Related papers: Some examples of quantum graphs
An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve a quantum computer into an entanglement of states which map onto the set of all possible paths…
We investigate a quantum generalization of the Mycielski construction for quantum graphs. In particular, we analyze the quantum symmetries of quantum Mycielskians and their relation to the symmetries of the underlying quantum graphs. We…
For a set of quantum states generated by the action of a group, we consider the graph obtained by considering two group elements adjacent whenever the corresponding states are non-orthogonal. We analyze the structure of the connected…
Graphs are topological spaces that include broader objects than discretized manifolds, making them interesting playgrounds for the study of quantum phases not realized by symmetry breaking. In particular they are known to support anyons of…
A fundamental problem in quantum information is to describe efficiently multipartite quantum states. An efficient representation in terms of graphs exists for several families of quantum states (graph, cluster, stabilizer states),…
We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of…
We develop an equivariant theory of graphs with respect to quantum symmetries and present a detailed exposition of various examples. We portray unitary tensor categories as a unifying framework encompassing all finite classical simple…
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…
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…
We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley…
We introduce the quantum Cayley graphs associated to quantum discrete groups and study them in the case of trees. We focus in particular on the notion of quantum ascending orientation and describe the associated space of edges at infinity,…
We define a notion of quantum automorphism group of Graph C*-algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of underlying…
We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups…
We describe a new class of scattering matrices for quantum graphs in which back-scattering is prohibited. We discuss some properties of quantum graphs with these scattering matrices and explain the advantages and interest in their study. We…
In this survey article we give basic introduction to the theory of quantum families of maps. We begin with a general look at non-commutative (or "quantum") topology. Then we formulate all our results in this language. Existence of quantum…
We introduce quantum association schemes. This allows to define distance regular and strongly regular quantum graphs. We bring examples thereof. In addition, we formulate the duality for translation quantum association schemes corresponding…
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…
Graph structures are ubiquitous throughout the natural sciences. Here we consider graph-structured quantum data and describe how to carry out its quantum machine learning via quantum neural networks. In particular, we consider training data…
We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p>>k$ the graph has no quantum symmetry, in the sense that the quantum…