Homomorphism Indistinguishability Relations induced by Quantum Groups
Tim Seppelt, Gian Luca Spitzer
TL;DR
This work extends the connection between quantum symmetries and graph isomorphism relaxations from planar graphs to all orthogonal easy quantum groups by introducing graph instantiations $Q(G)$ and analysing their intertwiners via partition categories. A central result shows that two graphs $G$ and $H$ are homomorphism indistinguishable over the unlabelled intertwiner graphs of a given orthogonal easy quantum group $Q$ if and only if there exists a quantum orthogonal matrix $oldsymbol{U}$ that intertwines adjacency tensors and all intertwiners, i.e., $oldsymbol{U} A_G = A_H oldsymbol{U}$ and $oldsymbol{U}^{5 ext{0}} M = M oldsymbol{U}^{5 ext{0}}$ for every intertwiner $M$ of $Q$. The authors also develop a category-theoretic bridge showing that weak isomorphisms between graph instantiations correspond to monoidal equivalences of their representation categories, yielding a unifying framework that reproduces MR’s planar result as a special case (quantum permutation matrices and planar graphs) and classifies the intertwiner-graph classes for all orthogonal easy quantum groups into four families (all graphs, planar graphs, cycles, or paths together with cycles). They conclude with concrete characterisations of homomorphism indistinguishability for several graph classes and discuss extensions to unitary easy quantum groups and multigraphs.
Abstract
Homomorphism indistinguishability is a way of characterising many natural equivalence relations on graphs. Two graphs $G$ and $H$ are called homomorphism indistinguishable over a graph class $\mathcal{F}$ if for each $F \in \mathcal{F}$, the number of homomorphisms from $F$ to $G$ equals the number of homomorphisms from $F$ to $H$. Examples of such equivalence relations include isomorphism and cospectrality, as well as equivalence with respect to many formal logics. Quantum groups are a generalisation of topological groups that describe "non-commutative symmetries" and, inter alia, have applications in quantum information theory. An important subclass are the easy quantum groups, which enjoy a combinatorial characterisation and have been fully classified by Raum and Weber. A recent connection between these seemingly distant concepts was made by Mančinska and Roberson, who showed that quantum isomorphism, a relaxation of classical isomorphism that can be phrased in terms of the quantum symmetric group, is equivalent to homomorphism indistinguishability over the class of planar graphs. We generalise Mančinska and Roberson's result to all orthogonal easy quantum groups. We obtain for each orthogonal easy quantum group a graph isomorphism relaxation $\approx$ and a graph class $\mathcal{F}$, such that homomorphism indistinguishability over $\mathcal{F}$ coincides with $\approx$. Our results include a full classification of the $(0, 0)$-intertwiners of the graph-theoretic quantum group obtained by adding the adjacency matrix of a graph to the intertwiners of an orthogonal easy quantum group.