We study the class of edge-coloured graphs arising from the graph-theoretic representation of quantum photonic experiments that generate multipartite W-states. Abstracting away physical amplitudes and phases, we introduce W-state graphs: matching-covered graphs equipped with a half-edge 2-colouring such that every perfect matching contains exactly one bichromatic edge and every vertex is incident with a red half-edge. Our main contribution is a complete structural characterization of W-state graphs. We show that a graph is a W-state graph if and only if each of its 3-connected components is a W-cone, a simple and rigid building block defined by a universal vertex and a factor-critical base. This characterization implies that no W-state graph is simple and yields a recognition algorithm running as fast as verifying whether a graph is matching-covered. We also show that the natural generalization to Dicke states encounters a complexity barrier: verifying one of the two Dicke state conditions is itself coNP-complete, resolving an open problem of Vardi and Zhang [IJCAI 2023]. Our results place W-state graphs firmly within classical matching theory and precisely delineate the combinatorial structures capable of realizing idealized W-states in the experiment-graph framework.
We study the class of edge-coloured graphs arising from the graph-theoretic representation of quantum photonic experiments that generate multipartite W-states. Abstracting away physical amplitudes and phases, we introduce W-state graphs: matching-covered graphs equipped with a half-edge 2-colouring such that every perfect matching contains exactly one bichromatic edge and every vertex is incident with a red half-edge. Our main contribution is a complete structural characterization of W-state graphs. We show that a graph is a W-state graph if and only if each of its 3-connected components is a W-cone, a simple and rigid building block defined by a universal vertex and a factor-critical base. This characterization implies that no W-state graph is simple and yields a recognition algorithm running as fast as verifying whether a graph is matching-covered. We also show that the natural generalization to Dicke states encounters a complexity barrier: verifying one of the two Dicke state conditions is itself coNP-complete, resolving an open problem of Vardi and Zhang [IJCAI 2023]. Our results place W-state graphs firmly within classical matching theory and precisely delineate the combinatorial structures capable of realizing idealized W-states in the experiment-graph framework.
Graph theory is a powerful tool in quantum computing, with applications ranging from quantum circuit synthesis and optimization to entanglement mapping. Recent work has shown how one can use edge-colored graphs to model photonic experiments that generate GHZ and W states. However, the latter work also proved that verifying that a graph models a Dicke state experiment is coNP-complete. In this work, we provide families of graphs that generate ∣Dka⟩⊗∣0⟩⊗b, where b=∣a−2k∣ is the number of spectator modes. The graph setup consists of a doubled complete subgraph on a vertices and a collection of auxiliary vertices. We prove that every coincidence carries exactly k excitations, every weight-k computational basis state on bitstrings of length a is realized, and each of those bitstrings is realized exactly (n/2)! times, where n=a+b. Since verifying the Dicke FORALL condition is coNP-complete in general, constructing explicit families that provably generate Dicke states is of interest.
Greenberger-Horne-Zeilinger (GHZ) states are multipartite entangled quantum states that play a central role in quantum information theory, with applications in quantum communication, cryptography, and distributed quantum computing. A major experimental goal is to create high-dimensional GHZ states using multiple entangled parties. Krenn, Gu and Zeilinger discovered a surprising bridge between these quantum-optical experiments and graph theory: certain photonic experiments can be represented by edge-coloured, edge-weighted graphs, and the dimension of the produced GHZ state equals the number of feasible monochromatic inherited vertex colourings. This led Krenn and Gu to conjecture that every GHZ graph with more than four vertices has dimension at most 2. A counterexample would give a new high-dimensional multipartite GHZ construction, while a proof would clarify fundamental resource limitations in quantum optics. It is known that the conjecture holds for graphs of vertex connectivity at most 2 [MFCS 2024], and that simple GHZ graphs with n>4 vertices have dimension less than n/2 [Quantum 2024]. However, no general upper bound depending only on vertex connectivity was known. We prove that every simple connected GHZ graph G with at least four vertices has dimension at most 2κ(G)−1+(2κ(G)), where κ(G) is the vertex connectivity of G. We also prove new bounds in terms of edge connectivity and maximum degree.
We show a surprising link between experimental setups to realize high-dimensional multipartite quantum states and graph theory. In these setups, the paths of photons are identified such that the photon-source information is never created. We find that each of these setups corresponds to an undirected graph, and every undirected graph corresponds to an experimental setup. Every term in the emerging quantum superposition corresponds to a perfect matching in the graph. Calculating the final quantum state is in the #P-complete complexity class, thus it cannot be done efficiently. To strengthen the link further, theorems from graph theory-such as Hall's marriage problem-are rephrased in the language of pair creation in quantum experiments. We show explicitly how this link allows one to answer questions about quantum experiments (such as which classes of entangled states can be created) with graph theoretical methods, and how to potentially simulate properties of graphs and networks with quantum experiments (such as critical exponents and phase transitions).
In a previous version of this document we misinterpreted the runtime of a part of the described algorithm. Indeed, the runtime is not better than the Grover-Algorithm. We therefor withdraw this work. We present a novel algorithmic approach to find a proper vertex colouring of a graph with d colours, if it exists. We associate a d-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime of the problem-specific part of the algorithm scales polynomially in the number of edges.
Many properties of classical graphs are defined in terms of subsets of the vertex set. Examples include connected components, which are subsets X⊆V(G) such that X×Xc and E(G) are disjoint, or independent sets, for which X×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.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.