Geometric properties of evolutionary graph states and their detection on a quantum computer

Geometric properties of evolutionary graph states of spin systems generated by the operator of evolution with Ising Hamiltonian are examined, using their relationship with fluctuations of energy. We find that the geometric characteristics of the graph states depend on properties of the corresponding graphs. Namely, it is obtained that the fluctuations of energy in graph states and therefore the velocity of quantum evolution, the curvature and the torsion of the states are related with the total number of edges, triangles and squares in the corresponding graphs. The obtained results give a possibility to quantify the number of edges, triangles and squares in a graph on a quantum devise and achieve quantum supremacy in solving this problem with the development of a multi-qubit quantum computer. Geometric characteristics of graph states corresponding to a chain, a triangle, and a square are detected on the basis of calculations on IBM's quantum computer ibmq-manila.