Pulsation of quantum walk between two arbitrary graphs with weakly connected bridge
Taisuke Hosaka, Etsuo Segawa
TL;DR
The paper addresses pulsation in a Grover walk on a finite graph composed of two arbitrary subgraphs connected by a weak bridge of strength $\epsilon$. Using spectral perturbation and Kato reduction, it derives explicit asymptotics for the time-averaged transfer probabilities $\mu_t(H_1)$ and $\mu_t(H_2)$, showing that their evolution depends only on the edge counts $|A_1|$ and $|A_2|$ and not on graph structure, and establishing a period $\tau(\epsilon)$ that scales as $\tau(\epsilon)=\left\lfloor\frac{\pi}{\sqrt{2}}\sqrt{R_{\mathrm{eff}}(|A_1|,|A_2|)}\,\epsilon^{-1/2}\right\rfloor$ with $R_{\mathrm{eff}}^{-1}=1/|A_1|+1/|A_2|$. Notably, when $|A_1|=|A_2|$, the walker is transferred almost completely between the graphs. These results reveal a universal pulsation phenomenon for quantum walks on finite graphs and connect to tunneling-like transport and effective resistances in electrical networks.
Abstract
We consider the Grover walk on a finite graph composed of two arbitrary simple graphs connected by one edge, referred to as a bridge. The parameter $ε>0$ assigned at the bridge represents the strength of connectivity: if $ε=0$, then the graph is completely separated. We show that for sufficiently small values of $ε$, a phenomenon called pulsation occurs. The pulsation is characterized by the periodic transfer of the quantum walker between the two graphs. An asymptotic expression with respect to small $ε$ for the probability of finding the walker on either of the two graphs is derived. This expression reveals that the pulsation depends solely on the number of edges in each graph, regardless of their structure. In addition, we obtain that the quantum walker is transferred periodically between the two graphs, with a period of order $O(ε^{-1/2})$. Furthermore, when the number of edges of two graphs is equal, the quantum walker is almost completely transferred.