Quantum Algorithm for Estimating Ollivier-Ricci Curvature

TL;DR

This paper presents a quantum algorithm that efficiently estimates Ollivier-Ricci curvature on graphs and metric spaces, potentially offering exponential speedups over classical methods for certain problems.

Contribution

The authors develop a quantum algorithm for computing Ollivier-Ricci curvature, demonstrating exponential speedup for specific input classes and advancing quantum approaches to geometric problems.

Findings

Achieves exponential speedup over classical algorithms for certain problem classes.

Applicable to point clouds with pairwise distances for curvature estimation.

Contributes to practical quantum algorithms in geometric analysis.

Abstract

We introduce a quantum algorithm for computing the Ollivier Ricci curvature, a discrete analogue of the Ricci curvature defined via optimal transport on graphs and general metric spaces. This curvature has seen applications ranging from signaling fragility in financial networks to serving as basic quantities in combinatorial quantum gravity. For inputs given as a point cloud with pairwise distances, we show that our algorithm can achieve an exponential speedup over the best-known classical methods for two particular classes of problem. Our work is another step toward quantum algorithms for geometrical problems that are capable of delivering practical value while also informing fundamental theory.