Graph isomorphism: Physical resources, optimization models, and algebraic characterizations [article]

Laura Mančinska, David E. Roberson, Antonios Varvitsiotis
2020 arXiv   pre-print
In the (G,H)-isomorphism game, a verifier interacts with two non-communicating players (called provers) by privately sending each of them a random vertex from either G or H, whose aim is to convince the verifier that two graphs G and H are isomorphic. In recent work along with Atserias, Šámal 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
more » ... um 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.
arXiv:2004.10893v1 fatcat:wnophwxi5jbhbcf4msnn56co3u