Graph isomorphism: Physical resources, optimization models, and algebraic characterizations
Abstract
In the -isomorphism game, a verifier interacts with two non-communicating players (called provers) by privately sending each of them a random vertex from either or , whose aim is to convince the verifier that two graphs and are isomorphic. In recent work along with Atserias, \v{S}\'amal and Severini [Journal of Combinatorial Theory, Series B, 136:89--328, 2019] we showed that a verifier can be convinced that two non-isomorphic graphs are isomorphic, if the provers are allowed to share quantum resources. In this paper we model classical and quantum graph isomorphism by linear constraints over certain complicated convex cones, which we then relax to a pair of tractable convex models (semidefinite programs). Our main result is a complete algebraic characterization of the corresponding equivalence relations on graphs in terms of appropriate matrix algebras. Our techniques are an interesting mix of algebra, combinatorics, optimization, and quantum information.
Cite
@article{arxiv.2004.10893,
title = {Graph isomorphism: Physical resources, optimization models, and algebraic characterizations},
author = {Laura Mančinska and David E. Roberson and Antonios Varvitsiotis},
journal= {arXiv preprint arXiv:2004.10893},
year = {2020}
}