Studies of properties of bipartite graphs with quantum programming

Abstract

Multi-qubit quantum states corresponding to bipartite graphs $G(U,V,E)$ are examined. These states are constructed by applying $CNOT$ gates to an arbitrary separable multi-qubit quantum state. The entanglement distance of the resulting states is derived analytically for an arbitrary bipartite graph structure. A relationship between entanglement and the vertex degree is established. Additionally, we identify how quantum correlators relate to the number of vertices with odd and even degrees in the sets $U$ and $V$. Based on these results, quantum protocols are proposed for quantifying the number of vertices with odd and even degrees in the sets $U$ and $V$. For a specific case where the bipartite graph is a star graph, we analytically calculate the dependence of entanglement distance on the state parameters. These results are also verified through quantum simulations on the AerSimulator, including noise models. Furthermore, we use quantum calculations to quantify the number of vertices with odd degrees in $U$ and $V$. The results agree with the theoretical predictions.