In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex- and edge-colored graph with certain properties, we construct an uncolored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the first known example of a graph that has quantum symmetry and finite quantum automorphism group. Furthermore, we construct a pair of quantum isomorphic, non-isomorphic graphs that both have no quantum symmetry.
Abstract 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 non‐trivial 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 non‐trivial 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 occurs as the quantum automorphism group of a finite graph. Combined with our main result, this shows that, for every finite group , there are graphs and that both have classical automorphism group isomorphic to 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.”
We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.
Given a graph
G, does it have quantum symmetries, or in other words, is the algebra of the quantum automorphism group non-commutative? In the world of ordinary groups, determining whether a finitely presented group is commutative is undecidable. This is an immediate consequence of the Adian-Rabin theorem, which shows that many natural properties of finitely presented groups are undecidable. In this paper, we prove an Adian-Rabin theorem for quantum automorphism groups of graphs, showing that many natural properties of the quantum automorphism group of a graph (including having quantum symmetries) are undecidable. We also show that it's undecidable to determine if the discrete quantum automorphism group is non-trivial. Our methods are based on solution groups of linear system nonlocal games, and we also prove an Adian-Rabin theorem in this context.
The computational cost of simulating quantum many-body systems can often be reduced by taking advantage of physical symmetries. While methods exist for specific symmetry classes, a general algorithm to find the full permutation symmetry group of an arbitrary Pauli Hamiltonian is notably lacking. This paper introduces a new method that identifies this symmetry group by establishing an isomorphism between the Hamiltonian's permutation symmetry group and the automorphism group of a coloured bipartite graph constructed from the Hamiltonian. We formally prove this isomorphism and show that for physical Hamiltonians with bounded locality and interaction degree, the resulting graph has a bounded degree, reducing the computational problem of finding the automorphism group to polynomial time. The algorithm's validity is empirically confirmed on various physical models with known symmetries. We further show that the problem of deciding whether two Hamiltonians are permutation-equivalent is polynomial-time reducible to the graph isomorphism problem using our graph representation. This work provides a general, structurally exact tool for algorithmic symmetry finding, enabling the scalable application of these symmetries to Hamiltonian simulation problems.
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(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(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,
S4+, 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
Z23 is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.