arXiv subjects

David E. Roberson

Publications and source records attributed to David E. Roberson.

At least 19 recordsLinked to original sources

arXiv2025

Free Inhomogeneous Wreath Product of Quantum Groups

We introduce the free inhomogeneous wreath product of compact matrix quantum groups, which generalizes the free wreath product (Bichon 2004). We use this to present a general technique to determine quantum automorphism groups of connected graphs in terms of their maximal biconnected subgraphs, provided that we have sufficient information about their quantum automorphism groups. We show that this requirement is met for forests, outerplanar graphs, and block graphs leading to algorithms to compute the quantum automorphism groups of these graphs.

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arXiv2025

Group Invariant Quantum Latin Squares

A quantum Latin square is an $n \times n$ array of unit vectors where each row and column forms an orthonormal basis of a fixed complex vector space. We introduce the notion of $(G,G')$-invariant quantum Latin squares for finite groups $G$ and $G'$. These are quantum Latin squares with rows and columns indexed by $G$ and $G'$ respectively such that the inner product of the $a,b$-entry with the $c,d$-entry depends only on $a^{-1}c \in G$ and $b^{-1}d \in G'$. This definition is motivated by the notion of group invariant bijective correlations introduced in [Roberson \& Schmidt (2020)], and every group invariant quantum Latin square produces a group invariant bijective correlation, though the converse does not hold. In this work we investigate these group invariant quantum Latin squares and their corresponding correlations. Our main result is that, up to applying a global isometry to every vector in a $(G,G')$-invariant quantum Latin square, there is a natural bijection between these objects and trace and conjugate transpose preserving isomorphisms between the group algebras of $G$ and $G'$. This in particular proves that a $(G,G')$-invariant quantum Latin square exists if and only if the multisets of degrees of irreducible representations are equal for $G$ and $G'$. Another motivation for this line of work is that whenever Cayley graphs for groups $G$ and $G'$ are quantum isomorphic, then there is a $(G,G')$-invariant quantum correlation witnessing this, and thus it suffices to consider such correlations when searching for quantum isomorphic Cayley graphs. Given a group invariant quantum correlation, we show how to construct all pairs of graphs for which it gives a quantum isomorphism.

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arXiv2025

Quantum Perfect Matchings

We investigate quantum and nonsignaling generalizations of perfect matchings in graphs using nonlocal games. Specifically, we introduce nonlocal games that test for $L$-perfect matchings in bipartite graphs, perfect matchings in general graphs and hypergraphs, and fractional perfect matchings. Our definitions come from the fact that these games are classical property tests for the corresponding matching conditions. We use the existence of perfect quantum and nonsignaling strategies for these games to define quantum and nonsignaling versions of perfect matchings. Finally, we provide characterizations of when graphs exhibit these extended properties: - For nonsignaling matchings, we give a complete combinatorial characterizations. In particular, a graph has a nonsignaling perfect matching if and only if it admits a fractional perfect matching that has bounded value on triangles. \item In bipartite graphs, the nonsignaling $L$-perfect matching property is achieved exactly when the left component of the graph can be split into two disjoint subgraphs: one with a classical $L$-perfect matching and another with left-degree 2. - In the quantum setting, we show that complete graphs $K_n$ with odd $n \geq 7$ have quantum perfect matchings. We prove that a graph has a quantum perfect matching if and only if the quantum independence number of its line graph is maximal, extending a classical relationship between perfect matchings and line graph independence numbers. - For bipartite graphs, we establish that the $L$-perfect matching game does not exhibit quantum pseudotelepathy, but we characterize the quantum advantage for complete bipartite graphs $K_{n,2}$. - Additionally, we prove that deciding quantum perfect matchings in hypergraphs is undecidable and leave open the question of its complexity in graphs.

