Since quantum spatial searches on networks have a strong network dependence, the question arises whether the universal perspective exists in this quantum algorithm for various networks. Here, we propose a universal scaling hypothesis of the quantum spatial search on complex networks such as small-world and scale-free networks. The average path length, a key quantity in the complex network science, is useful to expose this universal feature, where the collapse plot can be generated for the optimal time, the maximal finding probability, and the optimal hopping parameter. Based on the path integral method, we also clarify that the probability amplitude in the continuous-time quantum walk can be determined by the path length distribution. Published by the American Physical Society 2024
We study spatial search with continuous-time quantum walks on real-world complex networks. We use smaller replicas of the Internet network obtained with a recent geometric renormalization method introduced by Garc\'{\i}a-P\'erez et al., Nat. Phys. 14, 583 (2018). This allows us to infer for the first time the behavior of a quantum spatial search algorithm on a real-world complex network. By simulating numerically the dynamics and optimizing the coupling parameter, we study the optimality of the algorithm and its scaling with the size of the network, showing that on average it is considerably better than the classical scaling O(N), but it does not reach the ideal quadratic speedup O(N) that can be achieved, e.g. in complete graphs. However, the performance of the search algorithm strongly depends on the degree of the nodes and, in fact, the scaling is found to be very close to optimal when we consider the nodes below the 99th percentile ordered according to the degree.