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.