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arXiv2024

Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

We show that feasibility of the $t^\text{th}$ level of the Lasserre semidefinite programming hierarchy for graph isomorphism can be expressed as a homomorphism indistinguishability relation. In other words, we define a class $\mathcal{L}_t$ of graphs such that graphs $G$ and $H$ are not distinguished by the $t^\text{th}$ level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in $\mathcal{L}_t$. By analysing the treewidth of graphs in $\mathcal{L}_t$, we prove that the $3t^\text{th}$ level of Sherali--Adams linear programming hierarchy is as strong as the $t^\text{th}$ level of Lasserre. Moreover, we show that this is best possible in the sense that $3t$ cannot be lowered to $3t-1$ for any $t$. The same result holds for the Lasserre hierarchy with non-negativity constraints, which we similarly characterise in terms of homomorphism indistinguishability over a family $\mathcal{L}_t^+$ of graphs. Additionally, we give characterisations of level-$t$ Lasserre with non-negativity constraints in terms of logical equivalence and via a graph colouring algorithm akin to the Weisfeiler--Leman algorithm. This provides a polynomial time algorithm for determining if two given graphs are distinguished by the $t^\text{th}$ level of the Lasserre hierarchy with non-negativity constraints.

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arXiv2024

NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability

Man\v{c}inska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they admit the same number of homomorphisms from any planar graph. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general, which motivates the study of its relaxations. In the classical setting, Roberson and Seppelt [ICALP'23] characterized the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism in terms of equality of homomorphism counts from an appropriate graph class. The NPA hierarchy, a noncommutative generalization of the Lasserre hierarchy, provides a sequence of semidefinite programming relaxations for quantum isomorphism. In the quantum setting, we show that the feasibility of each level of the NPA hierarchy for quantum isomorphism is equivalent to equality of homomorphism counts from an appropriate class of planar graphs. Combining this characterization with the convergence of the NPA hierarchy, and noting that the union of these classes is the set of all planar graphs, we obtain a new proof of the result of Man\v{c}inska and Roberson [FOCS'20] that avoids the use of quantum groups. Moreover, this homomorphism indistinguishability characterization also yields a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.

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arXiv2024

Quantum Sabidussi's Theorem

Sabidussi's theorem [Duke Math. J. 28, 1961] gives necessary and sufficient conditions under which the automorphism group of a lexicographic product of two graphs is a wreath product of the respective automorphism groups. We prove a quantum version of Sabidussi's theorem for finite graphs, with the automorphism groups replaced by quantum automorphism groups and the wreath product replaced by the free wreath product of quantum groups. This extends the result of Chassaniol [J. Algebra 456, 2016], who proved it for regular graphs. Moreover, we apply our result to lexicographic products of quantum vertex transitive graphs, determining their quantum automorphism groups even when Sabidussi's conditions do not apply.

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arXiv2023

Quantum automorphism groups of trees

We give a characterisation of quantum automorphism groups of trees. In particular, for every tree, we show how to iteratively construct its quantum automorphism group using free products and free wreath products. This can be considered a quantum version of Jordan's theorem for the automorphism groups of trees. This is one of the first characterisations of quantum automorphism groups of a natural class of graphs with quantum symmetry.

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arXiv2023

Asymmetric graphs with quantum symmetry

We present an infinite sequence of finite graphs with trivial automorphism group and non-trivial quantum automorphism group. These are the first known examples of graphs with this property. Moreover, to the best of our knowledge, these are the first examples of any asymmetric classical space that has nontrivial quantum symmetries. Our construction is based on solution groups to (binary) linear systems, as defined by Cleve, Liu and Slofstra in the context of non-local games. We first show that the dual quantum group of every solution group occurs as the quantum automorphism group of some graph, and then construct an infinite sequence of systems whose solution groups are nontrivial perfect groups. This leads to the desired sequence of graphs. In addition to our main result, we prove a number of related results that allow us to answer several open problems from the literature. We prove a weak quantum analog of Frucht's theorem, namely that every finite classical group $\Gamma$ occurs as the quantum automorphism group of a finite graph. Combined with our main result, this shows that, for every finite group $\Gamma$, there are graphs $G_1$ and $G_2$ that both have classical automorphism group isomorphic to $\Gamma$ but one of them has quantum symmetry and the other does not. Therefore, the quantum automorphism group of a graph is never determined by its classical automorphism group, and there do not exist any "quantum excluding groups".

