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

TL;DR

This paper explores a quantum walk model with in- and out-flows on graphs, demonstrating high-probability detection of marked vertices through stationary and pulsation phases, with convergence and pulsation times scaling as O(N log N) and O(√N).

Contribution

It introduces a quantum walk with flows that achieves high-probability search on complete graphs, revealing pulsation phenomena and analyzing convergence using Kato's perturbation theory.

Findings

High probability of finding the marked vertex in stationary state.

Existence of pulsation with periodicity O(√N) before convergence.

Two opportunities to locate the marked vertex: during pulsation and in the stationary phase.

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.