Geometrical aspects of quantum walks on random two-dimensional structures

We study the transport properties of continuous-time quantum walks (CTQWs) over finite two-dimensional structures with a given number of randomly placed bonds and with different aspect ratios (ARs). Here, we focus on the transport from, say, the left side to the right side of the structure where absorbing sites are placed. We do so by analyzing the long-time average of the survival probability of CTQWs. We compare the results to the classical continuous-time random walk case (CTRW). For small ARs (landscape configurations) we observe only small differences between the quantum and the classical transport properties, i.e., roughly the same number of bonds is needed to facilitate the transport. However, with increasing ARs (portrait configurations) a much larger number of bonds is needed in the CTQW case than in the CTRW case. While for CTRWs the number of bonds needed decreases when going from small ARs to large ARs, for CTQWs this number is large for small ARs, has a minimum for the square configuration, and increases again for increasing ARs. We explain our findings by analyzing the average eigenstates of the corresponding structures: The participation ratios allow us to distinguish between localized and nonlocalized (average) eigenstates. In particular, for large ARs we find for CTQWs that the eigenstates are localized for bond numbers exceeding the bond numbers needed to facilitate transport in the CTRW case. Thus, a rather large number of bonds is needed in order for quantum transport to be efficient for large ARs.