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arXiv2022

Oddomorphisms and homomorphism indistinguishability over graphs of bounded degree

We introduce (weak) oddomorphisms of graphs which are homomorphisms with additional constraints based on parity. These maps turn out to have interesting properties (e.g., they preserve planarity), particularly in relation to homomorphism indistinguishability. Graphs $G$ and $H$ are *homomorphism indistinguishable* over a family $\mathcal{F}$ if $\hom(F,G) = \hom(F,H)$ for all $F \in \mathcal{F}$, where $\hom(F,G)$ is the number of homomorphisms from $F$ to $G$. A classical result of Lovász says that isomorphism is equivalent to homomorphism indistinguishability over the class of all graphs. In recent years it has been shown that many homomorphism indistinguishability relations have natural algebraic and/or logical formulations. Currently, much research in this area is focused on finding such reformulations. We aim to broaden the scope of current research on homomorphism indistinguishability by introducing new concepts/constructions and proposing several conjectures/questions. In particular, we conjecture that every family closed under disjoint unions and minors gives rise to a distinct homomorphism indistinguishability relation. We also show that if $\mathcal{F}$ is a family of graphs closed under disjoint unions, restrictions to connected components, and weak oddomorphisms, then $\mathcal{F}$ satisfies a certain maximality or closure property: homomorphism indistinguishability over $\mathcal{F}$ of $G$ and $H$ does not imply $\hom(F,G) = \hom(F,H)$ for any $F \notin \mathcal{F}$. This allows us to answer a question raised over ten years ago, showing that homomorphism indistinguishability over graphs of bounded degree is not equivalent to isomorphism.

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arXiv2021

Quantum symmetry vs nonlocal symmetry

We introduce the notion of nonlocal symmetry of a graph $G$, defined as a winning quantum correlation for the $G$-automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on $C(\text{Qut}(G))$ -- the algebra of functions on the quantum automorphism group of $G$. This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of $C(\text{Qut}(G))$. Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points, $S_4^+$, does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group $Γ$ and a permutation $π$ on $|Γ|$ elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations $π$ result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when $\mathbb{Z}_2^3$ is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.

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arXiv2020

Limit of Gaussian operations and measurements for Gaussian state discrimination, and its application to state comparison

We determine the optimal method of discriminating and comparing quantum states from a certain class of multimode Gaussian states and their mixtures when arbitrary global Gaussian operations and general Gaussian measurements are allowed. We consider the so-called constant-$\hat{p}$ displaced states which include mixtures of multimode coherent states arbitrarily displaced along a common axis. We first show that no global or local Gaussian transformations or generalized Gaussian measurements can lead to a better discrimination method than simple homodyne measurements applied to each mode separately and classical postprocessing of the results. This result is applied to binary state comparison problems. We show that homodyne measurements, separately performed on each mode, are the best Gaussian measurement for binary state comparison. We further compare the performance of the optimal Gaussian strategy for binary coherent states comparison with these of non-Gaussian strategies using photon detections.

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arXiv2020

Graph isomorphism: Physical resources, optimization models, and algebraic characterizations

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 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.

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arXiv2019

Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs

Over 50 years ago, Lovász proved that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph [Acta Math. Hungar. 18 (1967), pp. 321--328]. In this work we prove that two graphs are quantum isomorphic (in the commuting operator framework) if and only if they admit the same number of homomorphisms from any planar graph. As there exist pairs of non-isomorphic graphs that are quantum isomorphic, this implies that homomorphism counts from planar graphs do not determine a graph up to isomorphism. Another immediate consequence is that determining whether there exists some planar graph that has a different number of homomorphisms to two given graphs is an undecidable problem, since quantum isomorphism is known to be undecidable. Our characterization of quantum isomorphism is proven via a combinatorial characterization of the intertwiner spaces of the quantum automorphism group of a graph based on counting homomorphisms from planar graphs. This result inspires the definition of "graph categories" which are analogous to, and a generalization of, partition categories that are the basis of the definition of easy quantum groups. Thus we introduce a new class of "graph-theoretic quantum groups" whose intertwiner spaces are spanned by maps associated to (bi-labeled) graphs. Finally, we use our result on quantum isomorphism to prove an interesting reformulation of the Four Color Theorem: that any planar graph is 4-colorable if and only if it has a homomorphism to a specific Cayley graph on the symmetric group $S_4$ which contains a complete subgraph on four vertices but is not 4-colorable.

