Recovering a group from few orbits

TL;DR

This paper establishes precise bounds on the number of generic orbits required to uniquely identify an unknown finite automorphism group of a finite-dimensional Hilbert space, both up to isomorphism and as a set of automorphisms.

Contribution

It provides the first sharp bounds on the minimal number of orbits needed for group recovery in this context.

Findings

Derived sharp bounds for group recovery from orbits

Quantified the number of orbits needed for isomorphism classification

Analyzed the set-based recovery of automorphism groups

Abstract

For an unknown finite group $G$ of automorphisms of a finite-dimensional Hilbert space, we find sharp bounds on the number of generic $G$-orbits needed to recover $G$ up to group isomorphism, as well as the number needed to recover $G$ as a concrete set of automorphisms.