Search and state transfer between hubs by quantum walks

Search and state transfer between hubs, i.e., fully connected vertices, on otherwise arbitrary connected graph is investigated. Motivated by a recent result of Razzoli et al. J. Phys. A: Math. Theor. 55, 265303 (2022) on the universality of hubs in continuous-time quantum walks and spatial search, we extend the investigation to state transfer and also to the discrete-time case. We show that the continuous-time quantum walk allows for perfect state transfer between multiple hubs if the numbers of senders and receivers are close. Turning to the discrete-time case, we show that the search for hubs is successful provided that the initial state is locally modified to account for a degree of each individual vertex. Concerning state transfer using discrete-time quantum walk, it is shown that between a single sender and a single receiver one can transfer two orthogonal states in the same run time. Hence, it is possible to transfer an arbitrary quantum state of a qubit between two hubs. In addition, if the sender and the receiver know each other location, another linearly independent state can be transferred, allowing for exchange of a qutrit state. Finally, we consider the case of transfer between multiple senders and receivers. In this case we cannot transfer specific quantum states. Nevertheless, quantum walker can be transferred with high probability in two regimes---either when there is a similar number of senders and receivers, which is the same as for the continuous-time quantum walk, or when the number of receivers is considerably larger than the number of senders. Our investigation is based on dimensional reduction utilizing the invariant subspaces of the respective evolutions and the fact that for the appropriate choice of the loop weights the problem can be reduced to the complete graph with loops.