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.
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.
Abstract Problems based on the structure of graphs—for example, finding cliques, independent sets, or colorings—are of fundamental importance in classical complexity. Defining well-formulated decision problems for quantum graphs, which are an operator system generalization of graphs, presents several technical challenges. Consequently, the connections between quantum graphs and complexity have been underexplored. In this work, we introduce and study the clique problem for quantum graphs. Our approach utilizes a well-known connection between quantum graphs and quantum channels. The inputs for our problems are presented as quantum channels induced by circuits, which implicitly determine a corresponding quantum graph. We show that when quantifying over all channels, this problem is complete for QMA(2); in fact, it remains QMA(2) -complete when restricted to channels that are probabilistic mixtures of entanglement-breaking and partial trace channels. Quantifying over a subset of entanglementbreaking channels, this problem becomes QMA-complete, and restricting further to deterministic or classical noisy channels gives rise to complete problems for NP and MA, respectively. In this way, we exhibit a classical complexity problem whose natural quantization is QMA(2), rather than QMA, and provide the first problem that allows for a direct comparison of the classes QMA(2), QMA, MA, and NP by quantifying over increasingly larger families of instances. We use methods that are inspired by self-testing to provide a direct proof of QMA(2)-completeness, rather than reducing to a previously studied complete problem. We also give a new proof of the celebrated reduction of QMA( k ) to QMA(2). In parallel, we study a version of the closely related independent set problem for quantum graphs, and provide preliminary evidence that it may be in general weaker in complexity, contrasting to the classical case.
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.
We have generalised the concept of graph states to what we have called mixed graph states, which we define in terms of mixed graphs, that is graphs with both directed and undirected edges, as the density matrix stabilized by the associated stabilizer matrix defined by the mixed graph. We can interpret this matrix as a quantum object by making it part of a larger fully commuting matrix, i.e. where we choose the environment appropriately, and this will imply that our quantum object is a mixed state. We prove that, in the same way as (pure) graph states, the density matrix of a parent of mixed graph state can be written as sum of a few Pauli matrices, well defined from the mixed graph. We have proven that the set of matrices that appear in this sum is fully pair-wise commuting, and form a multiplicative group up to global constants, which is always of maximum size. Furthermore, the cardinality of the set depends solely of the miminum possible number of extension columns/rows, and the number of nodes of the mixed graph. We prove a formula for this cardinality. Finally, in the case of purely undirected graphs, this corresponds to the usual pure graph state. Also, we have developed a way of finding maximal commutative group of such Pauli matrices as a linear subspace problem, for any given mixed graph. We also have proven how the structure of maximal commutative groups is independent of the direction of the arrows of the mixed graph, and also of the undirected edges; this allows the simplification of the problem of finding these groups in general to finding them for a much smaller set of graphs.
We construct a new graph on 120 vertices whose quantum and classical independence numbers are different. At the same time, we construct an infinite family of graphs whose quantum chromatic numbers are smaller than the classical chromatic numbers. Furthermore, we discover the relation to Kochen-Specker sets that characterizes quantum cocliques that are strictly bigger than classical ones. Finally, we prove that for graphs with independence number is two, quantum and classical independence numbers coincide.
