arXiv subjects

Gian Luca Spitzer

Publications and source records attributed to Gian Luca Spitzer.

5 recordsLinked to original sources

arXiv2026

Quantum Graph Theory by Example

Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices $(A, B, C)$. The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components: $A$ and $C$ are described by a classical weighted graph called the strange graph, while $B$ provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.

math.OA↗

arXiv2026

Unification of Quantum Graph Properties

Many properties of classical graphs are defined in terms of subsets of the vertex set. Examples include connected components, which are subsets $X \subseteq V(G)$ such that $X \times X^c$ and $E(G)$ are disjoint, or independent sets, for which $X \times X$ and $E(G)$ are disjoint. Direct generalisations of these definitions to quantum graphs are difficult to achieve, since the natural notion of subsets of a quantum set is much too rigid. As a consequence, approaches to generalising these classical properties to the quantum setting have been eclectic. In some cases, multiple inequivalent definitions of the same notion are in use. We introduce a natural and well-motivated alternative definition of subsets of a quantum set. Building on this, we propose unified definitions of quantum graph properties as straightforward generalisations of the classical definitions. We recover this way the established notions of colourings and connected components. For independent sets and cliques, our approach suggests variations that diverge from existing definitions, but address some of their counterintuitive properties. We nevertheless show how to recover an important existing definition of independent sets in our framework.

math.OA↗

arXiv2026

Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth

We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio~(2021) proved that two graphs satisfy the same $k$-variable and quantifier-rank-$q$ sentences if and only if they are homomorphism indistinguishable over the class of graphs admitting a $k$-pebble forest cover of depth $q$. After reproving this result using elementary means, we provide a graph-theoretic analysis of this graph class. This allows us to separate it from the intersection of the class of all graphs of treewidth at most $k-1$ and the class of all graphs of treedepth at most $q$, provided that $q$ is sufficiently larger than $k$. We are able to lift this separation to a (semantic) separation of the respective homomorphism indistinguishability relations. We do this by showing that the graph classes of all graphs of treedepth at most $q$ and of graphs admitting a $k$-pebble forest cover of depth $q$ are homomorphism distinguishing closed, as conjectured by Roberson~(2022). In order to prove Roberson's conjecture for the class of graphs admitting a $k$-pebble forest cover of depth $q$ we characterise the class in terms of a monotone Cops-and-Robber game.The crux is to prove that if Cop has a winning strategy then Cop also has a winning strategy that is monotone.To that end, we show how to transform Cop's winning strategy into a pre-tree-decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first `cleaning up' procedure along the pre-tree-decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of rounds simultaneously across all branches of the decomposition via a vertex exchange argument.

cs.LO↗

arXiv2026

Homomorphism Indistinguishability Relations induced by Quantum Groups

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.

quant-ph↗

arXiv2023

Going Deep and Going Wide: Counting Logic and Homomorphism Indistinguishability over Graphs of Bounded Treedepth and Treewidth

We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment $\mathsf{C}^k_q$, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same $\mathsf{C}^k_q$-sentences if and only if they are homomorphism indistinguishable over the class $\mathcal{T}^k_q$ of graphs admitting a $k$-pebble forest cover of depth $q$. Their proof builds on the categorical framework of game comonads developed by Abramsky, Dawar, and Wang (2017). We reprove their result using elementary techniques inspired by Dvořák (2010). Using these techniques we also give a characterisation of guarded counting logic. Our main focus, however, is to provide a graph theoretic analysis of the graph class $\mathcal{T}^k_q$. This allows us to separate $\mathcal{T}^k_q$ from the intersection of the graph class $\mathcal{TW}_{k-1}$, that is graphs of treewidth less or equal $k-1$, and $\mathcal{TD}_q$, that is graphs of treedepth at most $q$ if $q$ is sufficiently larger than $k$. We are able to lift this separation to the semantic separation of the respective homomorphism indistinguishability relations. A part of this separation is to prove that the class $\mathcal{TD}_q$ is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).

cs.LO↗