Continuous-time quantum walks on directed bipartite graphs

This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $\alpha$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes $N$, time $t$, and the model parameter $\alpha$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $\alpha =\pi /2$. By proving two mirror symmetries around $\alpha =0$ and $\pi /2$ we show that these quantum walks have a period of $\pi$ in $\alpha$. We show that undirected edges lose their effect on the quantum walk at $\alpha =\pi /2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.

Figure 1

Bipartite circulant graphs with $N=10$ nodes. The two node types correspond to the graph's node partitions. The Möbius ladder graphs (c) and (d) are bipartite if $N/2$ is odd. Continuous-time quantum walks on ring graphs corresponding to (a) and (b) with $N=200$ nodes are shown in Fig. 2. Walks on Möbius ladder graphs (c) and (d) with $N=202$ nodes are shown in Fig. 3.

Figure 2

Time evolution of the quantum walk on a directed ring (a) and an undirected ring (b) for different values of the phase $\alpha =0$ (left column), $\alpha =\pi /4$ (middle column), and $\alpha =\pi /2$ (right column). Both rings have $N=200$ nodes. At $t=0$, the quantum walk is started at node $i_0=100$.

Figure 3

Time evolution of quantum walks on directed and undirected Möbius ladder graphs as shown in Figs. 1 and 1, but with $N=202$ nodes. The walks start at node $i_0=100$. The values of the phase $\alpha$ are $\alpha =0$ (left column), $\alpha =\pi /4$ (middle column), and $\alpha =\pi /2$ (right column).