Quantum Approximate Optimization Algorithm applied to multi-objective routing for large scale 6G networks

A multi-objective optimization problem involves optimizing two or more conflicting objectives simultaneously. This type of problem arises in many scientific and industrial areas and it is classified as NP-Hard. Network routing optimization with multiple objectives falls into this category. In the context of 6G networks, solving this problem will become even more challenging due to the exponential growth of Internet of Things devices and the high quality of service requirements. Finding good quality solutions for large-scale networks will be increasingly difficult. In this paper, we introduce a quantum-inspired routing optimization scheme in which noisy-intermediate scale quantum computers (NISQ) can be used to solve the Multi-Objective Routing Problem (MORP). We evaluate the application of the proposed scheme in detail by first developing the mathematical formulas for both single-objective and multi-objective routing and mapping the problem onto gate-based models by using the quadratic unconstrained binary optimization (QUBO) approach. To validate the proposed scheme, we use the quantum approximate optimization algorithm (QAOA), the go-to approach for solving combinatorial optimization problems that are classically intractable. For the simulation, we use the IBM-Qasm simulator and Qiskit framework. Additionally, we use the Chernoff Bound as a standard technique to estimate the sample complexity of QAOA. Finally, we provide a detailed numerical and theoretical analysis of the proposed scheme, including its time complexity, resource requirements, and the challenges associated with it. Our results demonstrate that the proposed approach operates with a time complexity of $O\left(E^2\right)$ per iteration in both single and multi-objective scenarios, with an overall runtime of (n_iteration + n_CB) $\cdot$ $O\left(E^2\right)$ influenced by the sampling overhead, significantly outperforming Dijkstra’s algorithm in the multi-objective case, where the complexity increases to $O\left(2^k(N(k+logN)+2^{k}E)\right)$.