Entanglement of multi-qubit quantum graph states and studies structural properties of tripartite graphs with quantum programming

TL;DR

This paper introduces a method to construct multi-qubit entangled states based on tripartite graphs, linking quantum properties with graph structures, and demonstrates their analysis through quantum simulations.

Contribution

It presents a novel approach to relate quantum graph states with tripartite graph structures, enabling structural analysis via quantum programming.

Findings

Quantum entanglement distance correlates with graph edge weights and vertex degrees.

Quantum correlators relate to structural graph properties like neighbors and cycles.

Simulations on noisy quantum hardware validate theoretical predictions.

Abstract

We propose a method for constructing multi-qubit entangled quantum states representing weighted tripartite graphs. An expression for the entanglement distance for multi-qubit states corresponding to arbitrary tripartite graph structures is obtained. The entanglement of a qubit with the rest of the system in a quantum graph state is determined by the weights of the edges in the closed neighborhood of the corresponding vertex and by its degree with respect to other sets. We also calculate quantum correlators in the general case of tripartite quantum graph states. We establish a relationship between these quantum properties and the structural properties of the corresponding tripartite graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. As an illustrative example, we consider a tripartite graph forming a triangle and compute the entanglement distance using quantum simulations on the AerSimulator with noise models. The numerical results are consistent with the theoretical predictions. The obtained results demonstrate that quantum graph states provide an effective framework for studying structural properties of tripartite graphs. They open up the possibility of investigating such properties using quantum programming. It is worth highlighting that tripartite graphs have applications in solving practical problems such as resource allocation, scheduling, and database and hypergraph modeling.