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

TL;DR

This paper compares quantum and classical Monte Carlo algorithms for estimating Betti numbers of clique complexes, introduces a modular framework, and demonstrates an exponential advantage of the quantum approach through theoretical bounds and simulations.

Contribution

It provides a unified modular framework for analyzing these algorithms and introduces a new quantum algorithm with exponentially better sample complexity.

Findings

Quantum algorithm shows exponential improvement in sample complexity.

Classical simulations confirm theoretical convergence bounds.

Empirical convergence occurs earlier than theoretical predictions.

Abstract

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. We derive upper bounds for the number of samples needed to reach a given level of precision, and use them to compare these algorithms. 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.