Studies of properties of bipartite graphs with quantum programming

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.

💡 Research Summary

This research presents a sophisticated framework for investigating the structural properties of bipartite graphs $G(U,V,E)$ through the lens of quantum information science. The core methodology involves the construction of multi-qubit quantum states that encode the topological features of a bipartite graph. By applying CNOT gates, corresponding to each edge $(l, k) \in E$, to an arbitrary separable multi-qubit state, the researchers successfully mapped the graph’s adjacency information into the entanglement structure of a quantum state.

The study achieves several significant scientific milestones. First, it provides an analytical derivation of the “entanglement distance” for arbitrary bipartite graph structures. A pivotal finding of this work is the establishment of a direct mathematical relationship between the entanglement distance of the generated quantum states and the vertex degree of the underlying graph. This demonstrates that the connectivity of the graph is intrinsically linked to the non-local correlations within the quantum system.

Second, the paper introduces a novel quantum protocol for graph property estimation. By analyzing quantum correlators, the researchers identified a clear relationship between these correlators and the parity (the count of odd and even degrees) of the vertices within the bipartite sets $U$ and $V$. This breakthrough allows for the quantification of complex graph-theoretic properties—specifically the number of odd and even degree vertices—using quantum measurement techniques, effectively turning quantum computing into a tool for topological graph analysis.

Third, the research provides a rigorous validation of the proposed theory. Using the star graph as a specific case study, the authors analytically calculated how the entanglement distance fluctuates based on the parameters of the initial quantum state. To ensure the practical relevance of these findings, the researchers performed quantum simulations using the IBM AerSimulator, incorporating realistic noise models. The simulation results demonstrated high fidelity with the theoretical predictions, confirming that the proposed protocols are robust even in the presence of environmental decoherence and gate errors.

In conclusion, this paper bridges the gap between discrete mathematics and quantum mechanics. By demonstrating that graph-theoretic properties like vertex degree parity can be extracted via quantum correlations, the study paves the way for new classes of quantum algorithms designed for network science, complex system analysis, and advanced graph-based computational problems.