ubaru, Shashanka
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 29.08.2024
• Subjects: Algorithms; Convergence; Estimation; Modular structures; Upper bounds

Several quantum and classical Monte Carlo algorithms for Betti Number Estimation (BNE) on clique complexes have recently been proposed, though it is unclear how their performances compare. We review these algorithms, emphasising their common Monte Carlo structure within a new modular framework. This framework allows us to directly compare these algorithms by calculating upper bounds on the minimum number of samples needed for convergence. By recombining the different modules, we create a new quantum algorithm with an exponentially-improved dependence in the sample complexity. We run classical simulations to verify convergence within the theoretical bounds and observe the predicted exponential separation, even though empirical convergence occurs substantially earlier than the conservative theoretical bounds.