Entanglement of multi-qubit quantum graph states and studies structural properties of tripartite graphs with quantum programming

Abstract: We propose a method for constructing multi-qubit entangled quantum states representing weighted tripartite graphs. An expression for the entanglement distance for multi-qubit states corresponding to arbitrary tripartite graph structures is obtained. The entanglement of a qubit with the rest of the system in a quantum graph state is determined by the weights of the edges in the closed neighborhood of the corresponding vertex and by its degree with respect to other sets. We also calculate quantum correlators in the general case of tripartite quantum graph states. We establish a relationship between these quantum properties and the structural properties of the corresponding tripartite graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. As an illustrative example, we consider a tripartite graph forming a triangle and compute the entanglement distance using quantum simulations on the AerSimulator with noise models. The numerical results are consistent with the theoretical predictions. The obtained results demonstrate that quantum graph states provide an effective framework for studying structural properties of tripartite graphs. They open up the possibility of investigating such properties using quantum programming. It is worth highlighting that tripartite graphs have applications in solving practical problems such as resource allocation, scheduling, and database and hypergraph modeling.

• The paper introduces a protocol to map weighted tripartite graphs onto entangled multi-qubit states using parameterized two-qubit gates.
• The paper derives analytical expressions for entanglement distance and quantum correlators that reflect graph structural properties such as vertex degree and 4-cycles.
• The paper validates theoretical predictions via IBM AerSimulator, demonstrating robustness against noise and potential for quantum-enhanced graph analysis.

Entanglement and Structural Analysis of Tripartite Quantum Graph States via Quantum Programming

Methodology for Constructing Tripartite Quantum Graph States

This paper introduces a systematic approach for constructing multi-qubit quantum states that mathematically encode weighted tripartite graphs. The mapping relies on applying parameterized two-qubit gates—specifically, $RXY_{kl}(\theta_{kl})$, $RXZ_{kl}(\theta_{kl})$, and $RYZ_{kl}(\theta_{kl})$—to an arbitrary separable initial state. Vertices of the graph correspond to qubits, and edges encode entangling interactions, with gate parameters derived from the adjacency matrix. The tripartite structure is maintained by enforcing that the sets $U$, $V$, and $W$ (the three partitions) have no internal edges, guaranteeing inter-set connectivity only.

The construction ensures commutativity among gates acting across different partitions, simplifying circuit implementation and analysis. Each entangled quantum graph state is uniquely associated with a specific weighted, directed tripartite graph, allowing the physical quantum state to mirror both topology and weight distribution.

Analytical Results: Entanglement Distance and Quantum Correlators

A central quantitative metric, the entanglement distance (EED), is derived for arbitrary tripartite quantum graph states. The EED for any qubit measures its entanglement with the rest of the system and is directly related to mean spin values, making it experimentally accessible. The explicit expressions reveal that the EED of a qubit representing a vertex is governed by the weights of adjacent edges and the degree of connectivity to vertices in other partitions.

For unweighted graphs and uniform initialization, entanglement distance simplifies to a function of inter-partition degree, highlighting that a vertex's quantum entanglement is structurally determined by its combinatorial connectivity.

Quantum correlators, both two-point and higher-order, are computed analytically. Their dependence on graph structural parameters—such as the number of non-overlapping neighbors (symmetric differences), common neighbors (intersections), and closed 4-cycles—is rigorously established. Notably, the number of 4-cycles involving two vertices from different partitions directly enters correlator expressions, underscoring a deep intertwining between quantum observables and classical graph 'motifs.'

Quantum Simulation: Validation and Numerical Results

A specific instance—a tripartite triangle graph—is selected for empirical validation. Quantum circuits implementing the proposed protocol are simulated using IBM's AerSimulator, incorporating realistic noise models (readout error $\sim 10^{-2}$, Pauli-X error $\sim 10^{-4}$, CNOT error $\sim 10^{-2}$). Mean values of Pauli operators are measured to compute EED for each qubit.

Numerical results show excellent concordance with analytical predictions, with absolute errors remaining within the expected range due to noise. Key numerical findings include:

• Across varying gate parameters, the EED exhibits predictable modulations driven by edge weights and initial state configuration.
• The protocol's robustness to noise affirms practical feasibility for quantum characterization of graph-theoretic properties.

Implications for Graph Theory, Quantum Computing, and Applications

This work provides a rigorous bridge between quantum entanglement theory and classical graph structural analysis, specifically for tripartite graphs. The analytical and quantum programming tools presented enable the extraction of structural graph invariants (degree distributions, neighborhood overlaps, cycle counts) from quantum measurements.

The framework offers direct applicability to practical scenarios where tripartite graphs arise, such as multi-resource allocation, scheduling, database modeling, and hypergraph analysis. Quantum computation can now be leveraged to study these classical objects, potentially enabling new quantum-accelerated algorithms for combinatorial optimization.

On a theoretical level, the tight correspondence between quantum correlators and local subgraph counts (like 4-cycles) opens avenues to quantum graph invariants, providing new tools for quantum graph theory and network analysis.

Future Directions

Potential directions for further research include:

• Extension to multipartite graphs and higher-order entanglement metrics.
• Exploration of quantum machine learning approaches for structural graph classification using entanglement-based features.
• Experimental realization on larger, noisy quantum hardware to study scalability, robustness, and error mitigation strategies.

The formalism may eventually underpin quantum-inspired algorithms for graph analysis and optimization tasks, bridging quantum information and combinatorial mathematics.

Conclusion

The paper introduces a well-defined method for constructing entangled quantum graph states representing weighted tripartite graphs. Analytical expressions link entanglement distance and quantum correlators to explicit structural properties (vertex degree, neighbor overlap, cycle counts) of underlying graphs. Quantum programming methods validate these results experimentally, demonstrating their practical accessibility and opening new directions for quantum-based graph analysis and applications in resource allocation and network modeling (2604.27829).