Toward fixed point and pulsation quantum search on graphs driven by quantum walks with in- and out-flows: a trial to the complete graph

Abstract

We treat a quantum walk model with in- and out- flows at every time step from the outside. We show that this quantum walk can find the marked vertex of the complete graph with a high probability in the stationary state. In exchange of the stability, the convergence time is estimated by $O(N\log N)$, where $N$ is the number of vertices. However until the time step $O(N)$, we show that there is a pulsation with the periodicity $O(\sqrt{N})$. We find the marked vertex with a high relative probability in this pulsation phase. This means that we have two chances to find the marked vertex with a high relative probability; the first chance visits in the pulsation phase at short time step $O(\sqrt{N})$ while the second chance visits in the stable phase after long time step $O(N\log N)$. The proofs are based on Kato's perturbation theory.