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Kh. Gnatenko, Geom. Properties of Quantum States Representing Directed Graphs on Quantum Computer

Abstract. We investigate multi-qubit states that can be described using directed graphs G(V,E). Our main focus is on examining their geometric properties, such as curvature and torsion [1]. It has been shown that the curvature of quantum states is determined by the sum of the weighted degrees of the vertices in the graphs, where the weights in G(V,E) are raised to the second and fourth powers [2]. Additionally, the curvature depends on sum of the products of the weights of edges that form squares within the graph G(V,E). Torsion is associated with the sum of the products of the weights of edges that create triangles in the graph G(V,E). These geometric properties were calculated on IBM's quantum computer for a quantum graph state corresponding to a chain [2]. The results of quantum calculations are in agreement with the theoretical ones. We have also examined quantum states that can be represented by directed networks [3]. We computed the geometric measure of entanglement of these states both analytically and through programming on the AerSimulator. The computations were done on the basis of the relation of the entanglement with the mean spin obtained in [4]. We found a relationship between the geometric measure of entanglement of a qubit with other qubits in the graph states with the weights of incoming and outgoing arcs, as well as the outdegree and indegree of the vertices corresponding to the qubit in the graph [3]. References [1] H. P. Laba, V. M. Tkachuk, Condens. Matter Phys. 20, 13003 (2017). [2] Kh. P. Gnatenko, Relation of curvature and torsion of weighted graph states with graph properties and its studies on a quantum computer, arXiv:2408.01511 (2024). [3] Kh. P. Gnatenko, Physics Letters A 521, 129815 (2024). [4] A. M. Frydryszak, M. I. Samar, V. M. Tkachuk, Eur. Phys. J. D 71, 233 (2017)