Properties of multiqubit variational quantum states representing weighted graphs and their computing with quantum programming

We study multiqubit variational quantum states that can be considered as weighted quantum graph states. These states are constructed as single-layer variational circuits with RX rotations and RZZ entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are directly related to the degrees of the corresponding vertices in graph. As an example, we analyze the state associated with the star graph K_1,4 using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of classical graphs via quantum computing.