Engineering the success of quantum walk search using weighted graphs

Continuous-time quantum walks are natural tools for spatial search, where one searches for a marked vertex in a graph. Sometimes the structure of the graph causes the walker to get trapped, such that the probability of finding the marked vertex is limited. We give an example with two linked cliques, proving that the captive probability can be liberated by increasing the weights of the links. This allows the search to succeed with probability 1 without increasing the energy scaling of the algorithm. Further increasing the weights, however, slows the runtime, so the optimal search requires weights that are neither too weak nor too strong.