Efficiency of quantum and classical transport on graphs

We propose a measure to quantify the efficiency of classical and quantum mechanical transport processes on graphs. The measure only depends on the density of states (DOS), which contains all the necessary information about the graph. For some given (continuous) DOS, the measure shows a power law behavior, where the exponent for the quantum transport is twice the exponent of its classical counterpart. For small-world networks, however, the measure shows rather a stretched exponential law but still the quantum transport outperforms the classical one. Some finite tree graphs have a few highly degenerate eigenvalues, such that, on the other hand, on them the classical transport may be more efficient than the quantum one.

Figure 1

(Color online) $\overline{p}\left(t\right)$ and ${\mid \overline{\alpha }\left(t\right)\mid }^2$ as well as the power laws given in Eqs. (7, 6) for (a) an infinite regular (1D) graph $(\nu =-1∕2)$ and (b) a random graph whose DOS obeys Wigner’s semicircle law $(\nu =1∕2)$.

Figure 2

(Color online) ${\overline{p}}_{\mathrm{discr}}\left(t\right)$ and ${\mid {\overline{\alpha }}_{\mathrm{discr}}\left(t\right)\mid }^2$ for (a) a finite regular (1D) graph of size $N=200$ with periodic boundary conditions and (b) a dendrimer of generation 10 having functionality $z=3$, i.e., $N=3\times 2^{10}-2$. Panel (a) contains also the power-law behavior for the infinite regular (1D) graph, see Fig. 1a.

Figure 3

(Color online) ${\overline{p}}_{\mathrm{discr}}\left(t\right)$, ${\overline{\pi }}_{\mathrm{discr}}\left(t\right)$, and ${\mid {\overline{\alpha }}_{\mathrm{discr}}\left(t\right)\mid }^2$ for a star with $N=10$.