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arXiv2019

Algebras, graphs and thetas

We extend the clique-coclique inequality, previously known to hold for graphs in association schemes and vertex-transitive graphs, to graphs in homogeneous coherent configurations and 1-walk regular graphs. We further generalize it to a stronger inequality involving the Lovász theta number of such graphs, and some theta variants, including characterizations of the equality.

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arXiv2019

Graph Homomorphisms via Vector Colorings

In this paper we study the existence of homomorphisms $G\to H$ using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number $t \ge 2$ for which there exists an assignment of unit vectors $i\mapsto p_i$ to its vertices such that $\langle p_i, p_j\rangle\le -1/(t-1),$ when $i\sim j$. Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for $n>2r$ the Kneser graph $K_{n:r}$ and the $q$-Kneser graph $qK_{n:r}$ are cores, and furthermore, that for $n/r = n'/r'$ there exists a homomorphism $K_{n:r}\to K_{n':r'}$ if and only if $n$ divides $n'$. In terms of new applications, we show that the even-weight component of the distance $k$-graph of the $n$-cube $H_{n,k}$ is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms $H_{n,k}\to H_{n',k'}$ when $n/k = n'/k'$. Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence's list of strongly regular graphs and found that at least 84% are cores.

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arXiv2018

Cores of Cubelike Graphs

A graph is $\textit{cubelike}$ if it is a Cayley graph for some elementary abelian $2$-group $\mathbb{Z}_2^n$. The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Nešetřil and Šámal (On tension-continuous mappings. $\textit{European J. Combin.,}$ 29(4):1025--1054, 2008) asked whether the core of a cubelike graph is cubelike, but since then very little progress has been made towards resolving the question. Here we investigate the structure of the core of a cubelike graph, deducing a variety of structural, spectral and group-theoretical properties that the core "inherits" from the host cubelike graph. These properties constrain the structure of the core quite severely --- even if the core of a cubelike graph is not actually cubelike, it must bear a very close resemblance to a cubelike graph. Moreover we prove the much stronger result that not only are these properties inherited by the core of a cubelike graph, but also by the orbital graphs of the core. Even though the core and its orbital graphs look very much like cubelike graphs, we are unable to show that this is sufficient to characterise cubelike graphs. However, our results are strong enough to eliminate all non-cubelike vertex-transitive graphs on up to $32$ vertices as potential cores of cubelike graphs (of any size). Thus, if one exists at all, a cubelike graph with a non-cubelike core has at least $128$ vertices and its core has at least $64$ vertices.

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arXiv2018

Nonlocal Games and Quantum Permutation Groups

We present a strong connection between quantum information and quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs $X$ and $Y$ are quantum isomorphic if and only if there exists $x \in V(X)$ and $y \in V(Y)$ that are in the same orbit of the quantum automorphism group of the disjoint union of $X$ and $Y$. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. We exploit this link by using ideas and results from quantum information in order to prove new results about quantum automorphism groups, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum group literature. Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other.

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arXiv2018

Vector Coloring the Categorical Product of Graphs

A vector $t$-coloring of a graph is an assignment of real vectors $p_1, \ldots, p_n$ to its vertices such that $p_i^Tp_i = t-1$ for all $i=1, \ldots, n$ and $p_i^Tp_j \le -1$ whenever $i$ and $j$ are adjacent. The vector chromatic number of $G$ is the smallest real number $t \ge 1$ for which a vector $t$-coloring of $G$ exists. For a graph $H$ and a vector $t$-coloring $p_1,\ldots,p_n$ of a graph $G$, the assignment $(i,\ell) \mapsto p_i$ is a vector $t$-coloring of the categorical product $G \times H$. It follows that the vector chromatic number of $G \times H$ is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring. Furthermore, we prove a necessary and sufficient condition for when all of the optimal vector colorings of the product can be expressed in terms of the optimal vector colorings of the factors. The vector chromatic number is closely related to the well-known Lovász theta function, and both of these parameters admit formulations as semidefinite programs. This connection to semidefinite programming is crucial to our work and the tools and techniques we develop could likely be of interest to others in this field.

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