Decision Support Systems 392 1.1 The problem of perfect end-to-end secrecy Quantum cryptography claims to bring perfect secrecy to a given line, but speaking honestly, it is no more than this.Using a carrier that is sufficiently fragile to rule out copying it, naturally raises the question of how much distance can be bridged?In fact, nowadays available quantum cryptography allows for communication over a distance of up to 144 km, as demonstrated by Schmitt-Manderbach et al. (2007), but arbitrary distances can yet not be bridged.Although theoretical results due to Lo & Chau (1999) indicate that the noise problem can be overcome, making arbitrary ranges theoretically possible, building networks is inevitable for a global roll-out.Existing solutions mostly rely on trusted relay for that matter.However, why attack the quantum line, if attacking a relay node is fully sufficient?Under the assumption of perfectly protected lines, recent results indicate that without preexisting secrets that are exclusively known to the sender and the receiver, end-to-endsecurity is only achievable under hard constraints on the network topology.To be more precise, let G be a graph that models a network.Let V(G), E(G) be the sets of vertices and edges of G, and assume the sender s and receiver r to be parts of G, that is {s, r} ⊆ V(G).The adversary can be modelled by a set A ⊆ 2 V(G)\{s, r} (the powerset of V(G)\{s, r}), that is we assume that a selection of subsets of vertices can be compromised.If k such sets can become conquered simultaneously, then we face a k-active adversary.An infected vertex v is assumed fully under the adversary's control, so a message passing through v can be read, blocked or modified and v is free to create as many new messages as desired.There is no limitation on computational power or knowledge of the adversary.If removing from G the vertices in any k sets in the adversary structure A cannot disconnect s and r in G, then we call the graph A (k) (s,r)-subconnected.If, by doing so, the network cannot be disconnected at all, then the graph is said to be A (k) -subconnected.Referring to these notions, a network permits perfectly secure message delivery from s to r if and only if the graph G is A (2) (s,r)-subconnected.The reader may consult Ashwin Kumar et al. (2002) for a proof.Different, yet no less stringent requirements are imposed by Wang & Desmedt (2008): among related results, the following necessary condition best highlights the difficulty of achieving unconditional security in a real-life network: if for u ≥ 1, 3(ku) + 1 ≥ k + 1 directed node-disjoint paths from s to r exist, then a necessary condition for perfectly secure message transmission from s to r against a k-active adversary is that there are u directed node disjoint paths (these u paths are also disjoint from the 3(ku) + 1 paths from s to r) from r to s.The described adversary model applies to many situations, as for example machines running certain software may all suffer from the same security holes.Networks equipped with devices from different vendors may be considered vulnerable if one vendor's devices turn out to be insecure.A k-active adversary would correspond to k vendors cooperating, or equivalently arise, if k vendors obtained the same malicious module from a single fraudulent manufacturer, turning a heterogeneous set of products into a possible backdoor for an adversary. Decision theory and system securityMany results either guarantee or rule out perfectly secret communication, but this might not be satisfactory.If perfectly secure communication is not possible, then how much is achievable with the given resources?A variety of security metrics has been proposed, but a measure of security is yet missing.This work summarizes a decision-theoretic approach to quantifying risk in terms that can be specified to best suit the application at hand.www.intechopen.
Selecting mutually compatible natural gas transportation contracts is a practically important optimization task in which operators must choose from many candidate agreements subject to temporal, infrastructural, and flow-related constraints. As the number of candidates grows, the resulting search space becomes difficult to explore exhaustively. We study a pairwise abstraction of this task, formulated as a Maximum Clique problem on a contract-compatibility graph, or equivalently as a Maximum Independent Set (MIS) problem on the complement graph.
Building on recent work, this paper studies a quantum-classical framework for solving large-scale MIS instances within the limitations of noisy quantum hardware. The approach combines iterative classical graph reduction with quantum-guided optimization to progressively simplify the search space while maintaining high solution quality. This enables large candidate spaces to be reduced to smaller subproblems that are more suitable for execution on current quantum computers.
We evaluate the approach on fifteen benchmark instances from the Quantum Optimization Benchmarking Library (QOBLIB), obtaining an average approximation ratio of 0.996 and recovering optimal solutions for fourteen instances, including graphs with up to 186 vertices. We further evaluate the algorithm on six synthetic pairwise contract-compatibility graphs containing up to 900 contracts, where the proposed method achieves an average approximation ratio of 0.989 and obtains optimal solutions in four cases. These experiments demonstrate the ability of the hybrid MIS solver to reduce industrially motivated graphs. The pairwise abstraction serves as the first step of a two-stage screening procedure that narrows the candidate contracts to a smaller set of mutually compatible ones, which can then be verified against pipeline-capacity constraints.