Evaluating the Impact of Urban Road Topology on Quantum Approximate Optimization: A Comparative Study of Planned and Organic Networks in Islamabad and Karachi(Pakistan)

The performance of shallow-depth quantum optimization algorithms is known to depend strongly on problem structure, yet the role of real-world network topology remains poorly understood. In this work, we study how urban graph structure influences the behaviour of the Quantum Approximate Optimization Algorithm (QAOA) at depth p=1. Using street-network subgraphs extracted from two cities in Pakistan with contrasting urban designs-a planned city (Islamabad) and an organically grown city (Lyari)-we analyze probability concentration, approximation quality, and performance variability on the minimum vertex cover problem. By comparing classical brute-force solutions with QAOA outcomes, we show that planned topologies yield more reliable convergence, while organic networks exhibit higher variance and a greater tendency toward trivial solutions. Our results suggest that urban structure primarily affects the robustness rather than the average quality of shallow QAOA solutions, highlighting the importance of higher-order structural heterogeneity in shaping low-depth quantum optimization landscapes. This research is vital because it bridges the gap between abstract quantum theory and the chaotic reality of our physical world, proving that the way we build our cities directly impacts our ability to optimize them. By identifying how "topological DNA" influences algorithmic success, this work enables the development of more resilient quantum solutions for critical infrastructure, such as smart power grids and emergency response routing. Ultimately, these insights benefit society by paving the way for more efficient, data-driven urban management that can reduce resource waste and improve the quality of life in both planned and organically growing metropolitan areas.